220 lines
6.2 KiB
Python
220 lines
6.2 KiB
Python
"""
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Various transforms used for by the 3D code
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"""
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import numpy as np
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from matplotlib import _api
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def world_transformation(xmin, xmax,
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ymin, ymax,
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zmin, zmax, pb_aspect=None):
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"""
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Produce a matrix that scales homogeneous coords in the specified ranges
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to [0, 1], or [0, pb_aspect[i]] if the plotbox aspect ratio is specified.
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"""
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dx = xmax - xmin
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dy = ymax - ymin
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dz = zmax - zmin
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if pb_aspect is not None:
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ax, ay, az = pb_aspect
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dx /= ax
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dy /= ay
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dz /= az
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return np.array([[1/dx, 0, 0, -xmin/dx],
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[ 0, 1/dy, 0, -ymin/dy],
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[ 0, 0, 1/dz, -zmin/dz],
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[ 0, 0, 0, 1]])
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def _rotation_about_vector(v, angle):
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"""
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Produce a rotation matrix for an angle in radians about a vector.
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"""
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vx, vy, vz = v / np.linalg.norm(v)
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s = np.sin(angle)
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c = np.cos(angle)
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t = 2*np.sin(angle/2)**2 # more numerically stable than t = 1-c
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R = np.array([
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[t*vx*vx + c, t*vx*vy - vz*s, t*vx*vz + vy*s],
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[t*vy*vx + vz*s, t*vy*vy + c, t*vy*vz - vx*s],
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[t*vz*vx - vy*s, t*vz*vy + vx*s, t*vz*vz + c]])
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return R
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def _view_axes(E, R, V, roll):
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"""
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Get the unit viewing axes in data coordinates.
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Parameters
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----------
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E : 3-element numpy array
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The coordinates of the eye/camera.
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R : 3-element numpy array
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The coordinates of the center of the view box.
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V : 3-element numpy array
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Unit vector in the direction of the vertical axis.
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roll : float
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The roll angle in radians.
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Returns
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-------
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u : 3-element numpy array
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Unit vector pointing towards the right of the screen.
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v : 3-element numpy array
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Unit vector pointing towards the top of the screen.
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w : 3-element numpy array
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Unit vector pointing out of the screen.
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"""
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w = (E - R)
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w = w/np.linalg.norm(w)
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u = np.cross(V, w)
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u = u/np.linalg.norm(u)
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v = np.cross(w, u) # Will be a unit vector
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# Save some computation for the default roll=0
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if roll != 0:
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# A positive rotation of the camera is a negative rotation of the world
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Rroll = _rotation_about_vector(w, -roll)
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u = np.dot(Rroll, u)
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v = np.dot(Rroll, v)
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return u, v, w
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def _view_transformation_uvw(u, v, w, E):
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"""
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Return the view transformation matrix.
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Parameters
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----------
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u : 3-element numpy array
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Unit vector pointing towards the right of the screen.
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v : 3-element numpy array
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Unit vector pointing towards the top of the screen.
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w : 3-element numpy array
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Unit vector pointing out of the screen.
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E : 3-element numpy array
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The coordinates of the eye/camera.
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"""
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Mr = np.eye(4)
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Mt = np.eye(4)
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Mr[:3, :3] = [u, v, w]
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Mt[:3, -1] = -E
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M = np.dot(Mr, Mt)
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return M
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def _persp_transformation(zfront, zback, focal_length):
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e = focal_length
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a = 1 # aspect ratio
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b = (zfront+zback)/(zfront-zback)
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c = -2*(zfront*zback)/(zfront-zback)
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proj_matrix = np.array([[e, 0, 0, 0],
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[0, e/a, 0, 0],
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[0, 0, b, c],
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[0, 0, -1, 0]])
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return proj_matrix
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def _ortho_transformation(zfront, zback):
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# note: w component in the resulting vector will be (zback-zfront), not 1
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a = -(zfront + zback)
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b = -(zfront - zback)
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proj_matrix = np.array([[2, 0, 0, 0],
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[0, 2, 0, 0],
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[0, 0, -2, 0],
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[0, 0, a, b]])
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return proj_matrix
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def _proj_transform_vec(vec, M):
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vecw = np.dot(M, vec.data)
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w = vecw[3]
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txs, tys, tzs = vecw[0]/w, vecw[1]/w, vecw[2]/w
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if np.ma.isMA(vec[0]): # we check each to protect for scalars
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txs = np.ma.array(txs, mask=vec[0].mask)
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if np.ma.isMA(vec[1]):
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tys = np.ma.array(tys, mask=vec[1].mask)
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if np.ma.isMA(vec[2]):
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tzs = np.ma.array(tzs, mask=vec[2].mask)
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return txs, tys, tzs
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def _proj_transform_vec_clip(vec, M, focal_length):
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vecw = np.dot(M, vec.data)
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w = vecw[3]
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txs, tys, tzs = vecw[0] / w, vecw[1] / w, vecw[2] / w
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if np.isinf(focal_length): # don't clip orthographic projection
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tis = np.ones(txs.shape, dtype=bool)
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else:
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tis = (-1 <= txs) & (txs <= 1) & (-1 <= tys) & (tys <= 1) & (tzs <= 0)
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if np.ma.isMA(vec[0]):
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tis = tis & ~vec[0].mask
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if np.ma.isMA(vec[1]):
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tis = tis & ~vec[1].mask
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if np.ma.isMA(vec[2]):
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tis = tis & ~vec[2].mask
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txs = np.ma.masked_array(txs, ~tis)
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tys = np.ma.masked_array(tys, ~tis)
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tzs = np.ma.masked_array(tzs, ~tis)
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return txs, tys, tzs, tis
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def inv_transform(xs, ys, zs, invM):
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"""
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Transform the points by the inverse of the projection matrix, *invM*.
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"""
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vec = _vec_pad_ones(xs, ys, zs)
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vecr = np.dot(invM, vec)
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if vecr.shape == (4,):
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vecr = vecr.reshape((4, 1))
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for i in range(vecr.shape[1]):
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if vecr[3][i] != 0:
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vecr[:, i] = vecr[:, i] / vecr[3][i]
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return vecr[0], vecr[1], vecr[2]
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def _vec_pad_ones(xs, ys, zs):
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if np.ma.isMA(xs) or np.ma.isMA(ys) or np.ma.isMA(zs):
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return np.ma.array([xs, ys, zs, np.ones_like(xs)])
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else:
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return np.array([xs, ys, zs, np.ones_like(xs)])
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def proj_transform(xs, ys, zs, M):
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"""
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Transform the points by the projection matrix *M*.
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"""
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vec = _vec_pad_ones(xs, ys, zs)
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return _proj_transform_vec(vec, M)
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@_api.deprecated("3.10")
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def proj_transform_clip(xs, ys, zs, M):
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return _proj_transform_clip(xs, ys, zs, M, focal_length=np.inf)
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def _proj_transform_clip(xs, ys, zs, M, focal_length):
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"""
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Transform the points by the projection matrix
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and return the clipping result
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returns txs, tys, tzs, tis
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"""
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vec = _vec_pad_ones(xs, ys, zs)
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return _proj_transform_vec_clip(vec, M, focal_length)
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def _proj_points(points, M):
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return np.column_stack(_proj_trans_points(points, M))
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def _proj_trans_points(points, M):
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points = np.asanyarray(points)
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xs, ys, zs = points[:, 0], points[:, 1], points[:, 2]
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return proj_transform(xs, ys, zs, M)
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