forked from s_ranjbar/city_retrofit
690 lines
21 KiB
Python
690 lines
21 KiB
Python
"""
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triangles.py
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-------------
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Functions for dealing with triangle soups in (n, 3, 3) float form.
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"""
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import numpy as np
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from . import util
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from .util import unitize, diagonal_dot
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from .points import point_plane_distance
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from .constants import tol
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def cross(triangles):
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"""
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Returns the cross product of two edges from input triangles
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Parameters
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--------------
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triangles: (n, 3, 3) float
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Vertices of triangles
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Returns
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--------------
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crosses : (n, 3) float
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Cross product of two edge vectors
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"""
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vectors = np.diff(triangles, axis=1)
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crosses = np.cross(vectors[:, 0], vectors[:, 1])
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return crosses
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def area(triangles=None, crosses=None, sum=False):
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"""
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Calculates the sum area of input triangles
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Parameters
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----------
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triangles : (n, 3, 3) float
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Vertices of triangles
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crosses : (n, 3) float or None
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As a speedup don't re- compute cross products
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sum : bool
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Return summed area or individual triangle area
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Returns
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----------
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area : (n,) float or float
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Individual or summed area depending on `sum` argument
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"""
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if crosses is None:
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crosses = cross(triangles)
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area = (np.sum(crosses**2, axis=1)**.5) * .5
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if sum:
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return np.sum(area)
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return area
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def normals(triangles=None, crosses=None):
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"""
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Calculates the normals of input triangles
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Parameters
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------------
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triangles : (n, 3, 3) float
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Vertex positions
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crosses : (n, 3) float
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Cross products of edge vectors
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Returns
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------------
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normals : (m, 3) float
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Normal vectors
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valid : (n,) bool
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Was the face nonzero area or not
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"""
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if crosses is None:
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crosses = cross(triangles)
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# unitize the cross product vectors
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unit, valid = unitize(crosses, check_valid=True)
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return unit, valid
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def angles(triangles):
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"""
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Calculates the angles of input triangles.
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Parameters
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------------
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triangles : (n, 3, 3) float
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Vertex positions
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Returns
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------------
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angles : (n, 3) float
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Angles at vertex positions in radians
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Degenerate angles will be returned as zero
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"""
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# don't copy triangles
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triangles = np.asanyarray(triangles, dtype=np.float64)
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# get a unit vector for each edge of the triangle
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u = unitize(triangles[:, 1] - triangles[:, 0])
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v = unitize(triangles[:, 2] - triangles[:, 0])
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w = unitize(triangles[:, 2] - triangles[:, 1])
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# run the cosine and per-row dot product
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result = np.zeros((len(triangles), 3), dtype=np.float64)
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# clip to make sure we don't float error past 1.0
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result[:, 0] = np.arccos(np.clip(diagonal_dot(u, v), -1, 1))
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result[:, 1] = np.arccos(np.clip(diagonal_dot(-u, w), -1, 1))
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# the third angle is just the remaining
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result[:, 2] = np.pi - result[:, 0] - result[:, 1]
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# a triangle with any zero angles is degenerate
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# so set all of the angles to zero in that case
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result[(result < tol.merge).any(axis=1), :] = 0.0
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return result
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def all_coplanar(triangles):
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"""
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Check to see if a list of triangles are all coplanar
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Parameters
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----------------
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triangles: (n, 3, 3) float
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Vertices of triangles
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Returns
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---------------
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all_coplanar : bool
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True if all triangles are coplanar
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"""
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triangles = np.asanyarray(triangles, dtype=np.float64)
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if not util.is_shape(triangles, (-1, 3, 3)):
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raise ValueError('Triangles must be (n, 3, 3)!')
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test_normal = normals(triangles)[0]
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test_vertex = triangles[0][0]
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distances = point_plane_distance(points=triangles[1:].reshape((-1, 3)),
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plane_normal=test_normal,
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plane_origin=test_vertex)
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all_coplanar = np.all(np.abs(distances) < tol.zero)
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return all_coplanar
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def any_coplanar(triangles):
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"""
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For a list of triangles if the FIRST triangle is coplanar
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with ANY of the following triangles, return True.
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Otherwise, return False.
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"""
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triangles = np.asanyarray(triangles, dtype=np.float64)
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if not util.is_shape(triangles, (-1, 3, 3)):
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raise ValueError('Triangles must be (n, 3, 3)!')
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test_normal = normals(triangles)[0]
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test_vertex = triangles[0][0]
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distances = point_plane_distance(points=triangles[1:].reshape((-1, 3)),
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plane_normal=test_normal,
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plane_origin=test_vertex)
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any_coplanar = np.any(
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np.all(np.abs(distances.reshape((-1, 3)) < tol.zero), axis=1))
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return any_coplanar
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def mass_properties(triangles,
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crosses=None,
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density=1.0,
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center_mass=None,
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skip_inertia=False):
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"""
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Calculate the mass properties of a group of triangles.
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Implemented from:
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http://www.geometrictools.com/Documentation/PolyhedralMassProperties.pdf
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Parameters
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----------
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triangles : (n, 3, 3) float
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Triangle vertices in space
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crosses : (n,) float
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Optional cross products of triangles
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density : float
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Optional override for density
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center_mass : (3,) float
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Optional override for center mass
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skip_inertia : bool
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if True will not return moments matrix
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Returns
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---------
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info : dict
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Mass properties
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"""
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triangles = np.asanyarray(triangles, dtype=np.float64)
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if not util.is_shape(triangles, (-1, 3, 3)):
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raise ValueError('Triangles must be (n, 3, 3)!')
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if crosses is None:
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crosses = cross(triangles)
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# these are the subexpressions of the integral
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# this is equvilant but 7x faster than triangles.sum(axis=1)
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f1 = triangles[:, 0, :] + triangles[:, 1, :] + triangles[:, 2, :]
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# for the the first vertex of every triangle:
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# triangles[:,0,:] will give rows like [[x0, y0, z0], ...]
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# for the x coordinates of every triangle
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# triangles[:,:,0] will give rows like [[x0, x1, x2], ...]
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f2 = (triangles[:, 0, :]**2 +
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triangles[:, 1, :]**2 +
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triangles[:, 0, :] * triangles[:, 1, :] +
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triangles[:, 2, :] * f1)
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f3 = ((triangles[:, 0, :]**3) +
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(triangles[:, 0, :]**2) * (triangles[:, 1, :]) +
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(triangles[:, 0, :]) * (triangles[:, 1, :]**2) +
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(triangles[:, 1, :]**3) +
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(triangles[:, 2, :] * f2))
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g0 = (f2 + (triangles[:, 0, :] + f1) * triangles[:, 0, :])
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g1 = (f2 + (triangles[:, 1, :] + f1) * triangles[:, 1, :])
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g2 = (f2 + (triangles[:, 2, :] + f1) * triangles[:, 2, :])
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integral = np.zeros((10, len(f1)))
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integral[0] = crosses[:, 0] * f1[:, 0]
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integral[1:4] = (crosses * f2).T
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integral[4:7] = (crosses * f3).T
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for i in range(3):
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triangle_i = np.mod(i + 1, 3)
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integral[i + 7] = crosses[:, i] * (
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(triangles[:, 0, triangle_i] * g0[:, i]) +
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(triangles[:, 1, triangle_i] * g1[:, i]) +
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(triangles[:, 2, triangle_i] * g2[:, i]))
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coefficients = 1.0 / np.array(
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[6, 24, 24, 24, 60, 60, 60, 120, 120, 120],
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dtype=np.float64)
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integrated = integral.sum(axis=1) * coefficients
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volume = integrated[0]
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if center_mass is None:
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if np.abs(volume) < tol.zero:
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center_mass = np.zeros(3)
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else:
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center_mass = integrated[1:4] / volume
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mass = density * volume
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result = {'density': density,
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'mass': mass,
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'volume': volume,
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'center_mass': center_mass}
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if skip_inertia:
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return result
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inertia = np.zeros((3, 3))
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inertia[0, 0] = integrated[5] + integrated[6] - \
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(volume * (center_mass[[1, 2]]**2).sum())
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inertia[1, 1] = integrated[4] + integrated[6] - \
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(volume * (center_mass[[0, 2]]**2).sum())
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inertia[2, 2] = integrated[4] + integrated[5] - \
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(volume * (center_mass[[0, 1]]**2).sum())
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inertia[0, 1] = (
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integrated[7] - (volume * np.product(center_mass[[0, 1]])))
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inertia[1, 2] = (
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integrated[8] - (volume * np.product(center_mass[[1, 2]])))
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inertia[0, 2] = (
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integrated[9] - (volume * np.product(center_mass[[0, 2]])))
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inertia[2, 0] = inertia[0, 2]
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inertia[2, 1] = inertia[1, 2]
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inertia[1, 0] = inertia[0, 1]
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inertia *= density
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result['inertia'] = inertia
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return result
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def windings_aligned(triangles, normals_compare):
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"""
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Given a list of triangles and a list of normals determine if the
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two are aligned
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Parameters
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----------
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triangles : (n, 3, 3) float
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Vertex locations in space
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normals_compare : (n, 3) float
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List of normals to compare
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Returns
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----------
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aligned : (n,) bool
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Are normals aligned with triangles
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"""
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triangles = np.asanyarray(triangles, dtype=np.float64)
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if not util.is_shape(triangles, (-1, 3, 3), allow_zeros=True):
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raise ValueError(
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'triangles must have shape (n, 3, 3), got %s' % str(triangles.shape))
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calculated, valid = normals(triangles)
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difference = diagonal_dot(calculated,
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normals_compare[valid])
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aligned = np.zeros(len(triangles), dtype=np.bool)
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aligned[valid] = difference > 0.0
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return aligned
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def bounds_tree(triangles):
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"""
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Given a list of triangles, create an r-tree for broad- phase
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collision detection
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Parameters
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---------
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triangles : (n, 3, 3) float
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Triangles in space
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Returns
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---------
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tree : rtree.Rtree
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One node per triangle
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"""
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triangles = np.asanyarray(triangles, dtype=np.float64)
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if not util.is_shape(triangles, (-1, 3, 3)):
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raise ValueError('Triangles must be (n, 3, 3)!')
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# the (n,6) interleaved bounding box for every triangle
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triangle_bounds = np.column_stack((triangles.min(axis=1),
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triangles.max(axis=1)))
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tree = util.bounds_tree(triangle_bounds)
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return tree
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def nondegenerate(triangles, areas=None, height=None):
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"""
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Find all triangles which have an oriented bounding box
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where both of the two sides is larger than a specified height.
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Degenerate triangles can be when:
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1) Two of the three vertices are colocated
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2) All three vertices are unique but colinear
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Parameters
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----------
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triangles : (n, 3, 3) float
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Triangles in space
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height : float
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Minimum edge length of a triangle to keep
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Returns
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----------
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nondegenerate : (n,) bool
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True if a triangle meets required minimum height
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"""
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triangles = np.asanyarray(triangles, dtype=np.float64)
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if not util.is_shape(triangles, (-1, 3, 3)):
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raise ValueError('Triangles must be (n, 3, 3)!')
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if height is None:
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height = tol.merge
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# if both edges of the triangles OBB are longer than tol.merge
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# we declare them to be nondegenerate
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ok = (extents(triangles=triangles,
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areas=areas) > height).all(axis=1)
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return ok
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def extents(triangles, areas=None):
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"""
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Return the 2D bounding box size of each triangle.
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Parameters
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----------
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triangles : (n, 3, 3) float
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Triangles in space
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areas : (n,) float
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Optional area of input triangles
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Returns
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----------
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box : (n, 2) float
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The size of each triangle's 2D oriented bounding box
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"""
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triangles = np.asanyarray(triangles, dtype=np.float64)
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if not util.is_shape(triangles, (-1, 3, 3)):
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raise ValueError('Triangles must be (n, 3, 3)!')
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if areas is None:
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areas = area(triangles=triangles,
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sum=False)
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# the edge vectors which define the triangle
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a = triangles[:, 1] - triangles[:, 0]
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b = triangles[:, 2] - triangles[:, 0]
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# length of the edge vectors
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length_a = (a**2).sum(axis=1)**.5
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length_b = (b**2).sum(axis=1)**.5
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# which edges are acceptable length
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nonzero_a = length_a > tol.merge
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nonzero_b = length_b > tol.merge
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# find the two heights of the triangle
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# essentially this is the side length of an
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# oriented bounding box, per triangle
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box = np.zeros((len(triangles), 2), dtype=np.float64)
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box[:, 0][nonzero_a] = (areas[nonzero_a] * 2) / length_a[nonzero_a]
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box[:, 1][nonzero_b] = (areas[nonzero_b] * 2) / length_b[nonzero_b]
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return box
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def barycentric_to_points(triangles, barycentric):
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"""
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Convert a list of barycentric coordinates on a list of triangles
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to cartesian points.
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Parameters
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------------
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triangles : (n, 3, 3) float
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Triangles in space
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barycentric : (n, 2) float
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Barycentric coordinates
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Returns
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-----------
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points : (m, 3) float
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Points in space
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"""
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barycentric = np.asanyarray(barycentric, dtype=np.float64)
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triangles = np.asanyarray(triangles, dtype=np.float64)
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if not util.is_shape(triangles, (-1, 3, 3)):
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raise ValueError('Triangles must be (n, 3, 3)!')
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if barycentric.shape == (2,):
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barycentric = np.ones((len(triangles), 2),
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dtype=np.float64) * barycentric
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if util.is_shape(barycentric, (len(triangles), 2)):
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barycentric = np.column_stack((barycentric,
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1.0 - barycentric.sum(axis=1)))
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elif not util.is_shape(barycentric, (len(triangles), 3)):
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raise ValueError('Barycentric shape incorrect!')
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barycentric /= barycentric.sum(axis=1).reshape((-1, 1))
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points = (triangles * barycentric.reshape((-1, 3, 1))).sum(axis=1)
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return points
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def points_to_barycentric(triangles,
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points,
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method='cramer'):
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"""
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Find the barycentric coordinates of points relative to triangles.
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The Cramer's rule solution implements:
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http://blackpawn.com/texts/pointinpoly
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The cross product solution implements:
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https://www.cs.ubc.ca/~heidrich/Papers/JGT.05.pdf
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Parameters
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-----------
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triangles : (n, 3, 3) float
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Triangles vertices in space
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points : (n, 3) float
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Point in space associated with a triangle
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method : str
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Which method to compute the barycentric coordinates with:
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- 'cross': uses a method using cross products, roughly 2x slower but
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different numerical robustness properties
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- anything else: uses a cramer's rule solution
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Returns
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-----------
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barycentric : (n, 3) float
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Barycentric coordinates of each point
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"""
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def method_cross():
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n = np.cross(edge_vectors[:, 0], edge_vectors[:, 1])
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denominator = diagonal_dot(n, n)
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barycentric = np.zeros((len(triangles), 3), dtype=np.float64)
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barycentric[:, 2] = diagonal_dot(
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np.cross(edge_vectors[:, 0], w), n) / denominator
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barycentric[:, 1] = diagonal_dot(
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np.cross(w, edge_vectors[:, 1]), n) / denominator
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barycentric[:, 0] = 1 - barycentric[:, 1] - barycentric[:, 2]
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return barycentric
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def method_cramer():
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dot00 = diagonal_dot(edge_vectors[:, 0], edge_vectors[:, 0])
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dot01 = diagonal_dot(edge_vectors[:, 0], edge_vectors[:, 1])
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dot02 = diagonal_dot(edge_vectors[:, 0], w)
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dot11 = diagonal_dot(edge_vectors[:, 1], edge_vectors[:, 1])
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dot12 = diagonal_dot(edge_vectors[:, 1], w)
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inverse_denominator = 1.0 / (dot00 * dot11 - dot01 * dot01)
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barycentric = np.zeros((len(triangles), 3), dtype=np.float64)
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barycentric[:, 2] = (dot00 * dot12 - dot01 *
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dot02) * inverse_denominator
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barycentric[:, 1] = (dot11 * dot02 - dot01 *
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dot12) * inverse_denominator
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barycentric[:, 0] = 1 - barycentric[:, 1] - barycentric[:, 2]
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return barycentric
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# establish that input triangles and points are sane
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triangles = np.asanyarray(triangles, dtype=np.float64)
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points = np.asanyarray(points, dtype=np.float64)
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if not util.is_shape(triangles, (-1, 3, 3)):
|
|
raise ValueError('triangles shape incorrect')
|
|
if not util.is_shape(points, (len(triangles), 3)):
|
|
raise ValueError('triangles and points must correspond')
|
|
|
|
edge_vectors = triangles[:, 1:] - triangles[:, :1]
|
|
w = points - triangles[:, 0].reshape((-1, 3))
|
|
|
|
if method == 'cross':
|
|
return method_cross()
|
|
return method_cramer()
|
|
|
|
|
|
def closest_point(triangles, points):
|
|
"""
|
|
Return the closest point on the surface of each triangle for a
|
|
list of corresponding points.
|
|
|
|
Implements the method from "Real Time Collision Detection" and
|
|
use the same variable names as "ClosestPtPointTriangle" to avoid
|
|
being any more confusing.
|
|
|
|
|
|
Parameters
|
|
----------
|
|
triangles : (n, 3, 3) float
|
|
Triangle vertices in space
|
|
points : (n, 3) float
|
|
Points in space
|
|
|
|
Returns
|
|
----------
|
|
closest : (n, 3) float
|
|
Point on each triangle closest to each point
|
|
"""
|
|
|
|
# check input triangles and points
|
|
triangles = np.asanyarray(triangles, dtype=np.float64)
|
|
points = np.asanyarray(points, dtype=np.float64)
|
|
if not util.is_shape(triangles, (-1, 3, 3)):
|
|
raise ValueError('triangles shape incorrect')
|
|
if not util.is_shape(points, (len(triangles), 3)):
|
|
raise ValueError('need same number of triangles and points!')
|
|
|
|
# store the location of the closest point
|
|
result = np.zeros_like(points)
|
|
# which points still need to be handled
|
|
remain = np.ones(len(points), dtype=np.bool)
|
|
|
|
# if we dot product this against a (n, 3)
|
|
# it is equivalent but faster than array.sum(axis=1)
|
|
ones = [1.0, 1.0, 1.0]
|
|
|
|
# get the three points of each triangle
|
|
# use the same notation as RTCD to avoid confusion
|
|
a = triangles[:, 0, :]
|
|
b = triangles[:, 1, :]
|
|
c = triangles[:, 2, :]
|
|
|
|
# check if P is in vertex region outside A
|
|
ab = b - a
|
|
ac = c - a
|
|
ap = points - a
|
|
# this is a faster equivalent of:
|
|
# diagonal_dot(ab, ap)
|
|
d1 = np.dot(ab * ap, ones)
|
|
d2 = np.dot(ac * ap, ones)
|
|
|
|
# is the point at A
|
|
is_a = np.logical_and(d1 < tol.zero, d2 < tol.zero)
|
|
if is_a.any():
|
|
result[is_a] = a[is_a]
|
|
remain[is_a] = False
|
|
|
|
# check if P in vertex region outside B
|
|
bp = points - b
|
|
d3 = np.dot(ab * bp, ones)
|
|
d4 = np.dot(ac * bp, ones)
|
|
|
|
# do the logic check
|
|
is_b = (d3 > -tol.zero) & (d4 <= d3) & remain
|
|
if is_b.any():
|
|
result[is_b] = b[is_b]
|
|
remain[is_b] = False
|
|
|
|
# check if P in edge region of AB, if so return projection of P onto A
|
|
vc = (d1 * d4) - (d3 * d2)
|
|
is_ab = ((vc < tol.zero) &
|
|
(d1 > -tol.zero) &
|
|
(d3 < tol.zero) & remain)
|
|
if is_ab.any():
|
|
v = (d1[is_ab] / (d1[is_ab] - d3[is_ab])).reshape((-1, 1))
|
|
result[is_ab] = a[is_ab] + (v * ab[is_ab])
|
|
remain[is_ab] = False
|
|
|
|
# check if P in vertex region outside C
|
|
cp = points - c
|
|
d5 = np.dot(ab * cp, ones)
|
|
d6 = np.dot(ac * cp, ones)
|
|
is_c = (d6 > -tol.zero) & (d5 <= d6) & remain
|
|
if is_c.any():
|
|
result[is_c] = c[is_c]
|
|
remain[is_c] = False
|
|
|
|
# check if P in edge region of AC, if so return projection of P onto AC
|
|
vb = (d5 * d2) - (d1 * d6)
|
|
is_ac = (vb < tol.zero) & (d2 > -tol.zero) & (d6 < tol.zero) & remain
|
|
if is_ac.any():
|
|
w = (d2[is_ac] / (d2[is_ac] - d6[is_ac])).reshape((-1, 1))
|
|
result[is_ac] = a[is_ac] + w * ac[is_ac]
|
|
remain[is_ac] = False
|
|
|
|
# check if P in edge region of BC, if so return projection of P onto BC
|
|
va = (d3 * d6) - (d5 * d4)
|
|
is_bc = ((va < tol.zero) &
|
|
((d4 - d3) > - tol.zero) &
|
|
((d5 - d6) > -tol.zero) & remain)
|
|
if is_bc.any():
|
|
d43 = d4[is_bc] - d3[is_bc]
|
|
w = (d43 / (d43 + (d5[is_bc] - d6[is_bc]))).reshape((-1, 1))
|
|
result[is_bc] = b[is_bc] + w * (c[is_bc] - b[is_bc])
|
|
remain[is_bc] = False
|
|
|
|
# any remaining points must be inside face region
|
|
if remain.any():
|
|
# point is inside face region
|
|
denom = 1.0 / (va[remain] + vb[remain] + vc[remain])
|
|
v = (vb[remain] * denom).reshape((-1, 1))
|
|
w = (vc[remain] * denom).reshape((-1, 1))
|
|
# compute Q through its barycentric coordinates
|
|
result[remain] = a[remain] + (ab[remain] * v) + (ac[remain] * w)
|
|
|
|
return result
|
|
|
|
|
|
def to_kwargs(triangles):
|
|
"""
|
|
Convert a list of triangles to the kwargs for the Trimesh
|
|
constructor.
|
|
|
|
Parameters
|
|
---------
|
|
triangles : (n, 3, 3) float
|
|
Triangles in space
|
|
|
|
Returns
|
|
---------
|
|
kwargs : dict
|
|
Keyword arguments for the trimesh.Trimesh constructor
|
|
Includes keys 'vertices' and 'faces'
|
|
|
|
Examples
|
|
---------
|
|
>>> mesh = trimesh.Trimesh(**trimesh.triangles.to_kwargs(triangles))
|
|
"""
|
|
triangles = np.asanyarray(triangles, dtype=np.float64)
|
|
if not util.is_shape(triangles, (-1, 3, 3)):
|
|
raise ValueError('Triangles must be (n, 3, 3)!')
|
|
|
|
vertices = triangles.reshape((-1, 3))
|
|
faces = np.arange(len(vertices)).reshape((-1, 3))
|
|
kwargs = {'vertices': vertices,
|
|
'faces': faces}
|
|
|
|
return kwargs
|