forked from s_ranjbar/city_retrofit
543 lines
17 KiB
Python
543 lines
17 KiB
Python
"""
|
|
A module providing some utility functions regarding Bezier path manipulation.
|
|
"""
|
|
|
|
import numpy as np
|
|
|
|
import matplotlib.cbook as cbook
|
|
from matplotlib.path import Path
|
|
|
|
|
|
class NonIntersectingPathException(ValueError):
|
|
pass
|
|
|
|
# some functions
|
|
|
|
|
|
def get_intersection(cx1, cy1, cos_t1, sin_t1,
|
|
cx2, cy2, cos_t2, sin_t2):
|
|
"""
|
|
Return the intersection between the line through (*cx1*, *cy1*) at angle
|
|
*t1* and the line through (*cx2*, *cy2*) at angle *t2*.
|
|
"""
|
|
|
|
# line1 => sin_t1 * (x - cx1) - cos_t1 * (y - cy1) = 0.
|
|
# line1 => sin_t1 * x + cos_t1 * y = sin_t1*cx1 - cos_t1*cy1
|
|
|
|
line1_rhs = sin_t1 * cx1 - cos_t1 * cy1
|
|
line2_rhs = sin_t2 * cx2 - cos_t2 * cy2
|
|
|
|
# rhs matrix
|
|
a, b = sin_t1, -cos_t1
|
|
c, d = sin_t2, -cos_t2
|
|
|
|
ad_bc = a * d - b * c
|
|
if np.abs(ad_bc) < 1.0e-12:
|
|
raise ValueError("Given lines do not intersect. Please verify that "
|
|
"the angles are not equal or differ by 180 degrees.")
|
|
|
|
# rhs_inverse
|
|
a_, b_ = d, -b
|
|
c_, d_ = -c, a
|
|
a_, b_, c_, d_ = [k / ad_bc for k in [a_, b_, c_, d_]]
|
|
|
|
x = a_ * line1_rhs + b_ * line2_rhs
|
|
y = c_ * line1_rhs + d_ * line2_rhs
|
|
|
|
return x, y
|
|
|
|
|
|
def get_normal_points(cx, cy, cos_t, sin_t, length):
|
|
"""
|
|
For a line passing through (*cx*, *cy*) and having an angle *t*, return
|
|
locations of the two points located along its perpendicular line at the
|
|
distance of *length*.
|
|
"""
|
|
|
|
if length == 0.:
|
|
return cx, cy, cx, cy
|
|
|
|
cos_t1, sin_t1 = sin_t, -cos_t
|
|
cos_t2, sin_t2 = -sin_t, cos_t
|
|
|
|
x1, y1 = length * cos_t1 + cx, length * sin_t1 + cy
|
|
x2, y2 = length * cos_t2 + cx, length * sin_t2 + cy
|
|
|
|
return x1, y1, x2, y2
|
|
|
|
|
|
# BEZIER routines
|
|
|
|
# subdividing bezier curve
|
|
# http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/Bezier/bezier-sub.html
|
|
|
|
|
|
def _de_casteljau1(beta, t):
|
|
next_beta = beta[:-1] * (1 - t) + beta[1:] * t
|
|
return next_beta
|
|
|
|
|
|
def split_de_casteljau(beta, t):
|
|
"""
|
|
Split a Bezier segment defined by its control points *beta* into two
|
|
separate segments divided at *t* and return their control points.
|
|
"""
|
|
beta = np.asarray(beta)
|
|
beta_list = [beta]
|
|
while True:
|
|
beta = _de_casteljau1(beta, t)
|
|
beta_list.append(beta)
|
|
if len(beta) == 1:
|
|
break
|
|
left_beta = [beta[0] for beta in beta_list]
|
|
right_beta = [beta[-1] for beta in reversed(beta_list)]
|
|
|
|
return left_beta, right_beta
|
|
|
|
|
|
@cbook._rename_parameter("3.1", "tolerence", "tolerance")
|
|
def find_bezier_t_intersecting_with_closedpath(
|
|
bezier_point_at_t, inside_closedpath, t0=0., t1=1., tolerance=0.01):
|
|
"""
|
|
Find the intersection of the Bezier curve with a closed path.
|
|
|
|
The intersection point *t* is approximated by two parameters *t0*, *t1*
|
|
such that *t0* <= *t* <= *t1*.
|
|
|
|
Search starts from *t0* and *t1* and uses a simple bisecting algorithm
|
|
therefore one of the end points must be inside the path while the other
|
|
doesn't. The search stops when the distance of the points parametrized by
|
|
*t0* and *t1* gets smaller than the given *tolerance*.
|
|
|
|
Parameters
|
|
----------
|
|
bezier_point_at_t : callable
|
|
A function returning x, y coordinates of the Bezier at parameter *t*.
|
|
It must have the signature::
|
|
|
|
bezier_point_at_t(t: float) -> Tuple[float, float]
|
|
|
|
inside_closedpath : callable
|
|
A function returning True if a given point (x, y) is inside the
|
|
closed path. It must have the signature::
|
|
|
|
inside_closedpath(point: Tuple[float, float]) -> bool
|
|
|
|
t0, t1 : float
|
|
Start parameters for the search.
|
|
|
|
tolerance : float
|
|
Maximal allowed distance between the final points.
|
|
|
|
Returns
|
|
-------
|
|
t0, t1 : float
|
|
The Bezier path parameters.
|
|
"""
|
|
start = bezier_point_at_t(t0)
|
|
end = bezier_point_at_t(t1)
|
|
|
|
start_inside = inside_closedpath(start)
|
|
end_inside = inside_closedpath(end)
|
|
|
|
if start_inside == end_inside and start != end:
|
|
raise NonIntersectingPathException(
|
|
"Both points are on the same side of the closed path")
|
|
|
|
while True:
|
|
|
|
# return if the distance is smaller than the tolerance
|
|
if np.hypot(start[0] - end[0], start[1] - end[1]) < tolerance:
|
|
return t0, t1
|
|
|
|
# calculate the middle point
|
|
middle_t = 0.5 * (t0 + t1)
|
|
middle = bezier_point_at_t(middle_t)
|
|
middle_inside = inside_closedpath(middle)
|
|
|
|
if start_inside ^ middle_inside:
|
|
t1 = middle_t
|
|
end = middle
|
|
end_inside = middle_inside
|
|
else:
|
|
t0 = middle_t
|
|
start = middle
|
|
start_inside = middle_inside
|
|
|
|
|
|
class BezierSegment:
|
|
"""
|
|
A 2-dimensional Bezier segment.
|
|
|
|
Parameters
|
|
----------
|
|
control_points : array-like (N, 2)
|
|
A list of the (x, y) positions of control points of the Bezier line.
|
|
This must contain N points, where N is the order of the Bezier line.
|
|
1 <= N <= 3 is supported.
|
|
"""
|
|
# Higher order Bezier lines can be supported by simplying adding
|
|
# corresponding values.
|
|
_binom_coeff = {1: np.array([1., 1.]),
|
|
2: np.array([1., 2., 1.]),
|
|
3: np.array([1., 3., 3., 1.])}
|
|
|
|
def __init__(self, control_points):
|
|
_o = len(control_points)
|
|
self._orders = np.arange(_o)
|
|
|
|
_coeff = BezierSegment._binom_coeff[_o - 1]
|
|
xx, yy = np.asarray(control_points).T
|
|
self._px = xx * _coeff
|
|
self._py = yy * _coeff
|
|
|
|
def point_at_t(self, t):
|
|
"""Return the point (x, y) at parameter *t*."""
|
|
tt = ((1 - t) ** self._orders)[::-1] * t ** self._orders
|
|
_x = np.dot(tt, self._px)
|
|
_y = np.dot(tt, self._py)
|
|
return _x, _y
|
|
|
|
|
|
@cbook._rename_parameter("3.1", "tolerence", "tolerance")
|
|
def split_bezier_intersecting_with_closedpath(
|
|
bezier, inside_closedpath, tolerance=0.01):
|
|
"""
|
|
Split a Bezier curve into two at the intersection with a closed path.
|
|
|
|
Parameters
|
|
----------
|
|
bezier : array-like(N, 2)
|
|
Control points of the Bezier segment. See `.BezierSegment`.
|
|
inside_closedpath : callable
|
|
A function returning True if a given point (x, y) is inside the
|
|
closed path. See also `.find_bezier_t_intersecting_with_closedpath`.
|
|
tolerance : float
|
|
The tolerance for the intersection. See also
|
|
`.find_bezier_t_intersecting_with_closedpath`.
|
|
|
|
Returns
|
|
-------
|
|
left, right
|
|
Lists of control points for the two Bezier segments.
|
|
"""
|
|
|
|
bz = BezierSegment(bezier)
|
|
bezier_point_at_t = bz.point_at_t
|
|
|
|
t0, t1 = find_bezier_t_intersecting_with_closedpath(
|
|
bezier_point_at_t, inside_closedpath, tolerance=tolerance)
|
|
|
|
_left, _right = split_de_casteljau(bezier, (t0 + t1) / 2.)
|
|
return _left, _right
|
|
|
|
|
|
@cbook.deprecated("3.1")
|
|
@cbook._rename_parameter("3.1", "tolerence", "tolerance")
|
|
def find_r_to_boundary_of_closedpath(
|
|
inside_closedpath, xy, cos_t, sin_t, rmin=0., rmax=1., tolerance=0.01):
|
|
"""
|
|
Find a radius r (centered at *xy*) between *rmin* and *rmax* at
|
|
which it intersect with the path.
|
|
|
|
Parameters
|
|
----------
|
|
inside_closedpath : callable
|
|
A function returning True if a given point (x, y) is inside the
|
|
closed path.
|
|
xy : float, float
|
|
The center of the radius.
|
|
cos_t, sin_t : float
|
|
Cosine and sine for the angle.
|
|
rmin, rmax : float
|
|
Starting parameters for the radius search.
|
|
"""
|
|
cx, cy = xy
|
|
|
|
def _f(r):
|
|
return cos_t * r + cx, sin_t * r + cy
|
|
|
|
find_bezier_t_intersecting_with_closedpath(
|
|
_f, inside_closedpath, t0=rmin, t1=rmax, tolerance=tolerance)
|
|
|
|
# matplotlib specific
|
|
|
|
|
|
@cbook._rename_parameter("3.1", "tolerence", "tolerance")
|
|
def split_path_inout(path, inside, tolerance=0.01, reorder_inout=False):
|
|
"""
|
|
Divide a path into two segments at the point where ``inside(x, y)`` becomes
|
|
False.
|
|
"""
|
|
path_iter = path.iter_segments()
|
|
|
|
ctl_points, command = next(path_iter)
|
|
begin_inside = inside(ctl_points[-2:]) # true if begin point is inside
|
|
|
|
ctl_points_old = ctl_points
|
|
|
|
concat = np.concatenate
|
|
|
|
iold = 0
|
|
i = 1
|
|
|
|
for ctl_points, command in path_iter:
|
|
iold = i
|
|
i += len(ctl_points) // 2
|
|
if inside(ctl_points[-2:]) != begin_inside:
|
|
bezier_path = concat([ctl_points_old[-2:], ctl_points])
|
|
break
|
|
ctl_points_old = ctl_points
|
|
else:
|
|
raise ValueError("The path does not intersect with the patch")
|
|
|
|
bp = bezier_path.reshape((-1, 2))
|
|
left, right = split_bezier_intersecting_with_closedpath(
|
|
bp, inside, tolerance)
|
|
if len(left) == 2:
|
|
codes_left = [Path.LINETO]
|
|
codes_right = [Path.MOVETO, Path.LINETO]
|
|
elif len(left) == 3:
|
|
codes_left = [Path.CURVE3, Path.CURVE3]
|
|
codes_right = [Path.MOVETO, Path.CURVE3, Path.CURVE3]
|
|
elif len(left) == 4:
|
|
codes_left = [Path.CURVE4, Path.CURVE4, Path.CURVE4]
|
|
codes_right = [Path.MOVETO, Path.CURVE4, Path.CURVE4, Path.CURVE4]
|
|
else:
|
|
raise AssertionError("This should never be reached")
|
|
|
|
verts_left = left[1:]
|
|
verts_right = right[:]
|
|
|
|
if path.codes is None:
|
|
path_in = Path(concat([path.vertices[:i], verts_left]))
|
|
path_out = Path(concat([verts_right, path.vertices[i:]]))
|
|
|
|
else:
|
|
path_in = Path(concat([path.vertices[:iold], verts_left]),
|
|
concat([path.codes[:iold], codes_left]))
|
|
|
|
path_out = Path(concat([verts_right, path.vertices[i:]]),
|
|
concat([codes_right, path.codes[i:]]))
|
|
|
|
if reorder_inout and not begin_inside:
|
|
path_in, path_out = path_out, path_in
|
|
|
|
return path_in, path_out
|
|
|
|
|
|
def inside_circle(cx, cy, r):
|
|
"""
|
|
Return a function that checks whether a point is in a circle with center
|
|
(*cx*, *cy*) and radius *r*.
|
|
|
|
The returned function has the signature::
|
|
|
|
f(xy: Tuple[float, float]) -> bool
|
|
"""
|
|
r2 = r ** 2
|
|
|
|
def _f(xy):
|
|
x, y = xy
|
|
return (x - cx) ** 2 + (y - cy) ** 2 < r2
|
|
return _f
|
|
|
|
|
|
# quadratic Bezier lines
|
|
|
|
def get_cos_sin(x0, y0, x1, y1):
|
|
dx, dy = x1 - x0, y1 - y0
|
|
d = (dx * dx + dy * dy) ** .5
|
|
# Account for divide by zero
|
|
if d == 0:
|
|
return 0.0, 0.0
|
|
return dx / d, dy / d
|
|
|
|
|
|
@cbook._rename_parameter("3.1", "tolerence", "tolerance")
|
|
def check_if_parallel(dx1, dy1, dx2, dy2, tolerance=1.e-5):
|
|
"""
|
|
Check if two lines are parallel.
|
|
|
|
Parameters
|
|
----------
|
|
dx1, dy1, dx2, dy2 : float
|
|
The gradients *dy*/*dx* of the two lines.
|
|
tolerance : float
|
|
The angular tolerance in radians up to which the lines are considered
|
|
parallel.
|
|
|
|
Returns
|
|
-------
|
|
is_parallel
|
|
- 1 if two lines are parallel in same direction.
|
|
- -1 if two lines are parallel in opposite direction.
|
|
- False otherwise.
|
|
"""
|
|
theta1 = np.arctan2(dx1, dy1)
|
|
theta2 = np.arctan2(dx2, dy2)
|
|
dtheta = np.abs(theta1 - theta2)
|
|
if dtheta < tolerance:
|
|
return 1
|
|
elif np.abs(dtheta - np.pi) < tolerance:
|
|
return -1
|
|
else:
|
|
return False
|
|
|
|
|
|
def get_parallels(bezier2, width):
|
|
"""
|
|
Given the quadratic Bezier control points *bezier2*, returns
|
|
control points of quadratic Bezier lines roughly parallel to given
|
|
one separated by *width*.
|
|
"""
|
|
|
|
# The parallel Bezier lines are constructed by following ways.
|
|
# c1 and c2 are control points representing the begin and end of the
|
|
# Bezier line.
|
|
# cm is the middle point
|
|
|
|
c1x, c1y = bezier2[0]
|
|
cmx, cmy = bezier2[1]
|
|
c2x, c2y = bezier2[2]
|
|
|
|
parallel_test = check_if_parallel(c1x - cmx, c1y - cmy,
|
|
cmx - c2x, cmy - c2y)
|
|
|
|
if parallel_test == -1:
|
|
cbook._warn_external(
|
|
"Lines do not intersect. A straight line is used instead.")
|
|
cos_t1, sin_t1 = get_cos_sin(c1x, c1y, c2x, c2y)
|
|
cos_t2, sin_t2 = cos_t1, sin_t1
|
|
else:
|
|
# t1 and t2 is the angle between c1 and cm, cm, c2. They are
|
|
# also a angle of the tangential line of the path at c1 and c2
|
|
cos_t1, sin_t1 = get_cos_sin(c1x, c1y, cmx, cmy)
|
|
cos_t2, sin_t2 = get_cos_sin(cmx, cmy, c2x, c2y)
|
|
|
|
# find c1_left, c1_right which are located along the lines
|
|
# through c1 and perpendicular to the tangential lines of the
|
|
# Bezier path at a distance of width. Same thing for c2_left and
|
|
# c2_right with respect to c2.
|
|
c1x_left, c1y_left, c1x_right, c1y_right = (
|
|
get_normal_points(c1x, c1y, cos_t1, sin_t1, width)
|
|
)
|
|
c2x_left, c2y_left, c2x_right, c2y_right = (
|
|
get_normal_points(c2x, c2y, cos_t2, sin_t2, width)
|
|
)
|
|
|
|
# find cm_left which is the intersecting point of a line through
|
|
# c1_left with angle t1 and a line through c2_left with angle
|
|
# t2. Same with cm_right.
|
|
try:
|
|
cmx_left, cmy_left = get_intersection(c1x_left, c1y_left, cos_t1,
|
|
sin_t1, c2x_left, c2y_left,
|
|
cos_t2, sin_t2)
|
|
cmx_right, cmy_right = get_intersection(c1x_right, c1y_right, cos_t1,
|
|
sin_t1, c2x_right, c2y_right,
|
|
cos_t2, sin_t2)
|
|
except ValueError:
|
|
# Special case straight lines, i.e., angle between two lines is
|
|
# less than the threshold used by get_intersection (we don't use
|
|
# check_if_parallel as the threshold is not the same).
|
|
cmx_left, cmy_left = (
|
|
0.5 * (c1x_left + c2x_left), 0.5 * (c1y_left + c2y_left)
|
|
)
|
|
cmx_right, cmy_right = (
|
|
0.5 * (c1x_right + c2x_right), 0.5 * (c1y_right + c2y_right)
|
|
)
|
|
|
|
# the parallel Bezier lines are created with control points of
|
|
# [c1_left, cm_left, c2_left] and [c1_right, cm_right, c2_right]
|
|
path_left = [(c1x_left, c1y_left),
|
|
(cmx_left, cmy_left),
|
|
(c2x_left, c2y_left)]
|
|
path_right = [(c1x_right, c1y_right),
|
|
(cmx_right, cmy_right),
|
|
(c2x_right, c2y_right)]
|
|
|
|
return path_left, path_right
|
|
|
|
|
|
def find_control_points(c1x, c1y, mmx, mmy, c2x, c2y):
|
|
"""
|
|
Find control points of the Bezier curve passing through (*c1x*, *c1y*),
|
|
(*mmx*, *mmy*), and (*c2x*, *c2y*), at parametric values 0, 0.5, and 1.
|
|
"""
|
|
cmx = .5 * (4 * mmx - (c1x + c2x))
|
|
cmy = .5 * (4 * mmy - (c1y + c2y))
|
|
return [(c1x, c1y), (cmx, cmy), (c2x, c2y)]
|
|
|
|
|
|
def make_wedged_bezier2(bezier2, width, w1=1., wm=0.5, w2=0.):
|
|
"""
|
|
Being similar to get_parallels, returns control points of two quadratic
|
|
Bezier lines having a width roughly parallel to given one separated by
|
|
*width*.
|
|
"""
|
|
|
|
# c1, cm, c2
|
|
c1x, c1y = bezier2[0]
|
|
cmx, cmy = bezier2[1]
|
|
c3x, c3y = bezier2[2]
|
|
|
|
# t1 and t2 is the angle between c1 and cm, cm, c3.
|
|
# They are also a angle of the tangential line of the path at c1 and c3
|
|
cos_t1, sin_t1 = get_cos_sin(c1x, c1y, cmx, cmy)
|
|
cos_t2, sin_t2 = get_cos_sin(cmx, cmy, c3x, c3y)
|
|
|
|
# find c1_left, c1_right which are located along the lines
|
|
# through c1 and perpendicular to the tangential lines of the
|
|
# Bezier path at a distance of width. Same thing for c3_left and
|
|
# c3_right with respect to c3.
|
|
c1x_left, c1y_left, c1x_right, c1y_right = (
|
|
get_normal_points(c1x, c1y, cos_t1, sin_t1, width * w1)
|
|
)
|
|
c3x_left, c3y_left, c3x_right, c3y_right = (
|
|
get_normal_points(c3x, c3y, cos_t2, sin_t2, width * w2)
|
|
)
|
|
|
|
# find c12, c23 and c123 which are middle points of c1-cm, cm-c3 and
|
|
# c12-c23
|
|
c12x, c12y = (c1x + cmx) * .5, (c1y + cmy) * .5
|
|
c23x, c23y = (cmx + c3x) * .5, (cmy + c3y) * .5
|
|
c123x, c123y = (c12x + c23x) * .5, (c12y + c23y) * .5
|
|
|
|
# tangential angle of c123 (angle between c12 and c23)
|
|
cos_t123, sin_t123 = get_cos_sin(c12x, c12y, c23x, c23y)
|
|
|
|
c123x_left, c123y_left, c123x_right, c123y_right = (
|
|
get_normal_points(c123x, c123y, cos_t123, sin_t123, width * wm)
|
|
)
|
|
|
|
path_left = find_control_points(c1x_left, c1y_left,
|
|
c123x_left, c123y_left,
|
|
c3x_left, c3y_left)
|
|
path_right = find_control_points(c1x_right, c1y_right,
|
|
c123x_right, c123y_right,
|
|
c3x_right, c3y_right)
|
|
|
|
return path_left, path_right
|
|
|
|
|
|
def make_path_regular(p):
|
|
"""
|
|
If the :attr:`codes` attribute of `Path` *p* is None, return a copy of *p*
|
|
with the :attr:`codes` set to (MOVETO, LINETO, LINETO, ..., LINETO);
|
|
otherwise return *p* itself.
|
|
"""
|
|
c = p.codes
|
|
if c is None:
|
|
c = np.full(len(p.vertices), Path.LINETO, dtype=Path.code_type)
|
|
c[0] = Path.MOVETO
|
|
return Path(p.vertices, c)
|
|
else:
|
|
return p
|
|
|
|
|
|
def concatenate_paths(paths):
|
|
"""Concatenate a list of paths into a single path."""
|
|
vertices = np.concatenate([p.vertices for p in paths])
|
|
codes = np.concatenate([make_path_regular(p).codes for p in paths])
|
|
return Path(vertices, codes)
|