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""" Extra math for implementing SVG on top of Kiva.
"""
import sys
from math import acos, sin, cos, hypot, ceil, sqrt, radians, degrees
import warnings
def bezier_arc(x1,y1, x2,y2, start_angle=0, extent=90):
""" Compute a cubic Bezier approximation of an elliptical arc.
(x1, y1) and (x2, y2) are the corners of the enclosing rectangle. The
coordinate system has coordinates that increase to the right and down.
Angles, measured in degress, start with 0 to the right (the positive X axis)
and increase counter-clockwise. The arc extends from start_angle to
start_angle+extent. I.e. start_angle=0 and extent=180 yields an openside-down
semi-circle.
The resulting coordinates are of the form (x1,y1, x2,y2, x3,y3, x4,y4)
such that the curve goes from (x1, y1) to (x4, y4) with (x2, y2) and
(x3, y3) as their respective Bezier control points.
"""
x1,y1, x2,y2 = min(x1,x2), max(y1,y2), max(x1,x2), min(y1,y2)
if abs(extent) <= 90:
frag_angle = float(extent)
nfrag = 1
else:
nfrag = int(ceil(abs(extent)/90.))
if nfrag == 0:
warnings.warn('Invalid value for extent: %r' % extent)
return []
frag_angle = float(extent) / nfrag
x_cen = (x1+x2)/2.
y_cen = (y1+y2)/2.
rx = (x2-x1)/2.
ry = (y2-y1)/2.
half_angle = radians(frag_angle) / 2
kappa = abs(4. / 3. * (1. - cos(half_angle)) / sin(half_angle))
if frag_angle < 0:
sign = -1
else:
sign = 1
point_list = []
for i in range(nfrag):
theta0 = radians(start_angle + i*frag_angle)
theta1 = radians(start_angle + (i+1)*frag_angle)
c0 = cos(theta0)
c1 = cos(theta1)
s0 = sin(theta0)
s1 = sin(theta1)
if frag_angle > 0:
signed_kappa = -kappa
else:
signed_kappa = kappa
point_list.append((x_cen + rx * c0,
y_cen - ry * s0,
x_cen + rx * (c0 + signed_kappa * s0),
y_cen - ry * (s0 - signed_kappa * c0),
x_cen + rx * (c1 - signed_kappa * s1),
y_cen - ry * (s1 + signed_kappa * c1),
x_cen + rx * c1,
y_cen - ry * s1))
return point_list
def angle(x1,y1, x2,y2):
""" The angle in degrees between two vectors.
"""
sign = 1.0
usign = (x1*y2 - y1*x2)
if usign < 0:
sign = -1.0
num = x1*x2 + y1*y2
den = hypot(x1,y1) * hypot(x2,y2)
ratio = min(max(num/den, -1.0), 1.0)
return sign * degrees(acos(ratio))
def transform_from_local(xp,yp,cphi,sphi,mx,my):
""" Transform from the local frame to absolute space.
"""
x = xp * cphi - yp * sphi + mx
y = xp * sphi + yp * cphi + my
return (x,y)
def elliptical_arc_to(path, rx, ry, phi, large_arc_flag, sweep_flag, x1, y1, x2, y2):
""" Add an elliptical arc to the kiva CompiledPath by approximating it with
Bezier curves or a line segment.
Algorithm taken from the SVG 1.1 Implementation Notes:
http://www.w3.org/TR/SVG/implnote.html#ArcImplementationNotes
"""
# Basic normalization.
rx = abs(rx)
ry = abs(ry)
phi = phi % 360
# Check for certain special cases.
if x1==x2 and y1==y2:
# Omit the arc.
# x1 and y1 can obviously remain the same for the next segment.
return []
if rx == 0 or ry == 0:
# Line segment.
path.line_to(x2,y2)
return []
rphi = radians(phi)
cphi = cos(rphi)
sphi = sin(rphi)
# Step 1: Rotate to the local coordinates.
dx = 0.5*(x1 - x2)
dy = 0.5*(y1 - y2)
x1p = cphi * dx + sphi * dy
y1p = -sphi * dx + cphi * dy
# Ensure that rx and ry are large enough to have a unique solution.
lam = (x1p/rx)**2 + (y1p/ry)**2
if lam > 1.0:
scale = sqrt(lam)
rx *= scale
ry *= scale
# Step 2: Solve for the center in the local coordinates.
num = max((rx*ry)**2 - (rx*y1p)**2 - (ry*x1p)**2, 0.0)
den = ((rx*y1p)**2 + (ry*x1p)**2)
a = sqrt(num / den)
cxp = a * rx*y1p/ry
cyp = -a * ry*x1p/rx
if large_arc_flag == sweep_flag:
cxp = -cxp
cyp = -cyp
# Step 3: Transform back.
mx = 0.5*(x1+x2)
my = 0.5*(y1+y2)
# Step 4: Compute the start angle and the angular extent of the arc.
# Note that theta1 is local to the phi-rotated coordinate space.
dx = (x1p-cxp) / rx
dy = (y1p-cyp) / ry
dx2 = (-x1p-cxp) / rx
dy2 = (-y1p-cyp) / ry
theta1 = angle(1,0,dx,dy)
dtheta = angle(dx,dy,dx2,dy2)
if not sweep_flag and dtheta > 0:
dtheta -= 360
elif sweep_flag and dtheta < 0:
dtheta += 360
# Step 5: Break it apart into Bezier arcs.
arcs = []
control_points = bezier_arc(cxp-rx,cyp-ry,cxp+rx,cyp+ry, theta1, dtheta)
for x1p,y1p, x2p,y2p, x3p,y3p, x4p,y4p in control_points:
# Transform them back to asbolute space.
args = (
transform_from_local(x2p,y2p,cphi,sphi,mx,my) +
transform_from_local(x3p,y3p,cphi,sphi,mx,my) +
transform_from_local(x4p,y4p,cphi,sphi,mx,my)
)
arcs.append(args)
return arcs
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