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"""Affine transforms, both in general and specific, named transforms."""
from math import cos, pi, sin, tan
import numpy as np
import shapely
__all__ = ["affine_transform", "rotate", "scale", "skew", "translate"]
def affine_transform(geom, matrix):
r"""Return a transformed geometry using an affine transformation matrix.
The coefficient matrix is provided as a list or tuple with 6 or 12 items
for 2D or 3D transformations, respectively.
For 2D affine transformations, the 6 parameter matrix is::
[a, b, d, e, xoff, yoff]
which represents the augmented matrix::
[x'] / a b xoff \ [x]
[y'] = | d e yoff | [y]
[1 ] \ 0 0 1 / [1]
or the equations for the transformed coordinates::
x' = a * x + b * y + xoff
y' = d * x + e * y + yoff
For 3D affine transformations, the 12 parameter matrix is::
[a, b, c, d, e, f, g, h, i, xoff, yoff, zoff]
which represents the augmented matrix::
[x'] / a b c xoff \ [x]
[y'] = | d e f yoff | [y]
[z'] | g h i zoff | [z]
[1 ] \ 0 0 0 1 / [1]
or the equations for the transformed coordinates::
x' = a * x + b * y + c * z + xoff
y' = d * x + e * y + f * z + yoff
z' = g * x + h * y + i * z + zoff
"""
if len(matrix) == 6:
ndim = 2
a, b, d, e, xoff, yoff = matrix
if geom.has_z:
ndim = 3
i = 1.0
c = f = g = h = zoff = 0.0
elif len(matrix) == 12:
ndim = 3
a, b, c, d, e, f, g, h, i, xoff, yoff, zoff = matrix
if not geom.has_z:
ndim = 2
else:
raise ValueError("'matrix' expects either 6 or 12 coefficients")
# if ndim == 2:
# A = np.array([[a, b], [d, e]], dtype=float)
# off = np.array([xoff, yoff], dtype=float)
# else:
# A = np.array([[a, b, c], [d, e, f], [g, h, i]], dtype=float)
# off = np.array([xoff, yoff, zoff], dtype=float)
def _affine_coords(coords):
# These are equivalent, but unfortunately not robust
# result = np.matmul(coords, A.T) + off
# result = np.matmul(A, coords.T).T + off
# Therefore, manual matrix multiplication is needed
if ndim == 2:
x, y = coords.T
xp = a * x + b * y + xoff
yp = d * x + e * y + yoff
result = np.stack([xp, yp]).T
elif ndim == 3:
x, y, z = coords.T
xp = a * x + b * y + c * z + xoff
yp = d * x + e * y + f * z + yoff
zp = g * x + h * y + i * z + zoff
result = np.stack([xp, yp, zp]).T
return result
return shapely.transform(geom, _affine_coords, include_z=ndim == 3)
def interpret_origin(geom, origin, ndim):
"""Return interpreted coordinate tuple for origin parameter.
This is a helper function for other transform functions.
The point of origin can be a keyword 'center' for the 2D bounding box
center, 'centroid' for the geometry's 2D centroid, a Point object or a
coordinate tuple (x0, y0, z0).
"""
# get coordinate tuple from 'origin' from keyword or Point type
if origin == "center":
# bounding box center
minx, miny, maxx, maxy = geom.bounds
origin = ((maxx + minx) / 2.0, (maxy + miny) / 2.0)
elif origin == "centroid":
origin = geom.centroid.coords[0]
elif isinstance(origin, str):
raise ValueError(f"'origin' keyword {origin!r} is not recognized")
elif getattr(origin, "geom_type", None) == "Point":
origin = origin.coords[0]
# origin should now be tuple-like
if len(origin) not in (2, 3):
raise ValueError("Expected number of items in 'origin' to be either 2 or 3")
if ndim == 2:
return origin[0:2]
else: # 3D coordinate
if len(origin) == 2:
return origin + (0.0,)
else:
return origin
def rotate(geom, angle, origin="center", use_radians=False):
r"""Return a rotated geometry on a 2D plane.
The angle of rotation can be specified in either degrees (default) or
radians by setting ``use_radians=True``. Positive angles are
counter-clockwise and negative are clockwise rotations.
The point of origin can be a keyword 'center' for the bounding box
center (default), 'centroid' for the geometry's centroid, a Point object
or a coordinate tuple (x0, y0).
The affine transformation matrix for 2D rotation is:
/ cos(r) -sin(r) xoff \
| sin(r) cos(r) yoff |
\ 0 0 1 /
where the offsets are calculated from the origin Point(x0, y0):
xoff = x0 - x0 * cos(r) + y0 * sin(r)
yoff = y0 - x0 * sin(r) - y0 * cos(r)
"""
if geom.is_empty:
return geom
if not use_radians: # convert from degrees
angle = angle * pi / 180.0
cosp = cos(angle)
sinp = sin(angle)
if abs(cosp) < 2.5e-16:
cosp = 0.0
if abs(sinp) < 2.5e-16:
sinp = 0.0
x0, y0 = interpret_origin(geom, origin, 2)
# fmt: off
matrix = (cosp, -sinp, 0.0,
sinp, cosp, 0.0,
0.0, 0.0, 1.0,
x0 - x0 * cosp + y0 * sinp, y0 - x0 * sinp - y0 * cosp, 0.0)
# fmt: on
return affine_transform(geom, matrix)
def scale(geom, xfact=1.0, yfact=1.0, zfact=1.0, origin="center"):
r"""Return a scaled geometry, scaled by factors along each dimension.
The point of origin can be a keyword 'center' for the 2D bounding box
center (default), 'centroid' for the geometry's 2D centroid, a Point
object or a coordinate tuple (x0, y0, z0).
Negative scale factors will mirror or reflect coordinates.
The general 3D affine transformation matrix for scaling is:
/ xfact 0 0 xoff \
| 0 yfact 0 yoff |
| 0 0 zfact zoff |
\ 0 0 0 1 /
where the offsets are calculated from the origin Point(x0, y0, z0):
xoff = x0 - x0 * xfact
yoff = y0 - y0 * yfact
zoff = z0 - z0 * zfact
"""
if geom.is_empty:
return geom
x0, y0, z0 = interpret_origin(geom, origin, 3)
# fmt: off
matrix = (xfact, 0.0, 0.0,
0.0, yfact, 0.0,
0.0, 0.0, zfact,
x0 - x0 * xfact, y0 - y0 * yfact, z0 - z0 * zfact)
# fmt: on
return affine_transform(geom, matrix)
def skew(geom, xs=0.0, ys=0.0, origin="center", use_radians=False):
r"""Return a skewed geometry, sheared by angles along x and y dimensions.
The shear angle can be specified in either degrees (default) or radians
by setting ``use_radians=True``.
The point of origin can be a keyword 'center' for the bounding box
center (default), 'centroid' for the geometry's centroid, a Point object
or a coordinate tuple (x0, y0).
The general 2D affine transformation matrix for skewing is:
/ 1 tan(xs) xoff \
| tan(ys) 1 yoff |
\ 0 0 1 /
where the offsets are calculated from the origin Point(x0, y0):
xoff = -y0 * tan(xs)
yoff = -x0 * tan(ys)
"""
if geom.is_empty:
return geom
if not use_radians: # convert from degrees
xs = xs * pi / 180.0
ys = ys * pi / 180.0
tanx = tan(xs)
tany = tan(ys)
if abs(tanx) < 2.5e-16:
tanx = 0.0
if abs(tany) < 2.5e-16:
tany = 0.0
x0, y0 = interpret_origin(geom, origin, 2)
# fmt: off
matrix = (1.0, tanx, 0.0,
tany, 1.0, 0.0,
0.0, 0.0, 1.0,
-y0 * tanx, -x0 * tany, 0.0)
# fmt: on
return affine_transform(geom, matrix)
def translate(geom, xoff=0.0, yoff=0.0, zoff=0.0):
r"""Return a translated geometry shifted by offsets along each dimension.
The general 3D affine transformation matrix for translation is:
/ 1 0 0 xoff \
| 0 1 0 yoff |
| 0 0 1 zoff |
\ 0 0 0 1 /
"""
if geom.is_empty:
return geom
# fmt: off
matrix = (1.0, 0.0, 0.0,
0.0, 1.0, 0.0,
0.0, 0.0, 1.0,
xoff, yoff, zoff)
# fmt: on
return affine_transform(geom, matrix)
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