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points

triwarp.points

Unstructured point-cloud geometry: fitting, second moments and normal estimation.

No connectivity here -- everything takes a bare (n,) array of positions. Two groups:

  • Whole-cloud fits. centroid, covariance and centered_covariance give the second moments; fit_line and fit_plane read the dominant and weakest eigenvector off them. gram_matrix is the uncentered form, for callers that want to center differently.
  • Per-point. estimate_normals fits a plane to each point's k-nearest neighbourhood, which is how an unoriented cloud acquires normals before reconstruction. outlier_probability and statistical_outlier_mask score the same neighbourhood for isolation, which is how a scanned cloud loses its stragglers before the normals are fitted. plane_basis and radial_sort then let a caller work in the tangent plane it defines.

  • Cleanup and subsampling. Four masks and one selector, all returning indices or wp.array[wp.bool] rather than a copied cloud, so a caller pays for the gather only if it wants one: point_finite_mask and point_duplicate_mask are the two exact predicates, radius_outlier_mask and statistical_outlier_mask the two density ones (an absolute floor and a cloud-relative threshold), and farthest_point_sample picks an exact count spread over the cloud's support. Run the exact predicates first: they are cheap and they remove the inputs the others are ill-defined on.

  • Approximate hull vertices. Which points are convex-hull vertices, without building a hull. Both entry points are exact-hull-free and run in a fixed number of parallel launches, and both are one-sided -- but in opposite directions, which is what the names say:

    • convex_subset_mask (and its point-returning form convex_subset) accumulates positive certificates: a direction a point is extremal along proves it is on the hull. Cheap and precise, but a hull vertex with no sampled certificate is dropped, so the result can be smaller than the hull-vertex set.
    • convex_superset_mask accumulates negative certificates: a tetrahedron of hull points strictly containing a point proves that point is interior. It keeps everything it cannot rule out, so the result is never smaller than the hull-vertex set -- a conservative filter, suitable as a prefilter before an exact hull.

Neither computes hull connectivity; for that use an exact hull library (scipy.spatial.ConvexHull or the qhull-backed mesh packages).

Normals from estimate_normals are unoriented -- a plane fit cannot pick a side. See triwarp.repair for orientation propagation.

One member here takes no cloud at all: vector_angle is the unsigned angle between two wp.vec3 arrays. It is geometry rather than array structure, so triwarp.array ("NumPy-style structural and elementwise ops") is no better a home, and there is no third candidate -- so it stays, and this sentence is why.

SUPPORT_TIE_SLACK module-attribute 

SUPPORT_TIE_SLACK = wp.constant(wp.float32(1e-06))

TETRAHEDRON_FLATNESS module-attribute 

TETRAHEDRON_FLATNESS = wp.constant(wp.float32(0.001))

point_plane_distance 

point_plane_distance(
    points: array[vec3],
    plane_normal: vec3,
    plane_origin: vec3 | None = None,
) -> wp.array[wp.float32]

Signed perpendicular distance of each point to a plane.

The sign follows plane_normal: positive on the side it points to, negative on the other, zero on the plane. An unsigned distance is wp.map(wp.abs, ...) over the result.

Parameters:

Name Type Description Default
points array[vec3]

(n,) query positions in space as wp.vec3.

 required
plane_normal vec3

Plane normal vector as wp.vec3; need not be unit length, as the distance is normalized by its magnitude.

 required
plane_origin vec3 | None

Point on the plane as wp.vec3. When None, the origin (0, 0, 0) is used.

 None

Returns:

Type Description
array[float32]

Length n signed distances from each point to the plane on points.device.
Source code in triwarp/points.py 
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def point_plane_distance(
    points: wp.array[wp.vec3], plane_normal: wp.vec3, plane_origin: wp.vec3 | None = None
) -> wp.array[wp.float32]:
    """
    Signed perpendicular distance of each point to a plane.

    The sign follows ``plane_normal``: positive on the side it points to, negative on the other,
    zero on the plane. An unsigned distance is ``wp.map(wp.abs, ...)`` over the result.

    Parameters
    ----------
    points
        ``(n,)`` query positions in space as ``wp.vec3``.
    plane_normal
        Plane normal vector as ``wp.vec3``; need not be unit length, as the
        distance is normalized by its magnitude.
    plane_origin
        Point on the plane as ``wp.vec3``. When ``None``, the origin
        ``(0, 0, 0)`` is used.

    Returns
    -------
    wp.array[wp.float32]
        Length ``n`` signed distances from each point to the plane on
        ``points.device``.
    """
    if plane_origin is None:
        plane_origin = wp.vec3(0.0, 0.0, 0.0)
    n = int(points.shape[0])
    out_distances = wp.empty(n, dtype=wp.float32, device=points.device)
    wp.map(
        kernel_points.point_plane_distance, points, plane_normal, plane_origin, out=out_distances
    )
    return out_distances

half_space_mask 

half_space_mask(
    points: array[vec3],
    plane_normal: vec3,
    plane_origin: vec3 | None = None,
) -> wp.array[wp.bool]

Flag the points strictly on the normal's side of a plane.

The unbounded selection primitive: one plane cuts space in two and this is the half the normal points into. Composes into a convex region by intersecting several masks, which is what makes it worth having next to the box query — a slab, a wedge or a frustum is a handful of these, and none of them needs a spatial index because every point is tested independently.

Parameters:

Name Type Description Default
points array[vec3]

(n,) positions in space as wp.vec3.

 required
plane_normal vec3

Plane normal as wp.vec3, pointing into the selected half. Need not be unit length: only the sign of the projection is read.

 required
plane_origin vec3 | None

Point on the plane as wp.vec3. When None, the origin (0, 0, 0) is used.

 None

Returns:

Type Description
array[bool]

Length-n mask on points.device. True marks a point to keep, the same sense as point_finite_mask.

Examples:

upper = tw.points.half_space_mask(v, wp.vec3(0.0, 0.0, 1.0))
kept = tw.array.gather(v, tw.array.flatnonzero(upper))
Notes

The test is strict, so a point exactly on the plane is excluded and the two masks for opposite normals are disjoint rather than overlapping -- with the plane z = 1, a point at z = 1 is in neither half -- and it makes the pair of masks a partition of the points off the plane.

See Also

point_plane_distance The signed distance this thresholds, when the magnitude is wanted too. triwarp.neighbors.query_bvh_box The bounded counterpart: selection by a box, through a BVH. triwarp.array.flatnonzero

Source code in triwarp/points.py 
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def half_space_mask(
    points: wp.array[wp.vec3], plane_normal: wp.vec3, plane_origin: wp.vec3 | None = None
) -> wp.array[wp.bool]:
    """
    Flag the points strictly on the normal's side of a plane.

    The unbounded selection primitive: one plane cuts space in two and this is the half the normal
    points into. Composes into a convex region by intersecting several masks, which is what makes it
    worth having next to the box query — a slab, a wedge or a frustum is a handful of these, and
    none of them needs a spatial index because every point is tested independently.

    Parameters
    ----------
    points
        ``(n,)`` positions in space as ``wp.vec3``.
    plane_normal
        Plane normal as ``wp.vec3``, pointing into the selected half. Need not be unit length: only
        the sign of the projection is read.
    plane_origin
        Point on the plane as ``wp.vec3``. When ``None``, the origin ``(0, 0, 0)`` is used.

    Returns
    -------
    wp.array[wp.bool]
        Length-``n`` mask on ``points.device``. ``True`` marks a point to **keep**, the same sense
        as [`point_finite_mask`][triwarp.points.point_finite_mask].

    Examples
    --------
    ```python
    upper = tw.points.half_space_mask(v, wp.vec3(0.0, 0.0, 1.0))
    kept = tw.array.gather(v, tw.array.flatnonzero(upper))
    ```

    Notes
    -----
    The test is **strict**, so a point exactly on the plane is excluded and the two masks for
    opposite normals are disjoint rather than overlapping -- with the plane ``z = 1``, a point at
    ``z = 1`` is in neither half -- and it makes the pair of masks a partition of the points off
    the plane.

    See Also
    --------
    [`point_plane_distance`][triwarp.points.point_plane_distance]
        The signed distance this thresholds, when the magnitude is wanted too.
    [`triwarp.neighbors.query_bvh_box`][triwarp.neighbors.query_bvh_box]
        The bounded counterpart: selection by a box, through a BVH.
    [`triwarp.array.flatnonzero`][triwarp.array.flatnonzero]
    """
    if plane_origin is None:
        plane_origin = wp.vec3(0.0, 0.0, 0.0)
    n = int(points.shape[0])
    out_mask = wp.empty(n, dtype=wp.bool, device=points.device)
    if n == 0:
        return out_mask

    wp.map(kernel_points.is_in_half_space, points, plane_normal, plane_origin, out=out_mask)
    return out_mask

centroid 

centroid(points: array[vec3]) -> wp.array[wp.vec3]

Mean position sum(points) / n as a (1,) device wp.vec3 array.

Parameters:

Name Type Description Default
points array[vec3]

(n,) positions in space as wp.vec3.

 required

Returns:

Type Description
array[vec3]

Shape (1,) device array holding the centroid on points.device. All-zeros when points is empty.
Source code in triwarp/points.py 
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def centroid(points: wp.array[wp.vec3]) -> wp.array[wp.vec3]:
    """
    Mean position ``sum(points) / n`` as a ``(1,)`` device ``wp.vec3`` array.

    Parameters
    ----------
    points
        ``(n,)`` positions in space as ``wp.vec3``.

    Returns
    -------
    wp.array[wp.vec3]
        Shape ``(1,)`` device array holding the centroid on ``points.device``.
        All-zeros when ``points`` is empty.
    """
    device = points.device
    n = int(points.shape[0])
    out = wp.zeros(1, dtype=wp.vec3, device=device)
    if n == 0:
        return out
    wp.launch_tiled(
        kernel_reduce.sum_vec3_1d_tiled,
        dim=[kernel_reduce.blocks_1d(n)],
        inputs=[points, out],
        block_dim=TILE_1D,
        device=device,
    )
    wp.map(wp.div, out, wp.float32(n), out=out)
    return out

gram_matrix 

gram_matrix(points: array[vec3]) -> wp.array[wp.mat33]

Uncentered Gram (scatter) matrix G = sum_k outer(x_k, x_k).

Parameters:

Name Type Description Default
points array[vec3]

(n,) positions in space as wp.vec3.

 required

Returns:

Type Description
array[mat33]

Shape (1,) device array holding the 3x3 Gram matrix on points.device. All-zeros when points is empty.
Source code in triwarp/points.py 
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def gram_matrix(points: wp.array[wp.vec3]) -> wp.array[wp.mat33]:
    """
    Uncentered Gram (scatter) matrix ``G = sum_k outer(x_k, x_k)``.

    Parameters
    ----------
    points
        ``(n,)`` positions in space as ``wp.vec3``.

    Returns
    -------
    wp.array[wp.mat33]
        Shape ``(1,)`` device array holding the ``3x3`` Gram matrix on
        ``points.device``. All-zeros when ``points`` is empty.
    """
    # The uncentred Gram matrix is the scatter matrix around a zero center.
    zero_center = wp.zeros(1, dtype=wp.vec3, device=points.device)
    return centered_covariance(points, center=zero_center)

fit_line 

fit_line(points: array[vec3]) -> wp.vec3

Major axis of a point set, weighted across all three singular directions.

The result is normalize(S @ V) over the SVD of the uncentered point matrix -- a sum of all three right singular vectors weighted by their singular values. It is not the first principal axis, and the two part company as soon as the cloud is neither strongly elongated nor centred on the origin: measured against the leading eigenvector of the covariance, |dot| is 1.000 on a 1000:1 needle but 0.939 on a 3:1:0.2 cloud and 0.812 once that cloud is offset from the origin. For the principal frame, use principal_axes.

Parameters:

Name Type Description Default
points array[vec3]

(n,) positions in space as wp.vec3.

 required

Returns:

Type Description
vec3

Unit vector along the weighted major axis. The result is direction-only: its sign is governed by the SVD convention and is not meaningful.
Notes

This is the quantity trimesh.points.major_axis computes. Because the weighted sum mixes all three singular vectors, its value depends on each one's sign, which no SVD fixes -- so the two implementations agree to 1.000 where one singular value dominates and to 0.992 on a moderate cloud.

See Also

principal_axes The first principal axis, and the full frame, of the centred cloud. fit_plane The complementary weakest direction. trimesh.points.major_axis

Source code in triwarp/points.py 
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def fit_line(points: wp.array[wp.vec3]) -> wp.vec3:
    """
    Major axis of a point set, weighted across all three singular directions.

    The result is ``normalize(S @ V)`` over the SVD of the **uncentered** point matrix -- a sum of
    all three right singular vectors weighted by their singular values. It is **not** the first
    principal axis, and the two
    part company as soon as the cloud is neither strongly elongated nor centred on the origin:
    measured against the leading eigenvector of the covariance, ``|dot|`` is 1.000 on a 1000:1
    needle but **0.939** on a 3:1:0.2 cloud and **0.812** once that cloud is offset from the origin.
    For the principal frame, use
    [`principal_axes`][triwarp.points.principal_axes].

    Parameters
    ----------
    points
        ``(n,)`` positions in space as ``wp.vec3``.

    Returns
    -------
    wp.vec3
        Unit vector along the weighted major axis. The result is
        direction-only: its sign is governed by the SVD convention and is
        not meaningful.

    Notes
    -----
    This is the quantity [`trimesh.points.major_axis`][] computes. Because the weighted sum mixes
    all three singular vectors, its value depends on each one's sign, which no SVD fixes -- so the
    two implementations agree to ``1.000`` where one singular value dominates and to ``0.992`` on a
    moderate cloud.

    See Also
    --------
    [`principal_axes`][triwarp.points.principal_axes]
        The first principal axis, and the full frame, of the *centred* cloud.
    [`fit_plane`][triwarp.points.fit_plane]
        The complementary weakest direction.
    [`trimesh.points.major_axis`][]
    """
    device = points.device
    n = int(points.shape[0])
    if n == 0:
        return wp.vec3(0.0, 0.0, 0.0)

    # Pass 1: accumulate the (uncentred) Gram matrix G = sum_j outer(x_j, x_j).
    gram = gram_matrix(points)

    # Pass 2: SVD of the 3x3 matrix and axis extraction (single thread).
    out_axis = wp.empty(1, dtype=wp.vec3, device=device)
    wp.launch(kernel_points.finalize_fit_line, dim=1, inputs=[gram, out_axis], device=device)
    return out_axis.list()[0]

centered_covariance 

centered_covariance(
    points: array[vec3], center: array[vec3] | None = None
) -> wp.array[wp.mat33]

Centered scatter matrix C = sum_k outer(x_k - mu, x_k - mu) (no 1/n).

Centering happens inside the outer-product loop (rather than via the sum(x x^T) - n mu mu^T identity) to avoid float32 catastrophic cancellation.

Parameters:

Name Type Description Default
points array[vec3]

(n,) positions in space as wp.vec3.

 required
center array[vec3] | None

Optional precomputed centroid as a (1,) wp.vec3 device array. When None it is computed on-device as sum(points) / n.

 None

Returns:

Type Description
array[mat33]

Shape (1,) device array holding the centered 3x3 scatter matrix on points.device. All-zeros when points is empty.

Raises:

Type Description
RuntimeError

If points and center are not all on one device.
Source code in triwarp/points.py 
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def centered_covariance(
    points: wp.array[wp.vec3], center: wp.array[wp.vec3] | None = None
) -> wp.array[wp.mat33]:
    """
    Centered scatter matrix ``C = sum_k outer(x_k - mu, x_k - mu)`` (no ``1/n``).

    Centering happens inside the outer-product loop (rather than via the
    ``sum(x x^T) - n mu mu^T`` identity) to avoid float32 catastrophic
    cancellation.

    Parameters
    ----------
    points
        ``(n,)`` positions in space as ``wp.vec3``.
    center
        Optional precomputed centroid as a ``(1,)`` ``wp.vec3`` device array.
        When ``None`` it is computed on-device as ``sum(points) / n``.

    Returns
    -------
    wp.array[wp.mat33]
        Shape ``(1,)`` device array holding the centered ``3x3`` scatter matrix
        on ``points.device``. All-zeros when ``points`` is empty.

    Raises
    ------
    RuntimeError
        If ``points`` and ``center`` are not all on one device.
    """
    require_same_device(points=points, center=center)
    device = points.device
    n = int(points.shape[0])
    out = wp.zeros(1, dtype=wp.mat33, device=device)
    if n == 0:
        return out
    if center is None:
        center = centroid(points)
    wp.launch_tiled(
        kernel_points.centered_covariance,
        dim=[kernel_reduce.blocks_1d(n)],
        inputs=[points, center, out],
        block_dim=TILE_1D,
        device=device,
    )
    return out

fit_plane 

fit_plane(points: array[vec3]) -> tuple[wp.vec3, wp.vec3]

Fit a plane to a point set using SVD.

The plane origin is the centroid of the points and the normal is the singular vector with the smallest singular value of the centered covariance matrix.

Parameters:

Name Type Description Default
points array[vec3]

(n,) positions in space as wp.vec3.

 required

Returns:

Type Description
tuple[vec3, vec3]

(normal, centroid) where normal is the plane's unit normal and centroid is a point on it. The normal is sign-ambiguous: its orientation is governed by the SVD convention. Orientation first, position last, matching principal_axes and bounds.oriented_bounding_box -- and so the pair splats directly into the (plane_normal, plane_origin) argument order every plane entry point takes. trimesh.points.plane_fit returns the two the other way round.
See Also

principal_axes All three axes at once; this normal is its third row. plane_basis Two in-plane axes completing this normal into a frame.

Source code in triwarp/points.py 
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def fit_plane(points: wp.array[wp.vec3]) -> tuple[wp.vec3, wp.vec3]:
    """
    Fit a plane to a point set using SVD.

    The plane origin is the centroid of the points and the normal is the
    singular vector with the smallest singular value of the centered
    covariance matrix.

    Parameters
    ----------
    points
        ``(n,)`` positions in space as ``wp.vec3``.

    Returns
    -------
    tuple[wp.vec3, wp.vec3]
        ``(normal, centroid)`` where ``normal`` is the plane's unit normal and ``centroid`` is a
        point on it. The normal is sign-ambiguous: its orientation is governed by the SVD
        convention. Orientation first, position last, matching
        [`principal_axes`][triwarp.points.principal_axes] and
        [`bounds.oriented_bounding_box`][triwarp.bounds.oriented_bounding_box] -- and so the pair
        splats directly into the ``(plane_normal, plane_origin)`` argument order every plane entry
        point takes. [`trimesh.points.plane_fit`][] returns the two the other way round.

    See Also
    --------
    [`principal_axes`][triwarp.points.principal_axes]
        All three axes at once; this normal is its third row.
    [`plane_basis`][triwarp.points.plane_basis]
        Two in-plane axes completing this normal into a frame.
    """
    device = points.device
    n = int(points.shape[0])
    if n == 0:
        # Both zero, so the order is readability rather than behaviour: normal, then centroid.
        return wp.vec3(0.0, 0.0, 0.0), wp.vec3(0.0, 0.0, 0.0)

    # Pass 1: centroid (plane origin) on-device.
    center = centroid(points)

    # Pass 2: covariance matrix of the centred points. Centring before the
    # outer products (rather than via the sum(x x^T) - n c c^T identity) avoids
    # float32 catastrophic cancellation.
    cov = centered_covariance(points, center=center)

    # Pass 3: SVD of the 3x3 covariance and centroid/normal extraction (single thread).
    out_centroid = wp.empty(1, dtype=wp.vec3, device=device)
    out_normal = wp.empty(1, dtype=wp.vec3, device=device)
    wp.launch(
        kernel_points.finalize_fit_plane,
        dim=1,
        inputs=[center, cov, out_centroid, out_normal],
        device=device,
    )
    return (out_normal.list()[0], out_centroid.list()[0])

plane_basis 

plane_basis(normal: vec3) -> tuple[wp.vec3, wp.vec3]

Right-handed orthonormal basis (u, v) spanning the plane with the given normal.

Parameters:

Name Type Description Default
normal vec3

Plane normal as wp.vec3; need not be unit length.

 required

Returns:

Type Description
tuple[vec3, vec3]

(u, v) unit vectors perpendicular to each other and to normal, such that (u, v, normalize(normal)) is right-handed.
Source code in triwarp/points.py 
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def plane_basis(normal: wp.vec3) -> tuple[wp.vec3, wp.vec3]:
    """
    Right-handed orthonormal basis ``(u, v)`` spanning the plane with the given ``normal``.

    Parameters
    ----------
    normal
        Plane normal as ``wp.vec3``; need not be unit length.

    Returns
    -------
    tuple[wp.vec3, wp.vec3]
        ``(u, v)`` unit vectors perpendicular to each other and to ``normal``, such that
        ``(u, v, normalize(normal))`` is right-handed.
    """
    unit_normal = wp.normalize(normal)
    axis = wp.vec3(1.0, 0.0, 0.0)
    if abs(unit_normal[0]) > 0.9:
        axis = wp.vec3(0.0, 1.0, 0.0)
    u = wp.normalize(wp.cross(axis, unit_normal))
    v = wp.cross(unit_normal, u)
    return u, v

covariance 

covariance(
    points: array[vec3], ddof: int = 1
) -> wp.array[wp.mat33]

Sample covariance matrix (1 / (n - ddof)) sum_k outer(x_k - mu, x_k - mu).

Matches numpy.cov(points.T, ddof=ddof) for the default ddof=1.

Parameters:

Name Type Description Default
points array[vec3]

(n,) positions in space as wp.vec3.

 required
ddof int

Delta degrees of freedom; the divisor is n - ddof. Defaults to 1.

 1

Returns:

Type Description
array[mat33]

Shape (1,) device array holding the 3x3 covariance matrix on points.device.

Raises:

Type Description
ValueError

If n - ddof <= 0.
Source code in triwarp/points.py 
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def covariance(points: wp.array[wp.vec3], ddof: int = 1) -> wp.array[wp.mat33]:
    """
    Sample covariance matrix ``(1 / (n - ddof)) sum_k outer(x_k - mu, x_k - mu)``.

    Matches ``numpy.cov(points.T, ddof=ddof)`` for the default ``ddof=1``.

    Parameters
    ----------
    points
        ``(n,)`` positions in space as ``wp.vec3``.
    ddof
        Delta degrees of freedom; the divisor is ``n - ddof``. Defaults to ``1``.

    Returns
    -------
    wp.array[wp.mat33]
        Shape ``(1,)`` device array holding the ``3x3`` covariance matrix on
        ``points.device``.

    Raises
    ------
    ValueError
        If ``n - ddof <= 0``.
    """
    n = int(points.shape[0])
    if n - ddof <= 0:
        raise ValueError(f"covariance requires n > ddof, got n={n}, ddof={ddof}")
    out = centered_covariance(points)
    wp.map(wp.div, out, wp.float32(n - ddof), out=out)
    return out

principal_axes 

principal_axes(
    points: array[vec3],
) -> tuple[wp.mat33, wp.vec3, wp.vec3]

Principal frame of a point cloud: its three axes, spreads and centroid.

The eigenvectors of the centred covariance, ordered widest first, as the rows of a proper rotation -- so rotation * (p - centroid) are the coordinates of p in the principal frame, and wp.transpose(rotation) maps back. This is the fit fit_line and fit_plane each read one direction of: the first row is the true major axis and the third is the plane normal.

Parameters:

Name Type Description Default
points array[vec3]

(n,) positions in space as wp.vec3.

 required

Returns:

Name Type Description
rotation mat33

Principal axes as rows, in descending order of spread. A proper rotation: orthonormal with determinant +1, which fixes the third row's sign, so only the first two are ambiguous.
eigenvalues vec3

The three spreads, descending. These are the eigenvalues of the scatter matrix centered_covariance returns, so they carry no 1 / n; divide by n - 1 for the sample variances along each axis.
centroid vec3

Mean position, the origin of the frame.
Notes

Matches pyvista.principal_axes up to the sign of the first two axes, which no eigendecomposition fixes.

Examples:

rotation, eigenvalues, centroid = tw.points.principal_axes(v)
See Also

fit_line A different estimator, and the reason this exists; see its docstring. fit_plane covariance triwarp.bounds.oriented_bounding_box A searched box rather than a covariance fit, and tighter for it.

Source code in triwarp/points.py 
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def principal_axes(points: wp.array[wp.vec3]) -> tuple[wp.mat33, wp.vec3, wp.vec3]:
    """
    Principal frame of a point cloud: its three axes, spreads and centroid.

    The eigenvectors of the centred covariance, ordered widest first, as the **rows** of a proper
    rotation -- so ``rotation * (p - centroid)`` are the coordinates of ``p`` in the principal
    frame, and ``wp.transpose(rotation)`` maps back. This is the fit
    [`fit_line`][triwarp.points.fit_line] and [`fit_plane`][triwarp.points.fit_plane] each read one
    direction of: the first row is the true major axis and the third is the plane normal.

    Parameters
    ----------
    points
        ``(n,)`` positions in space as ``wp.vec3``.

    Returns
    -------
    rotation : wp.mat33
        Principal axes as rows, in descending order of spread. A proper rotation: orthonormal with
        determinant ``+1``, which fixes the third row's sign, so only the first two are ambiguous.
    eigenvalues : wp.vec3
        The three spreads, descending. These are the eigenvalues of the *scatter* matrix
        [`centered_covariance`][triwarp.points.centered_covariance] returns, so they carry no
        ``1 / n``; divide by ``n - 1`` for the sample variances along each axis.
    centroid : wp.vec3
        Mean position, the origin of the frame.

    Notes
    -----
    Matches ``pyvista.principal_axes`` up to the sign of the first two axes, which no
    eigendecomposition fixes.

    Examples
    --------
    ```python
    rotation, eigenvalues, centroid = tw.points.principal_axes(v)
    ```

    See Also
    --------
    [`fit_line`][triwarp.points.fit_line]
        A different estimator, and the reason this exists; see its docstring.
    [`fit_plane`][triwarp.points.fit_plane]
    [`covariance`][triwarp.points.covariance]
    [`triwarp.bounds.oriented_bounding_box`][triwarp.bounds.oriented_bounding_box]
        A searched box rather than a covariance fit, and tighter for it.
    """
    device = points.device
    n = int(points.shape[0])
    if n == 0:
        return wp.mat33(1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 1.0), wp.vec3(), wp.vec3()

    center = centroid(points)
    scatter = centered_covariance(points, center=center)
    out_rotation = wp.empty(1, dtype=wp.mat33, device=device)
    out_eigenvalues = wp.empty(1, dtype=wp.vec3, device=device)
    out_centroid = wp.empty(1, dtype=wp.vec3, device=device)
    wp.launch(
        kernel_points.finalize_principal_axes,
        dim=1,
        inputs=[center, scatter, out_rotation, out_eigenvalues, out_centroid],
        device=device,
    )
    return (out_rotation.list()[0], out_eigenvalues.list()[0], out_centroid.list()[0])

estimate_normals 

estimate_normals(
    points: array[vec3],
    neighbor_idx: Array2dInt32,
    *,
    orient_reference: vec3 | None = None,
    camera_location: vec3 | None = None,
) -> wp.array[wp.vec3]

Estimate per-point normals by PCA over each point's neighbourhood.

Each normal is the eigenvector of the smallest eigenvalue of the local covariance matrix accumulated over the point's neighbours — the standard choice, which Open3D's FastEigen3x3 also makes, so the result matches the references up to sign. The neighbourhood is supplied by the caller as neighbor_idx: build it with query_nearest using a plain k for a k-nearest (KNN) neighbourhood, or with max_radius set for a radius-bounded (hybrid) neighbourhood — mirroring the two neighbour modes of Open3D's estimate_normals(max_nn, radius).

Parameters:

Name Type Description Default
points array[vec3]

(n,) point positions on the target device.

 required
neighbor_idx Array2dInt32

(n, k) int32 table of neighbour indices per point, as returned by query_nearest (unused slots marked -1). A self-query table includes each point itself once, which is counted normally.

 required
orient_reference vec3 | None

When given, normals are flipped to align with this fixed direction (dot(normal, orient_reference) >= 0), matching Open3D's orient_normals_to_align_with_direction. Mutually exclusive with camera_location.

 None
camera_location vec3 | None

When given, normals are flipped to point toward this location (dot(normal, camera_location - point) >= 0), matching Open3D's orient_normals_towards_camera_location. Mutually exclusive with orient_reference.

 None

Returns:

Type Description
array[vec3]

Length n unit normals on points.device. When neither orientation argument is given, normals are oriented outward from the whole-cloud centroid — a best-effort global orientation valid for star-shaped clouds (Open3D leaves the sign arbitrary instead). Points with fewer than two valid neighbours (or a degenerate neighbourhood) receive a fallback (0, 0, 1) normal.

Raises:

Type Description
ValueError

If both orient_reference and camera_location are given, or if neighbor_idx does not have one row per point.
RuntimeError

If points and neighbor_idx are not all on one device.
See Also

triwarp.points.fit_plane triwarp.reconstruction.triangulate_point_cloud

Source code in triwarp/points.py 
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def estimate_normals(
    points: wp.array[wp.vec3],
    neighbor_idx: twt.Array2dInt32,
    *,
    orient_reference: wp.vec3 | None = None,
    camera_location: wp.vec3 | None = None,
) -> wp.array[wp.vec3]:
    """
    Estimate per-point normals by PCA over each point's neighbourhood.

    Each normal is the eigenvector of the smallest eigenvalue of the local
    covariance matrix accumulated over the point's neighbours — the standard
    choice, which Open3D's ``FastEigen3x3`` also makes, so the result matches the
    references up to sign. The neighbourhood is supplied by
    the caller as ``neighbor_idx``: build it with
    [`query_nearest`][triwarp.neighbors.query_nearest] using a plain
    ``k`` for a k-nearest (KNN) neighbourhood, or with ``max_radius`` set for a
    radius-bounded (hybrid) neighbourhood — mirroring the two neighbour modes of
    Open3D's ``estimate_normals(max_nn, radius)``.

    Parameters
    ----------
    points
        ``(n,)`` point positions on the target device.
    neighbor_idx
        ``(n, k)`` int32 table of neighbour indices per point, as returned by
        [`query_nearest`][triwarp.neighbors.query_nearest] (unused slots
        marked ``-1``). A self-query table includes each point itself once, which
        is counted normally.
    orient_reference
        When given, normals are flipped to align with this fixed direction
        (``dot(normal, orient_reference) >= 0``), matching Open3D's
        ``orient_normals_to_align_with_direction``. Mutually exclusive with
        ``camera_location``.
    camera_location
        When given, normals are flipped to point toward this location
        (``dot(normal, camera_location - point) >= 0``), matching Open3D's
        ``orient_normals_towards_camera_location``. Mutually exclusive with
        ``orient_reference``.

    Returns
    -------
    wp.array[wp.vec3]
        Length ``n`` unit normals on ``points.device``. When neither orientation
        argument is given, normals are oriented outward from the whole-cloud
        centroid — a best-effort global orientation valid for star-shaped clouds
        (Open3D leaves the sign arbitrary instead). Points with fewer than two
        valid neighbours (or a degenerate neighbourhood) receive a fallback
        ``(0, 0, 1)`` normal.

    Raises
    ------
    ValueError
        If both ``orient_reference`` and ``camera_location`` are given, or if ``neighbor_idx`` does
        not have one row per point.
    RuntimeError
        If ``points`` and ``neighbor_idx`` are not all on one device.

    See Also
    --------
    [`triwarp.points.fit_plane`][triwarp.points.fit_plane]
    [`triwarp.reconstruction.triangulate_point_cloud`][triwarp.reconstruction.triangulate_point_cloud]
    """
    require_same_device(points=points, neighbor_idx=neighbor_idx)
    if orient_reference is not None and camera_location is not None:
        raise ValueError("pass at most one of orient_reference and camera_location")

    twt.ensure_ndim(neighbor_idx, 2, dtype=wp.int32)

    device = points.device
    n = int(points.shape[0])
    # The launch is one thread per *point* and each reads its own row, so a table with fewer rows
    # than the cloud is an out-of-bounds read rather than a short answer -- and on the CPU device a
    # Warp array is host heap, so that is heap corruption with no exception in release mode
    # (CLAUDE.md section 12.1). The sibling ``outlier_probability`` checks the same pair of shapes;
    # this one only checked the rank.
    if int(neighbor_idx.shape[0]) != n:
        raise ValueError(
            f"neighbor_idx must have one row per point, got {neighbor_idx.shape} for {n} points"
        )
    if n == 0:
        return wp.empty(0, dtype=wp.vec3, device=device)

    if camera_location is not None:
        orient_mode = kernel_points.ORIENT_CAMERA
        reference = camera_location
    elif orient_reference is not None:
        orient_mode = kernel_points.ORIENT_DIRECTION
        reference = orient_reference
    else:
        orient_mode = kernel_points.ORIENT_CENTROID
        reference = wp.vec3(0.0, 0.0, 0.0)

    center = centroid(points)
    out_normals = wp.empty(n, dtype=wp.vec3, device=device)
    wp.launch(
        kernel_points.estimate_point_normals,
        dim=n,
        inputs=[points, neighbor_idx, center, orient_mode, reference, out_normals],
        device=device,
    )
    return out_normals

outlier_probability 

outlier_probability(
    neighbor_idx: Array2dInt32,
    neighbor_distance: Array2dFloat32,
    *,
    scale: float = 3.0,
) -> wp.array[wp.float32]

Local Outlier Probability (LoOP) of each point, in [0, 1].

A point is an outlier to the extent that its own neighbourhood is stretched relative to the neighbourhoods of the points in it. Following Kriegel et al., each point gets a "standard distance" sigma (the RMS distance to its neighbours), a local factor plof = sigma / mean(sigma over the neighbourhood) - 1, and a probability max(0, erf(plof / (nplof * sqrt(2)))) where nplof = scale * sqrt(mean(plof^2)) over the whole cloud. Because the normalization is cloud-wide, the score is comparable across points but not across clouds — it is a rank, calibrated so that a threshold near 0.8 selects the tail.

This is the measure behind MeshLab's compute_selection_point_cloud_outliers, whose own propthreshold default is 0.8; threshold the result to reproduce a selection:

from triwarp.kernels import array as kernel_array

probability = tw.points.outlier_probability(neighbor_idx, neighbor_distance)
outlier_mask = wp.empty(probability.shape, dtype=wp.bool, device=probability.device)
wp.map(kernel_array.greater, probability, wp.float32(0.8), out=outlier_mask)
outliers = tw.array.flatnonzero(outlier_mask)

Parameters:

Name Type Description Default
neighbor_idx Array2dInt32

(n, k) int32 neighbour table, as returned by query_nearest (unused slots marked -1). A self-query table includes each point itself, which is counted normally — the same convention MeshLab's k-d tree query uses.

 required
neighbor_distance Array2dFloat32

(n, k) float32 distances aligned with neighbor_idx; unused slots are inf.

 required
scale float

The LoOP normalization factor lambda. Larger values make the score more conservative (fewer points near 1). MeshLab fixes it at 3, which is the default here.

 3.0

Returns:

Type Description
array[float32]

Length-n probabilities on neighbor_idx.device. A point whose neighbour row is empty scores 0. All-zeros when the cloud has no spread at all.

Raises:

Type Description
ValueError

If scale <= 0, or the two tables disagree in shape.
RuntimeError

If neighbor_idx and neighbor_distance are not all on one device.
See Also

statistical_outlier_mask estimate_normals triwarp.neighbors.query_nearest

Source code in triwarp/points.py 
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def outlier_probability(
    neighbor_idx: twt.Array2dInt32, neighbor_distance: twt.Array2dFloat32, *, scale: float = 3.0
) -> wp.array[wp.float32]:
    """
    Local Outlier Probability (LoOP) of each point, in ``[0, 1]``.

    A point is an outlier to the extent that its own neighbourhood is *stretched* relative to the
    neighbourhoods of the points in it. Following Kriegel et al., each point gets a "standard
    distance" ``sigma`` (the RMS distance to its neighbours), a local factor
    ``plof = sigma / mean(sigma over the neighbourhood) - 1``, and a probability
    ``max(0, erf(plof / (nplof * sqrt(2))))`` where ``nplof = scale * sqrt(mean(plof^2))`` over the
    whole cloud. Because the normalization is cloud-wide, the score is comparable across points but
    **not** across clouds — it is a rank, calibrated so that a threshold near ``0.8`` selects the
    tail.

    This is the measure behind MeshLab's ``compute_selection_point_cloud_outliers``, whose own
    ``propthreshold`` default is ``0.8``; threshold the result to reproduce a selection:

    ```python
    from triwarp.kernels import array as kernel_array

    probability = tw.points.outlier_probability(neighbor_idx, neighbor_distance)
    outlier_mask = wp.empty(probability.shape, dtype=wp.bool, device=probability.device)
    wp.map(kernel_array.greater, probability, wp.float32(0.8), out=outlier_mask)
    outliers = tw.array.flatnonzero(outlier_mask)
    ```

    Parameters
    ----------
    neighbor_idx
        ``(n, k)`` int32 neighbour table, as returned by
        [`query_nearest`][triwarp.neighbors.query_nearest] (unused slots marked ``-1``).
        A self-query table includes each point itself, which is counted normally — the same
        convention MeshLab's k-d tree query uses.
    neighbor_distance
        ``(n, k)`` float32 distances aligned with ``neighbor_idx``; unused slots are ``inf``.
    scale
        The LoOP normalization factor ``lambda``. Larger values make the score more conservative
        (fewer points near ``1``). MeshLab fixes it at ``3``, which is the default here.

    Returns
    -------
    wp.array[wp.float32]
        Length-``n`` probabilities on ``neighbor_idx.device``. A point whose neighbour row is empty
        scores ``0``. All-zeros when the cloud has no spread at all.

    Raises
    ------
    ValueError
        If ``scale <= 0``, or the two tables disagree in shape.
    RuntimeError
        If ``neighbor_idx`` and ``neighbor_distance`` are not all on one device.

    See Also
    --------
    [`statistical_outlier_mask`][triwarp.points.statistical_outlier_mask]
    [`estimate_normals`][triwarp.points.estimate_normals]
    [`triwarp.neighbors.query_nearest`][triwarp.neighbors.query_nearest]
    """
    require_same_device(neighbor_idx=neighbor_idx, neighbor_distance=neighbor_distance)
    twt.ensure_ndim(neighbor_idx, 2, dtype=wp.int32)
    twt.ensure_ndim(neighbor_distance, 2, dtype=wp.float32)
    if neighbor_idx.shape != neighbor_distance.shape:
        raise ValueError(
            "neighbor_idx and neighbor_distance must have the same shape, got "
            f"{neighbor_idx.shape} and {neighbor_distance.shape}"
        )
    if scale <= 0.0:
        raise ValueError(f"scale must be positive, got {scale}")

    device = neighbor_idx.device
    n = int(neighbor_idx.shape[0])
    if n == 0:
        return wp.empty(0, dtype=wp.float32, device=device)

    _mean, standard_distance, _count = _neighbor_distance_moments(neighbor_distance)
    plof = wp.empty(n, dtype=wp.float32, device=device)
    wp.launch(
        kernel_points.local_outlier_factor,
        dim=n,
        inputs=[standard_distance, neighbor_idx, plof],
        device=device,
    )

    # nplof = scale * sqrt(E[plof^2]) over the whole cloud: one fused sum-of-squares reduction and
    # one readback, because the normalizer is a scalar every point divides by and Warp cannot pass
    # a device scalar as a uniform argument.
    normalizer = scale * math.sqrt(wp.utils.array_inner(plof, plof) / n)
    out_probability = wp.zeros(n, dtype=wp.float32, device=device)
    if normalizer <= 0.0:
        return out_probability  # a cloud with no spread: every plof is zero
    wp.map(
        kernel_points.outlier_probability,
        plof,
        wp.float32(1.0 / (normalizer * math.sqrt(2.0))),
        out=out_probability,
    )
    return out_probability

statistical_outlier_mask 

statistical_outlier_mask(
    neighbor_distance: Array2dFloat32,
    *,
    std_ratio: float = 2.0,
) -> wp.array[wp.bool]

Flag points whose mean neighbour distance exceeds mean + std_ratio * std over the cloud.

Open3D's remove_statistical_outlier criterion, and the cheaper cousin of outlier_probability: one global threshold on a single per-point statistic rather than a neighbourhood-relative one. It is the right choice for a cloud of roughly uniform density and the wrong one for a cloud that is dense in places and sparse in others, where the sparse region is entirely above a global threshold.

Parameters:

Name Type Description Default
neighbor_distance Array2dFloat32

(n, k) float32 distances from query_nearest; unused slots are inf and are excluded from the per-point mean. A self-query table contributes one zero distance per row, exactly as Open3D's SearchKNN does, so pass the same k Open3D gets as nb_neighbors.

 required
std_ratio float

Multiplier on the cloud-wide standard deviation of the per-point means. Lower is stricter. Open3D's own examples use 2.0, the default here.

 2.0

Returns:

Type Description
array[bool]

Length-n mask on neighbor_distance.device; True marks an outlier (the complement of Open3D's keep mask). A point with an empty or fully coincident neighbourhood is marked as an outlier, matching Open3D's avg > 0 guard.

Raises:

Type Description
ValueError

If neighbor_distance is not a rank-2 float32 array.
See Also

outlier_probability triwarp.neighbors.query_nearest

Source code in triwarp/points.py 
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def statistical_outlier_mask(
    neighbor_distance: twt.Array2dFloat32, *, std_ratio: float = 2.0
) -> wp.array[wp.bool]:
    """
    Flag points whose mean neighbour distance exceeds ``mean + std_ratio * std`` over the cloud.

    Open3D's ``remove_statistical_outlier`` criterion, and the cheaper cousin of
    [`outlier_probability`][triwarp.points.outlier_probability]: one global threshold on a single
    per-point statistic rather than a neighbourhood-relative one. It is the right choice for a
    cloud of roughly uniform density and the wrong one for a cloud that is dense in places and
    sparse in others, where the sparse region is entirely above a global threshold.

    Parameters
    ----------
    neighbor_distance
        ``(n, k)`` float32 distances from
        [`query_nearest`][triwarp.neighbors.query_nearest]; unused slots are ``inf`` and
        are excluded from the per-point mean. A self-query table contributes one zero distance per
        row, exactly as Open3D's ``SearchKNN`` does, so pass the same ``k`` Open3D gets as
        ``nb_neighbors``.
    std_ratio
        Multiplier on the cloud-wide standard deviation of the per-point means. Lower is stricter.
        Open3D's own examples use ``2.0``, the default here.

    Returns
    -------
    wp.array[wp.bool]
        Length-``n`` mask on ``neighbor_distance.device``; ``True`` marks an **outlier** (the
        complement of Open3D's *keep* mask). A point with an empty or fully coincident
        neighbourhood is marked as an outlier, matching Open3D's ``avg > 0`` guard.

    Raises
    ------
    ValueError
        If ``neighbor_distance`` is not a rank-2 float32 array.

    See Also
    --------
    [`outlier_probability`][triwarp.points.outlier_probability]
    [`triwarp.neighbors.query_nearest`][triwarp.neighbors.query_nearest]
    """
    twt.ensure_ndim(neighbor_distance, 2, dtype=wp.float32)

    device = neighbor_distance.device
    n = int(neighbor_distance.shape[0])
    # `wp.empty`, not `wp.zeros`: every branch below (the `n == 0` return aside) fills every slot
    # of `out_mask` via a `wp.map` over the full `n`-length row, so nothing ever reads the
    # zero-fill.
    out_mask = wp.empty(n, dtype=wp.bool, device=device)
    if n == 0:
        return out_mask

    mean_distance, _rms, count = _neighbor_distance_moments(neighbor_distance)
    # Cloud mean and (ddof=1) deviation over the *counted* rows only, exactly as Open3D divides by
    # its ``valid_distances``. Empty rows contribute zero to both sums, so a plain reduction works.
    counted_mask = wp.empty(n, dtype=wp.bool, device=device)
    wp.map(kernel_array.greater, count, wp.int32(0), out=counted_mask)
    counted = int(tw.reduce.sum(counted_mask))
    if counted < 2:
        # Too few counted rows for a cloud deviation, but the docstring's "empty or fully
        # coincident neighbourhood is an outlier" needs no cloud statistic at all -- that half of
        # `is_statistical_outlier`'s predicate (`count == 0 or mean_distance <= 0.0`) still applies
        # per point. `threshold=inf` disables only the distance-based third of it, rather than
        # returning an all-`False` mask that silently drops the other two.
        wp.map(
            kernel_points.is_statistical_outlier,
            mean_distance,
            count,
            wp.float32(math.inf),
            out=out_mask,
        )
        return out_mask
    cloud_mean = float(tw.reduce.sum(mean_distance)) / float(counted)
    deviations = wp.empty(n, dtype=wp.float32, device=device)
    wp.map(
        kernel_points.centered_square_if_counted,
        mean_distance,
        count,
        wp.float32(cloud_mean),
        out=deviations,
    )
    cloud_std = math.sqrt(float(tw.reduce.sum(deviations)) / float(counted - 1))

    wp.map(
        kernel_points.is_statistical_outlier,
        mean_distance,
        count,
        wp.float32(cloud_mean + std_ratio * cloud_std),
        out=out_mask,
    )
    return out_mask

radius_outlier_mask 

radius_outlier_mask(
    points: array[vec3],
    radius: float,
    min_neighbors: int,
    *,
    grid: HashGrid | None = None,
) -> wp.array[wp.bool]

Flag points with at most min_neighbors other points within radius.

Open3D's remove_radius_outlier criterion, and the local counterpart of statistical_outlier_mask: an absolute density floor rather than a cloud-wide threshold on a per-point statistic. That makes it the right choice where the density is known in advance (a scanner's nominal spacing) and the wrong one where it varies across the cloud, which is the trade the two run in opposite directions.

Parameters:

Name Type Description Default
points array[vec3]

(n,) point positions as wp.array[wp.vec3].

 required
radius float

Search radius, inclusive at exactly radius. Must be > 0.

 required
min_neighbors int

A point survives when it has strictly more than this many neighbours, counting itself — Open3D's nb_points rule exactly, so the same number gives the same answer. Must be >= 1.

 required
grid HashGrid | None

Optional pre-built hash grid over points, from hashgrid_from_points. Pass it to hoist the build out of a sweep over several radii.

 None

Returns:

Type Description
array[bool]

Length-n mask on points.device; True marks an outlier (the complement of Open3D's keep mask), matching statistical_outlier_mask.

Raises:

Type Description
ValueError

If radius <= 0 or min_neighbors < 1.
RuntimeError

If points and grid are not all on one device.
Notes

The counts are exact, and the rule is the reference's — but Open3D's own remove_radius_outlier shares one KDTreeFlann across an OpenMP loop and its radius search is not thread-safe under that sharing, so it returns a different answer run to run: measured three distinct keep sets (43 / 44 / 45 points) over eight repetitions of one 500-point cloud, differing by one or two points each time. Querying the same tree serially reproduces this function exactly on that cloud. Expect a comparison against the filter to disagree on a handful of borderline points, and do not read that as a difference in the criterion.

Examples:

mask = tw.points.radius_outlier_mask(v, 1.0, 3)
outlier_indices = tw.array.flatnonzero(mask)
See Also

statistical_outlier_mask outlier_probability triwarp.neighbors.query_ball_count

Source code in triwarp/points.py 
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def radius_outlier_mask(
    points: wp.array[wp.vec3], radius: float, min_neighbors: int, *, grid: wp.HashGrid | None = None
) -> wp.array[wp.bool]:
    """
    Flag points with at most ``min_neighbors`` other points within ``radius``.

    Open3D's ``remove_radius_outlier`` criterion, and the local counterpart of
    [`statistical_outlier_mask`][triwarp.points.statistical_outlier_mask]: an *absolute* density
    floor rather than a cloud-wide threshold on a per-point statistic. That makes it the right
    choice where the density is known in advance (a scanner's nominal spacing) and the wrong one
    where it varies across the cloud, which is the trade the two run in opposite directions.

    Parameters
    ----------
    points
        ``(n,)`` point positions as ``wp.array[wp.vec3]``.
    radius
        Search radius, inclusive at exactly ``radius``. Must be ``> 0``.
    min_neighbors
        A point survives when it has **strictly more** than this many neighbours, counting
        *itself* — Open3D's ``nb_points`` rule exactly, so the same number gives the same answer.
        Must be ``>= 1``.
    grid
        Optional pre-built hash grid over ``points``, from
        [`hashgrid_from_points`][triwarp.neighbors.hashgrid_from_points]. Pass it to hoist the
        build out of a sweep over several radii.

    Returns
    -------
    wp.array[wp.bool]
        Length-``n`` mask on ``points.device``; ``True`` marks an **outlier** (the complement of
        Open3D's *keep* mask), matching
        [`statistical_outlier_mask`][triwarp.points.statistical_outlier_mask].

    Raises
    ------
    ValueError
        If ``radius <= 0`` or ``min_neighbors < 1``.
    RuntimeError
        If ``points`` and ``grid`` are not all on one device.

    Notes
    -----
    The counts are exact, and the rule is the reference's — but Open3D's own
    ``remove_radius_outlier`` shares one ``KDTreeFlann`` across an OpenMP loop and its radius search
    is not thread-safe under that sharing, so it returns a *different answer run to run*: measured
    three distinct keep sets (43 / 44 / 45 points) over eight repetitions of one 500-point cloud,
    differing by one or two points each time. Querying the same tree serially reproduces this
    function exactly on that cloud. Expect a comparison against the filter to disagree on a handful
    of borderline points, and do not read that as a difference in the criterion.

    Examples
    --------
    ```python
    mask = tw.points.radius_outlier_mask(v, 1.0, 3)
    outlier_indices = tw.array.flatnonzero(mask)
    ```

    See Also
    --------
    [`statistical_outlier_mask`][triwarp.points.statistical_outlier_mask]
    [`outlier_probability`][triwarp.points.outlier_probability]
    [`triwarp.neighbors.query_ball_count`][triwarp.neighbors.query_ball_count]
    """
    require_same_device(points=points, grid=grid)
    if radius <= 0.0:
        raise ValueError(f"radius must be > 0, got {radius}")
    if min_neighbors < 1:
        raise ValueError(f"min_neighbors must be >= 1, got {min_neighbors}")

    device = points.device
    n = int(points.shape[0])
    out_mask = wp.zeros(n, dtype=wp.bool, device=device)
    if n == 0:
        return out_mask

    # A self-query counts the point itself once, at distance 0 -- which is what makes
    # ``min_neighbors`` comparable with Open3D's ``nb_points`` without an off-by-one correction.
    counts = tw.neighbors.query_ball_count(points, points, radius, accelerator=grid)
    wp.map(kernel_array.less_equal, counts, wp.int32(min_neighbors), out=out_mask)
    return out_mask

point_finite_mask 

point_finite_mask(points: array[vec3]) -> wp.array[wp.bool]

Flag points whose three coordinates are all finite.

Open3D's remove_non_finite_points predicate. A single NaN or infinity poisons every reduction the point enters — a bounding box, a covariance, a BVH build — so this is the first pass over a cloud read off a scanner, before any of the fits in this module.

Parameters:

Name Type Description Default
points array[vec3]

(n,) point positions as wp.array[wp.vec3].

 required

Returns:

Type Description
array[bool]

Length-n mask on points.device. True marks a point to keep, which is the opposite sense from the outlier and duplicate masks in this module — the name is the tell: this one is named for what it selects.

Examples:

finite = tw.points.point_finite_mask(v)
cleaned = tw.array.gather(v, tw.array.flatnonzero(finite))
See Also

point_duplicate_mask The other exact-predicate cleanup pass; run this one first, since a NaN position is never equal to itself under Open3D's rule. triwarp.array.flatnonzero

Source code in triwarp/points.py 
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def point_finite_mask(points: wp.array[wp.vec3]) -> wp.array[wp.bool]:
    """
    Flag points whose three coordinates are all finite.

    Open3D's ``remove_non_finite_points`` predicate. A single ``NaN`` or infinity poisons every
    reduction the point enters — a bounding box, a covariance, a BVH build — so this is the first
    pass over a cloud read off a scanner, before any of the fits in this module.

    Parameters
    ----------
    points
        ``(n,)`` point positions as ``wp.array[wp.vec3]``.

    Returns
    -------
    wp.array[wp.bool]
        Length-``n`` mask on ``points.device``. ``True`` marks a point to **keep**, which is the
        opposite sense from the outlier and duplicate masks in this module — the name is the tell:
        this one is named for what it selects.

    Examples
    --------
    ```python
    finite = tw.points.point_finite_mask(v)
    cleaned = tw.array.gather(v, tw.array.flatnonzero(finite))
    ```

    See Also
    --------
    [`point_duplicate_mask`][triwarp.points.point_duplicate_mask]
        The other exact-predicate cleanup pass; run this one first, since a ``NaN`` position is
        never equal to itself under Open3D's rule.
    [`triwarp.array.flatnonzero`][triwarp.array.flatnonzero]
    """
    device = points.device
    n = int(points.shape[0])
    out_mask = wp.empty(n, dtype=wp.bool, device=device)
    if n == 0:
        return out_mask

    wp.map(kernel_points.is_finite_point, points, out=out_mask)
    return out_mask

point_duplicate_mask 

point_duplicate_mask(
    points: array[vec3],
) -> wp.array[wp.bool]

Flag every point that repeats an exactly equal position seen earlier in the cloud.

Open3D's remove_duplicated_points rule: bit-for-bit coordinate equality, keeping the first occurrence of each distinct position. This is deliberately not triwarp.grouping.unique_rows, whose wp.vec3 key buckets each coordinate to about 2.4e-4 relative and so merges positions that are merely close; reach for that one when a tolerance is what you want, and for this one when equality is.

Two injective 64-bit key rounds do it: the x and y bit patterns pack side by side into one int64, that key's equivalence class packs against the z bits, and the second class identifies the position exactly. Each half is exactly 32 bits, so neither round can collide.

Parameters:

Name Type Description Default
points array[vec3]

(n,) point positions as wp.array[wp.vec3].

 required

Returns:

Type Description
array[bool]

Length-n mask on points.device; True marks a duplicate, i.e. every occurrence but the first of each distinct position. Gathering by the complement keeps one representative of each in first-occurrence order, as Open3D does.
Notes

-0.0 and +0.0 merge, because IEEE-754 equality holds between them and the reference compares with ==; the packing folds the two bit patterns together to reproduce that. NaN is the one case that does not match the reference: NaN != NaN makes every NaN row its own class for Open3D, where a bit pattern is a bit pattern here and identical NaN rows merge. Run point_finite_mask first and the question does not arise. Positions are compared at float32, so two points that differ only below float32 resolution are one position here and two for a float64 reference.

Examples:

duplicates = tw.points.point_duplicate_mask(v)
n_distinct = int(v.shape[0]) - int(tw.reduce.sum(duplicates))
See Also

point_finite_mask triwarp.grouping.unique_rows The tolerance-bucketed sibling, and the right choice when exact equality is too strict. triwarp.grouping.first_occurrence_indices

Source code in triwarp/points.py 
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def point_duplicate_mask(points: wp.array[wp.vec3]) -> wp.array[wp.bool]:
    """
    Flag every point that repeats an **exactly** equal position seen earlier in the cloud.

    Open3D's ``remove_duplicated_points`` rule: bit-for-bit coordinate equality, keeping the
    *first* occurrence of each distinct position. This is deliberately not
    [`triwarp.grouping.unique_rows`][triwarp.grouping.unique_rows], whose ``wp.vec3`` key buckets
    each coordinate to about ``2.4e-4`` relative and so merges positions that are merely close;
    reach for that one when a *tolerance* is what you want, and for this one when equality is.

    Two injective 64-bit key rounds do it: the ``x`` and ``y`` bit patterns pack side by side into
    one ``int64``, that key's equivalence class packs against the ``z`` bits, and the second class
    identifies the position exactly. Each half is exactly 32 bits, so neither round can collide.

    Parameters
    ----------
    points
        ``(n,)`` point positions as ``wp.array[wp.vec3]``.

    Returns
    -------
    wp.array[wp.bool]
        Length-``n`` mask on ``points.device``; ``True`` marks a **duplicate**, i.e. every
        occurrence but the first of each distinct position. Gathering by the complement keeps one
        representative of each in first-occurrence order, as Open3D does.

    Notes
    -----
    ``-0.0`` and ``+0.0`` merge, because IEEE-754 equality holds between them and the reference
    compares with ``==``; the packing folds the two bit patterns together to reproduce that.
    ``NaN`` is the one case that does **not** match the reference: ``NaN != NaN`` makes every
    ``NaN`` row its own class for Open3D, where a bit pattern is a bit pattern here and identical
    ``NaN`` rows merge. Run [`point_finite_mask`][triwarp.points.point_finite_mask] first and the
    question does not arise. Positions are compared at ``float32``, so two points that differ only
    below ``float32`` resolution are one position here and two for a ``float64`` reference.

    Examples
    --------
    ```python
    duplicates = tw.points.point_duplicate_mask(v)
    n_distinct = int(v.shape[0]) - int(tw.reduce.sum(duplicates))
    ```

    See Also
    --------
    [`point_finite_mask`][triwarp.points.point_finite_mask]
    [`triwarp.grouping.unique_rows`][triwarp.grouping.unique_rows]
        The tolerance-bucketed sibling, and the right choice when exact equality is too strict.
    [`triwarp.grouping.first_occurrence_indices`][triwarp.grouping.first_occurrence_indices]
    """
    device = points.device
    n = int(points.shape[0])
    out_mask = wp.empty(n, dtype=wp.bool, device=device)
    if n == 0:
        return out_mask

    key_xy = wp.empty(n, dtype=wp.int64, device=device)
    wp.map(kernel_points.pack_xy_bits, points, out=key_xy)
    _unique_xy, class_xy = tw.grouping.unique_1d(key_xy, return_inverse=True)

    key_xyz = wp.empty(n, dtype=wp.int64, device=device)
    wp.map(kernel_points.pack_class_z_bits, class_xy, points, out=key_xyz)
    unique_xyz, class_xyz = tw.grouping.unique_1d(key_xyz, return_inverse=True)

    # The class count is the length of the unique-key array in hand, so the representative pass
    # needs no reduction of its own.
    first = tw.grouping.first_occurrence_indices(class_xyz, int(unique_xyz.shape[0]))
    wp.map(kernel_array.mask_not, tw.array.indices_to_mask(first, n), out=out_mask)
    return out_mask

farthest_point_sample 

farthest_point_sample(
    points: array[vec3], count: int, *, start: int = 0
) -> wp.array[wp.int32]

Greedily pick count points, each as far as possible from those already picked.

Open3D's farthest_point_down_sample: the classic maximin subsample, which spreads its output over the cloud's support rather than its density — so unlike triwarp.voxels.voxel_down_sample it returns an exact count, and unlike a random subsample it cannot leave a hole. The greedy choice makes it inherently sequential in count, and the whole sweep runs as one persistent block on the device: each round folds the newest sample into every point's running distance and takes the global arg-max as a block reduction, so nothing is launched or read back per round.

Parameters:

Name Type Description Default
points array[vec3]

(n,) point positions as wp.array[wp.vec3].

 required
count int

Number of points to select; must satisfy 0 <= count <= n. Zero returns an empty array, as Open3D's num_samples=0 does.

 required
start int

Index of the first sample, Open3D's start_index. Defaults to 0, which is its default too.

 0

Returns:

Type Description
array[int32]

Length-count indices into points, on points.device, in selection order: entry 0 is start and each later entry is the point farthest from every earlier one. Open3D returns the selected points instead, and its SelectByIndex emits them in ascending index order, so the sequence here is strictly more information than the reference gives back — the sets are the same.

Raises:

Type Description
ValueError

If count is not in [0, n], or start is not a valid index into points.
Notes

Ties are broken towards the lower index, matching the reference's strict > arg-max, and that is what makes the answer reproducible: the per-iteration arg-max is a single wp.atomic_max over a key that packs the squared distance against the complemented index, so the winner does not depend on the order the atomics land in. Distances are compared in float32 where Open3D uses float64, so a cloud with two points at nearly equal distance from the chosen set can diverge from that reference at the point where the tie is resolved — and then, being greedy, for the rest of the sequence.

Examples:

indices = tw.points.farthest_point_sample(v, 4)
spread = tw.array.gather(v, indices)
See Also

triwarp.sample.sample_surface_blue_noise The same "well-spread points" goal from a mesh, by dart throwing rather than greedily. triwarp.voxels.voxel_down_sample triwarp.neighbors.nearest_neighbor_distance

Source code in triwarp/points.py 
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def farthest_point_sample(
    points: wp.array[wp.vec3], count: int, *, start: int = 0
) -> wp.array[wp.int32]:
    """
    Greedily pick ``count`` points, each as far as possible from those already picked.

    Open3D's ``farthest_point_down_sample``: the classic maximin subsample, which spreads its
    output over the cloud's *support* rather than its density — so unlike
    [`triwarp.voxels.voxel_down_sample`][triwarp.voxels.voxel_down_sample] it returns an exact
    count, and unlike a random subsample it cannot leave a hole. The greedy choice makes it
    inherently sequential in ``count``, and the whole sweep runs as **one persistent block** on the
    device: each round folds the newest sample into every point's running distance and takes the
    global arg-max as a block reduction, so nothing is launched or read back per round.

    Parameters
    ----------
    points
        ``(n,)`` point positions as ``wp.array[wp.vec3]``.
    count
        Number of points to select; must satisfy ``0 <= count <= n``. Zero returns an empty array,
        as Open3D's ``num_samples=0`` does.
    start
        Index of the first sample, Open3D's ``start_index``. Defaults to ``0``, which is its
        default too.

    Returns
    -------
    wp.array[wp.int32]
        Length-``count`` indices into ``points``, on ``points.device``, in **selection order**:
        entry 0 is ``start`` and each later entry is the point farthest from every earlier one.
        Open3D returns the selected *points* instead, and its ``SelectByIndex`` emits them in
        ascending index order, so the sequence here is strictly more information than the reference
        gives back — the sets are the same.

    Raises
    ------
    ValueError
        If ``count`` is not in ``[0, n]``, or ``start`` is not a valid index into ``points``.

    Notes
    -----
    Ties are broken towards the lower index, matching the reference's strict ``>`` arg-max, and that
    is what makes the answer reproducible: the per-iteration arg-max is a single ``wp.atomic_max``
    over a key that packs the squared distance against the complemented index, so the winner does
    not depend on the order the atomics land in. Distances are compared in ``float32`` where Open3D
    uses ``float64``, so a cloud with two points at nearly equal distance from the chosen set can
    diverge from that reference at the point where the tie is resolved — and then, being greedy, for
    the rest of the sequence.

    Examples
    --------
    ```python
    indices = tw.points.farthest_point_sample(v, 4)
    spread = tw.array.gather(v, indices)
    ```

    See Also
    --------
    [`triwarp.sample.sample_surface_blue_noise`][triwarp.sample.sample_surface_blue_noise]
        The same "well-spread points" goal from a *mesh*, by dart throwing rather than greedily.
    [`triwarp.voxels.voxel_down_sample`][triwarp.voxels.voxel_down_sample]
    [`triwarp.neighbors.nearest_neighbor_distance`][triwarp.neighbors.nearest_neighbor_distance]
    """
    device = points.device
    n = int(points.shape[0])
    if not 0 <= count <= n:
        raise ValueError(f"count must be in [0, {n}], got {count}")

    out_selected = wp.empty(count, dtype=wp.int32, device=device)
    if count == 0:
        return out_selected
    if not 0 <= start < n:
        raise ValueError(f"start must be in [0, {n}), got {start}")

    # Scratch for the running squared distance to the chosen set; the kernel initializes it.
    min_distance_sq = wp.empty(n, dtype=wp.float32, device=device)
    if n >= kernel_points.FARTHEST_BLOCK_LARGE_FROM:
        block_dim = kernel_points.FARTHEST_BLOCK_LARGE
    elif n >= kernel_points.FARTHEST_BLOCK_MID_FROM:
        block_dim = kernel_points.FARTHEST_BLOCK_MID
    else:
        block_dim = kernel_points.FARTHEST_BLOCK_SMALL
    # The whole greedy sweep runs as one persistent block; see the kernel for why.
    wp.launch_tiled(
        kernel_points.farthest_point_sample_block,
        dim=(1,),
        inputs=[points, wp.int32(start), wp.int32(count), min_distance_sq, out_selected],
        block_dim=block_dim,
        device=device,
    )
    return out_selected

convex_subset_mask 

convex_subset_mask(
    points: array[vec3],
    n_directions: int = 128,
    tolerance: float = 1e-06,
) -> wp.array[wp.bool]

Approximate the convex-hull vertices of a point cloud as a boolean mask.

For any direction n, the point maximizing ⟨n, p⟩ is a vertex of the convex hull, and the point minimizing it is the hull vertex farthest along -n. A single dot-product sweep over the points therefore yields the two hull vertices supporting +n and -n. Directions are drawn on the positive-z hemisphere with the deterministic Fibonacci spiral (sample_fibonacci_hemisphere); because the hemisphere and its reflection tile the full sphere, taking both the max and min per direction covers all 2 * n_directions antipodal orientations at half the dot-product cost of sampling the full sphere.

The extrema are computed by one thread per (direction, point slice), each reducing a strided slice of ITEMS_PER_SLICE_CUDA points (ITEMS_PER_SLICE_CPU on the CPU device) and committing one atomic, then a second pass marks the maximizers and minimizers. At most roughly 2 * n_directions points (plus ties) can be marked.

The result is an inner approximation, not a superset

Marking a point requires a certificate -- a sampled direction it is extremal along -- so a hull vertex with no such direction is dropped. The mask is therefore a subset of the hull boundary, never a conservative superset of the hull vertices; see Notes for the exact guarantee and how to raise recall. When losing a hull vertex is unacceptable, use convex_superset_mask, which errs the other way by construction.

Parameters:

Name Type Description Default
points array[vec3]

(n_points,) point positions on the target device.

 required
n_directions int

Number of Fibonacci hemisphere directions. Each covers two antipodal orientations, so the effective coverage is 2 * n_directions. Larger values recover more of the hull vertices.

 128
tolerance float

Relative slack on the support test; a point is marked when its dot product with a direction is within tolerance * (max - min) of that direction's maximum or minimum, where max - min is the cloud's support extent along the direction. Scaling the slack with the extent captures coplanar ties and float32 roundoff at any coordinate scale.

 1e-06

Returns:

Type Description
array[bool]

Length-n_points mask on points.device; True for points selected as approximate hull vertices. Empty when there are no points.
Notes

What is guaranteed. Every marked point lies on the convex-hull boundary, to within the tolerance slack. That is the invariant that holds for every input. The stronger statement -- every marked point is a hull vertex -- holds only when no support direction ties, which is the generic case for a cloud in general position but fails on structured data: on a 5x5 grid over each face of a cube (98 points, 8 hull vertices) the mask returns 14 points at any n_directions, the 8 corners plus 6 face-edge midpoints that tie with a corner along a face normal. Raising tolerance widens that effect deliberately.

What is not guaranteed. A hull vertex is recovered only when a sampled direction falls inside its normal cone, so recall degrades with the flatness of the hull around a vertex, not with the point count. On 500 standard-normal points (31 hull vertices) the recovered fraction measures 0.61 at n_directions=32, 0.81 at 128, 0.87 at 256, 0.94 at 512 and 1.00 at 16384; the four vertices missed at 256 have normal cones spanning 3e-5 to 2e-3 of the sphere, so 512 antipodal orientations are expected to hit them 0.02 to 1.1 times. Increasing n_directions is the only knob that raises recall -- tolerance trades precision for it and is not a substitute. When all hull vertices are required, use an exact hull (scipy.spatial.ConvexHull and the qhull-backed mesh libraries): no setting of these parameters makes this function conservative.

See Also

convex_subset convex_superset_mask sample_fibonacci_hemisphere flatnonzero scipy.spatial.ConvexHull

Source code in triwarp/points.py 
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def convex_subset_mask(
    points: wp.array[wp.vec3], n_directions: int = 128, tolerance: float = 1e-6
) -> wp.array[wp.bool]:
    """
    Approximate the convex-hull vertices of a point cloud as a boolean mask.

    For any direction ``n``, the point maximizing ``⟨n, p⟩`` is a vertex of the
    convex hull, and the point minimizing it is the hull vertex farthest along
    ``-n``. A single dot-product sweep over the points therefore yields the two
    hull vertices supporting ``+n`` and ``-n``. Directions are drawn on the
    positive-``z`` hemisphere with the deterministic Fibonacci spiral
    ([`sample_fibonacci_hemisphere`][triwarp.sample.sample_fibonacci_hemisphere]);
    because the hemisphere and its reflection tile the full sphere, taking both
    the max and min per direction covers all ``2 * n_directions`` antipodal
    orientations at half the dot-product cost of sampling the full sphere.

    The extrema are computed by one thread per ``(direction, point slice)``, each reducing a
    strided slice of [`ITEMS_PER_SLICE_CUDA`][triwarp.constants.ITEMS_PER_SLICE_CUDA] points
    (``ITEMS_PER_SLICE_CPU`` on the CPU device) and committing one atomic, then a second pass marks
    the maximizers and minimizers. At most roughly ``2 * n_directions`` points (plus ties) can be
    marked.

    !!! warning "The result is an inner approximation, not a superset"

        Marking a point requires a *certificate* -- a sampled direction it is extremal
        along -- so a hull vertex with no such direction is **dropped**. The mask is
        therefore a subset of the hull boundary, never a conservative superset of the
        hull vertices; see Notes for the exact guarantee and how to raise recall. When
        losing a hull vertex is unacceptable, use
        [`convex_superset_mask`][triwarp.points.convex_superset_mask], which errs the
        other way by construction.

    Parameters
    ----------
    points
        ``(n_points,)`` point positions on the target device.
    n_directions
        Number of Fibonacci hemisphere directions. Each covers two antipodal
        orientations, so the effective coverage is ``2 * n_directions``. Larger
        values recover more of the hull vertices.
    tolerance
        Relative slack on the support test; a point is marked when its dot product
        with a direction is within ``tolerance * (max - min)`` of that direction's
        maximum or minimum, where ``max - min`` is the cloud's support extent along
        the direction. Scaling the slack with the extent captures coplanar ties and
        float32 roundoff at any coordinate scale.

    Returns
    -------
    wp.array[wp.bool]
        Length-``n_points`` mask on ``points.device``; ``True`` for points selected
        as approximate hull vertices. Empty when there are no points.

    Notes
    -----
    **What is guaranteed.** Every marked point lies on the convex-hull *boundary*, to
    within the ``tolerance`` slack. That is the invariant that holds for every input.
    The stronger statement -- every marked point is a hull *vertex* -- holds only when
    no support direction ties, which is the generic case for a cloud in general
    position but fails on structured data: on a 5x5 grid over each face of a cube
    (98 points, 8 hull vertices) the mask returns 14 points at any ``n_directions``,
    the 8 corners plus 6 face-edge midpoints that tie with a corner along a face
    normal. Raising ``tolerance`` widens that effect deliberately.

    **What is not guaranteed.** A hull vertex is recovered only when a sampled
    direction falls inside its *normal cone*, so recall degrades with the flatness of
    the hull around a vertex, not with the point count. On 500 standard-normal points
    (31 hull vertices) the recovered fraction measures 0.61 at ``n_directions=32``,
    0.81 at 128, 0.87 at 256, 0.94 at 512 and 1.00 at 16384; the four vertices missed
    at 256 have normal cones spanning 3e-5 to 2e-3 of the sphere, so 512 antipodal
    orientations are expected to hit them 0.02 to 1.1 times. Increasing
    ``n_directions`` is the only knob that raises recall -- ``tolerance`` trades
    precision for it and is not a substitute. When *all* hull vertices are required,
    use an exact hull ([`scipy.spatial.ConvexHull`][] and the qhull-backed mesh
    libraries): no setting of these parameters makes this function conservative.

    See Also
    --------
    [`convex_subset`][triwarp.points.convex_subset]
    [`convex_superset_mask`][triwarp.points.convex_superset_mask]
    [`sample_fibonacci_hemisphere`][triwarp.sample.sample_fibonacci_hemisphere]
    [`flatnonzero`][triwarp.array.flatnonzero]
    [`scipy.spatial.ConvexHull`][]
    """
    device = points.device
    n_points = int(points.shape[0])
    if n_points == 0:
        return wp.empty(0, dtype=wp.bool, device=device)

    n_dir = int(n_directions)
    directions = tw.sample.sample_fibonacci_hemisphere(n_dir, device=device)
    best_max, best_min = _support_extremes(points, directions)

    out_mask = wp.zeros(n_points, dtype=wp.bool, device=device)
    wp.launch(
        kernel_points.mark_hull_support,
        dim=(n_dir, n_points),
        inputs=[points, directions, best_max, best_min, wp.float32(tolerance), out_mask],
        device=device,
    )
    return out_mask

convex_subset 

convex_subset(
    points: array[vec3],
    n_directions: int = 128,
    tolerance: float = 1e-06,
) -> wp.array[wp.vec3]

Approximate the convex-hull vertices of a point cloud as a point subset.

Convenience wrapper around convex_subset_mask that returns the selected points directly, gathered from points in ascending index order. The accuracy is the mask's: the returned subset lies on the hull boundary but can omit hull vertices, and is not a conservative superset of them.

Parameters:

Name Type Description Default
points array[vec3]

(n_points,) point positions on the target device.

 required
n_directions int

Number of Fibonacci hemisphere directions (see convex_subset_mask).

 128
tolerance float

Relative slack on the support test (see convex_subset_mask).

 1e-06

Returns:

Type Description
array[vec3]

The subset of points selected as approximate hull vertices, on points.device. Empty when there are no points.
See Also

convex_subset_mask gather scipy.spatial.ConvexHull

Source code in triwarp/points.py 
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def convex_subset(
    points: wp.array[wp.vec3], n_directions: int = 128, tolerance: float = 1e-6
) -> wp.array[wp.vec3]:
    """
    Approximate the convex-hull vertices of a point cloud as a point subset.

    Convenience wrapper around
    [`convex_subset_mask`][triwarp.points.convex_subset_mask] that returns the
    selected points directly, gathered from ``points`` in ascending index order. The
    accuracy is the mask's: the returned subset lies on the hull boundary but can omit
    hull vertices, and is not a conservative superset of them.

    Parameters
    ----------
    points
        ``(n_points,)`` point positions on the target device.
    n_directions
        Number of Fibonacci hemisphere directions (see
        [`convex_subset_mask`][triwarp.points.convex_subset_mask]).
    tolerance
        Relative slack on the support test (see
        [`convex_subset_mask`][triwarp.points.convex_subset_mask]).

    Returns
    -------
    wp.array[wp.vec3]
        The subset of ``points`` selected as approximate hull vertices, on
        ``points.device``. Empty when there are no points.

    See Also
    --------
    [`convex_subset_mask`][triwarp.points.convex_subset_mask]
    [`gather`][triwarp.array.gather]
    [`scipy.spatial.ConvexHull`][]
    """
    mask = convex_subset_mask(points, n_directions=n_directions, tolerance=tolerance)
    indices = tw.array.flatnonzero(mask)
    return tw.array.gather(points, indices)

convex_superset_mask 

convex_superset_mask(
    points: array[vec3],
    subdivisions: int = 2,
    margin: float = 1e-05,
) -> wp.array[wp.bool]

Conservatively discard interior points, keeping a superset of the convex-hull vertices.

Unlike convex_subset_mask, which keeps only points it can prove are on the hull, this keeps every point it cannot prove is interior. The certificate is a tetrahedron: the support point of the cloud along each direction of an icosphere is a hull vertex, so for every triangle (a, b, c) of the icosphere the tetrahedron (centroid, s_a, s_b, s_c) has all four corners in the hull and therefore lies inside it. Any point strictly inside such a tetrahedron is strictly inside the hull and cannot be a hull vertex; every other point is kept.

Because a hull vertex lies on the hull boundary, it is in no tetrahedron's strict interior, so no hull vertex is ever discarded -- for any input, any subdivisions, and any margin. The parameters trade only how many interior points survive. That makes this a prefilter: run it, then hand the survivors to an exact hull, which then does its superlinear work on a small fraction of the cloud.

Cost is one support sweep (n_points times the 10 * 4 ** subdivisions + 2 icosphere directions) plus one pass testing each point against the 20 * 4 ** subdivisions tetrahedra, whose precomputed face planes every thread reads in lockstep.

Parameters:

Name Type Description Default
points array[vec3]

(n_points,) point positions on the target device.

 required
subdivisions int

Icosphere refinement level for the direction set, which also fixes the tetrahedron count. Higher values wrap the hull more tightly and so discard more interior points, at a proportionally higher cost; the guarantee is unaffected.

 2
margin float

Distance, as a fraction of the shell radius, by which a point must clear all four faces of a tetrahedron to count as strictly inside it. Raising it keeps more points, so it can only weaken the filter, never the guarantee. The default is set by float32 arithmetic rather than by taste: a point lying exactly on a face must not be read as inside. See Notes.

 1e-05

Returns:

Type Description
array[bool]

Length-n_points mask on points.device; True for points that may be hull vertices, which includes every actual hull vertex. Empty when there are no points.
Notes

Selectivity depends strongly on the cloud's shape: a near-spherical cloud keeps a small fraction of its points (a triangulated inner shell hugs it tightly), while a flat-faced cloud keeps far more, because the shell cannot hug a plane and the survivors form a thin slab under each face.

The guarantee is exact in real arithmetic; in float32 it rests on margin covering the error in the plane evaluation. Values of margin below 1e-6 can discard a true hull vertex -- always a point lying essentially exactly on a tetrahedron face, where the computed distance straddles zero -- while 1e-6 and above are clean; the 1e-5 default sits well above that boundary, at a small cost in selectivity. TETRAHEDRON_FLATNESS is the other half of the same protection: it discards sliver tetrahedra whose face normals are too ill-conditioned to trust, and without it no margin in this range is safe.

Degenerate input is handled by the same conservative logic rather than by a special case. A coplanar or collinear cloud makes every tetrahedron flat; flat tetrahedra are rejected as ill-conditioned, so nothing is certified interior and every point is kept, which is a valid (if useless) superset. Fewer than four points returns an all-True mask directly.

See Also

convex_subset_mask icosphere flatnonzero scipy.spatial.ConvexHull

Source code in triwarp/points.py 
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def convex_superset_mask(
    points: wp.array[wp.vec3], subdivisions: int = 2, margin: float = 1e-5
) -> wp.array[wp.bool]:
    """
    Conservatively discard interior points, keeping a superset of the convex-hull vertices.

    Unlike [`convex_subset_mask`][triwarp.points.convex_subset_mask], which keeps only points it can
    *prove* are on the hull, this keeps every point it cannot prove is *interior*. The certificate
    is a tetrahedron: the support point of the cloud along each direction of an
    [`icosphere`][triwarp.creation.icosphere] is a hull vertex, so for every triangle ``(a, b, c)``
    of the icosphere the tetrahedron ``(centroid, s_a, s_b, s_c)`` has all four corners in the hull
    and therefore lies inside it. Any point strictly inside such a tetrahedron is strictly inside
    the hull and cannot be a hull vertex; every other point is kept.

    Because a hull vertex lies on the hull *boundary*, it is in no tetrahedron's strict interior, so
    **no hull vertex is ever discarded** -- for any input, any ``subdivisions``, and any ``margin``.
    The parameters trade only how many interior points survive. That makes this a prefilter: run it,
    then hand the survivors to an exact hull, which then does its superlinear work on a small
    fraction of the cloud.

    Cost is one support sweep (``n_points`` times the ``10 * 4 ** subdivisions + 2`` icosphere
    directions) plus one pass testing each point against the ``20 * 4 ** subdivisions``
    tetrahedra, whose precomputed face planes every thread reads in lockstep.

    Parameters
    ----------
    points
        ``(n_points,)`` point positions on the target device.
    subdivisions
        Icosphere refinement level for the direction set, which also fixes the tetrahedron count.
        Higher values wrap the hull more tightly and so discard more interior points, at a
        proportionally higher cost; the guarantee is unaffected.
    margin
        Distance, as a fraction of the shell radius, by which a point must clear all four faces of a
        tetrahedron to count as strictly inside it. Raising it keeps *more* points, so it can only
        weaken the filter, never the guarantee. The default is set by float32 arithmetic rather than
        by taste: a point lying *exactly* on a face must not be read as inside. See Notes.

    Returns
    -------
    wp.array[wp.bool]
        Length-``n_points`` mask on ``points.device``; ``True`` for points that may be hull
        vertices, which includes every actual hull vertex. Empty when there are no points.

    Notes
    -----
    Selectivity depends strongly on the cloud's shape: a near-spherical cloud keeps a small
    fraction of its points (a triangulated inner shell hugs it tightly), while a flat-faced cloud
    keeps far more, because the shell cannot hug a plane and the survivors form a thin slab under
    each face.

    The guarantee is exact in real arithmetic; in float32 it rests on ``margin`` covering the error
    in the plane evaluation. Values of ``margin`` below ``1e-6`` can discard a true hull vertex --
    always a point lying *essentially exactly* on a tetrahedron face, where the computed distance
    straddles zero -- while ``1e-6`` and above are clean; the ``1e-5`` default sits well above that
    boundary, at a small cost in selectivity. ``TETRAHEDRON_FLATNESS`` is the other half of the
    same protection: it discards sliver tetrahedra whose face normals are too ill-conditioned to
    trust, and without it no margin in this range is safe.

    Degenerate input is handled by the same conservative logic rather than by a special case. A
    coplanar or collinear cloud makes every tetrahedron flat; flat tetrahedra are rejected as
    ill-conditioned, so nothing is certified interior and every point is kept, which is a valid
    (if useless) superset. Fewer than four points returns an all-``True`` mask directly.

    See Also
    --------
    [`convex_subset_mask`][triwarp.points.convex_subset_mask]
    [`icosphere`][triwarp.creation.icosphere]
    [`flatnonzero`][triwarp.array.flatnonzero]
    [`scipy.spatial.ConvexHull`][]
    """
    device = points.device
    n_points = int(points.shape[0])
    if n_points == 0:
        return wp.empty(0, dtype=wp.bool, device=device)
    if n_points < 4:
        # No tetrahedron exists, so nothing can be certified interior.
        return wp.full(n_points, value=True, dtype=wp.bool, device=device)

    directions, shell_faces = tw.creation.icosphere(subdivisions=int(subdivisions), device=device)
    n_dir = int(directions.shape[0])
    n_tetra = int(shell_faces.shape[0]) // 3
    best_max, best_min = _support_extremes(points, directions)

    # Seeded with the last index rather than a sentinel: a direction that somehow marks nothing
    # then yields a real point, which keeps the gather in range and the tetrahedra valid.
    support = wp.full(n_dir, value=n_points - 1, dtype=wp.int32, device=device)
    wp.launch(
        kernel_points.support_indices,
        dim=(n_dir, n_points),
        inputs=[points, directions, best_max, best_min, SUPPORT_TIE_SLACK, support],
        device=device,
    )

    shell_vertices = tw.array.gather(points, support)
    centroid = wp.empty(1, dtype=wp.vec3, device=device)
    radius = wp.empty(1, dtype=wp.float32, device=device)
    wp.launch(
        kernel_points.shell_bounds, dim=1, inputs=[shell_vertices, centroid, radius], device=device
    )

    planes = wp.zeros((n_tetra, 4), dtype=wp.vec4, device=device)
    valid = wp.empty(n_tetra, dtype=wp.bool, device=device)
    wp.launch(
        kernel_points.tetrahedron_planes,
        dim=n_tetra,
        inputs=[shell_vertices, shell_faces, centroid, TETRAHEDRON_FLATNESS, planes, valid],
        device=device,
    )

    out_mask = wp.empty(n_points, dtype=wp.bool, device=device)
    wp.launch(
        kernel_points.mark_hull_superset,
        dim=n_points,
        inputs=[points, planes, valid, radius, wp.float32(margin), out_mask],
        device=device,
    )
    return out_mask

vector_angle 

vector_angle(
    a: array[vec3], b: array[vec3]
) -> wp.array[wp.float32]

Unsigned angle in radians between pairs of vectors.

For each index i, computes atan2(norm(cross(a[i], b[i])), dot(a[i], b[i])), which lies in [0, pi]. Matches trimesh.geometry.vector_angle on stacked pairs, and agrees with its arccos(dot) to round-off on unit input while staying accurate for nearly parallel or nearly antiparallel pairs, where arccos loses most of its digits. Being scale-free, it also accepts unnormalized vectors, and pairs where either vector is zero read 0.0.

Parameters:

Name Type Description Default
a array[vec3]

Length-n vectors on the target device.

 required
b array[vec3]

Length-n vectors on the same device as a.

 required

Returns:

Type Description
array[float32]

Length-n unsigned angles in radians on a.device. Empty when n == 0.

Raises:

Type Description
ValueError

If a and b have different lengths.
RuntimeError

If a and b are not all on one device.
See Also

trimesh.geometry.vector_angle

Source code in triwarp/points.py 
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def vector_angle(a: wp.array[wp.vec3], b: wp.array[wp.vec3]) -> wp.array[wp.float32]:
    """
    Unsigned angle in radians between pairs of vectors.

    For each index ``i``, computes ``atan2(norm(cross(a[i], b[i])), dot(a[i], b[i]))``, which lies
    in ``[0, pi]``. Matches [`trimesh.geometry.vector_angle`][] on stacked pairs, and agrees with
    its ``arccos(dot)`` to round-off on unit input while staying accurate for nearly parallel or
    nearly antiparallel pairs, where ``arccos`` loses most of its digits. Being scale-free, it also
    accepts unnormalized vectors, and pairs where either vector is zero read ``0.0``.

    Parameters
    ----------
    a
        Length-``n`` vectors on the target device.
    b
        Length-``n`` vectors on the same device as ``a``.

    Returns
    -------
    wp.array[wp.float32]
        Length-``n`` unsigned angles in radians on ``a.device``. Empty when ``n == 0``.

    Raises
    ------
    ValueError
        If ``a`` and ``b`` have different lengths.
    RuntimeError
        If ``a`` and ``b`` are not all on one device.

    See Also
    --------
    [`trimesh.geometry.vector_angle`][]
    """
    require_same_device(a=a, b=b)
    device = a.device
    n = int(a.shape[0])
    if n != int(b.shape[0]):
        raise ValueError(f"a and b must have the same length, got {n} and {b.shape[0]}")

    if n == 0:
        return wp.empty(0, dtype=wp.float32, device=device)

    out_angles = wp.empty(n, dtype=wp.float32, device=device)
    wp.map(kernel_predicates.vector_angle, a, b, out=out_angles)
    return out_angles

radial_sort 

radial_sort(
    points: array[vec3],
    origin: vec3,
    normal: vec3,
    start: vec3 | None = None,
) -> wp.array[wp.vec3]

Sort points radially (by angle) around an axis and return them reordered.

Points are projected onto two axes perpendicular to normal and ordered by the angle atan2 of the projection, in descending order (matching trimesh.points.radial_sort).

Parameters:

Name Type Description Default
points array[vec3]

(n,) positions in space as wp.vec3.

 required
origin vec3

Point to sort around as wp.vec3.

 required
normal vec3

Axis to sort around as wp.vec3.

 required
start vec3 | None

Optional wp.vec3 specifying the start position in counter-clockwise order when viewed along normal. Must not be parallel with normal. When None, an arbitrary perpendicular axis is used.

 None

Returns:

Type Description
array[vec3]

Length n array of the input points reordered by descending angle, on points.device. Empty when points is empty.

Raises:

Type Description
ValueError

If start is provided and is (near-)parallel with normal.
Source code in triwarp/points.py 
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def radial_sort(
    points: wp.array[wp.vec3], origin: wp.vec3, normal: wp.vec3, start: wp.vec3 | None = None
) -> wp.array[wp.vec3]:
    """
    Sort points radially (by angle) around an axis and return them reordered.

    Points are projected onto two axes perpendicular to ``normal`` and ordered by
    the angle ``atan2`` of the projection, in **descending** order (matching
    [`trimesh.points.radial_sort`][]).

    Parameters
    ----------
    points
        ``(n,)`` positions in space as ``wp.vec3``.
    origin
        Point to sort around as ``wp.vec3``.
    normal
        Axis to sort around as ``wp.vec3``.
    start
        Optional ``wp.vec3`` specifying the start position in counter-clockwise
        order when viewed along ``normal``. Must not be parallel with ``normal``.
        When ``None``, an arbitrary perpendicular axis is used.

    Returns
    -------
    wp.array[wp.vec3]
        Length ``n`` array of the input points reordered by descending angle, on
        ``points.device``. Empty when ``points`` is empty.

    Raises
    ------
    ValueError
        If ``start`` is provided and is (near-)parallel with ``normal``.
    """
    device = points.device
    n = int(points.shape[0])
    if n == 0:
        return wp.empty(0, dtype=wp.vec3, device=device)

    # Build two axes perpendicular to each other and the normal, onto which the
    # points are projected to recover an angle. Done on the host since the axes
    # are a single O(1) setup shared by every point.
    if start is None:
        # Cross with whichever of x/y `normal` is *less* aligned with, so the cross product is
        # never near-degenerate (the same construction as `kernels.tangent_space.any_perpendicular`
        # -- but that one returns a specific unit chirality this needs to preserve exactly at
        # `normal = +z`, so it's spelled out here rather than called).
        #
        # The formula this replaces, `wp.vec3(normal[0], normal[2], -normal[1])`, is a faithful
        # port of ``trimesh.points.radial_sort``'s own axis0, and both share the same bug: for any
        # `normal` with a nonzero x-component, `dot(normal, axis0) == normal.x**2 != 0`, so axis0 is
        # not actually perpendicular to `normal` and carries a leftover component along it into
        # every point's angle -- exactly zero only at `normal = (0, *, *)`, which is why the
        # existing regression tests (fixed at `normal = (0, 0, 1)`) never caught it. This
        # construction is perpendicular to `normal` for every `normal`, and reduces to the replaced
        # formula's exact axis0/axis1 at `normal = (0, 0, 1)`: `abs(normal[0]) > abs(normal[1])` is
        # then `0 > 0`, False, so `helper = (1, 0, 0)` and
        # `cross((0, 0, 1), (1, 0, 0)) == (0, 1, 0)`, matching it bit for bit.
        helper = wp.vec3(1.0, 0.0, 0.0)
        if abs(normal[0]) > abs(normal[1]):
            helper = wp.vec3(0.0, 1.0, 0.0)
        axis0 = wp.cross(normal, helper)
        axis1 = wp.cross(normal, axis0)
    else:
        unit_normal = wp.normalize(normal)
        unit_start = wp.normalize(start)
        if abs(1.0 - abs(wp.dot(unit_normal, unit_start))) < TOLERANCE_ZERO:
            raise ValueError("start must not be parallel with normal")
        axis0 = wp.cross(unit_start, unit_normal)
        axis1 = wp.cross(axis0, unit_normal)

    out_keys = wp.empty(n, dtype=wp.float32, device=device)
    wp.map(kernel_points.radial_sort_key, points, origin, axis0, axis1, out=out_keys)

    # Ascending radix sort of the negated angles yields the descending-angle order.
    _sorted_keys, order = tw.array.sort_and_argsort(out_keys, fill_value=n)
    return tw.array.gather(points, order)