def get_arc_center(self):
"""
Looks at the normals to the first two
anchors, and finds their intersection points
"""
# First two anchors and handles
a1, h, a2 = self.get_points()[:3]
# Tangent vectors
t1 = h - a1
t2 = h - a2
# Normals
n1 = rotate_vector(t1, TAU / 4)
n2 = rotate_vector(t2, TAU / 4)
return find_intersection(a1, n1, a2, n2)
def get_quadratic_approximation_of_cubic(a0: npt.ArrayLike, h0: npt.ArrayLike,
h1: npt.ArrayLike,
a1: npt.ArrayLike) -> np.ndarray:
a0 = np.array(a0, ndmin=2)
h0 = np.array(h0, ndmin=2)
h1 = np.array(h1, ndmin=2)
a1 = np.array(a1, ndmin=2)
# Tangent vectors at the start and end.
T0 = h0 - a0
T1 = a1 - h1
# Search for inflection points. If none are found, use the
# midpoint as a cut point.
# Based on http://www.caffeineowl.com/graphics/2d/vectorial/cubic-inflexion.html
has_infl = np.ones(len(a0), dtype=bool)
p = h0 - a0
q = h1 - 2 * h0 + a0
r = a1 - 3 * h1 + 3 * h0 - a0
a = cross2d(q, r)
b = cross2d(p, r)
c = cross2d(p, q)
disc = b * b - 4 * a * c
has_infl &= (disc > 0)
sqrt_disc = np.sqrt(np.abs(disc))
settings = np.seterr(all='ignore')
ti_bounds = []
for sgn in [-1, +1]:
ti = (-b + sgn * sqrt_disc) / (2 * a)
ti[a == 0] = (-c / b)[a == 0]
ti[(a == 0) & (b == 0)] = 0
ti_bounds.append(ti)
ti_min, ti_max = ti_bounds
np.seterr(**settings)
ti_min_in_range = has_infl & (0 < ti_min) & (ti_min < 1)
ti_max_in_range = has_infl & (0 < ti_max) & (ti_max < 1)
# Choose a value of t which starts at 0.5,
# but is updated to one of the inflection points
# if they lie between 0 and 1
t_mid = 0.5 * np.ones(len(a0))
t_mid[ti_min_in_range] = ti_min[ti_min_in_range]
t_mid[ti_max_in_range] = ti_max[ti_max_in_range]
m, n = a0.shape
t_mid = t_mid.repeat(n).reshape((m, n))
# Compute bezier point and tangent at the chosen value of t
mid = bezier([a0, h0, h1, a1])(t_mid)
Tm = bezier([h0 - a0, h1 - h0, a1 - h1])(t_mid)
# Intersection between tangent lines at end points
# and tangent in the middle
i0 = find_intersection(a0, T0, mid, Tm)
i1 = find_intersection(a1, T1, mid, Tm)
m, n = np.shape(a0)
result = np.zeros((6 * m, n))
result[0::6] = a0
result[1::6] = i0
result[2::6] = mid
result[3::6] = mid
result[4::6] = i1
result[5::6] = a1
return result
def get_quadratic_approximation_of_cubic(a0, h0, h1, a1):
a0 = np.array(a0, ndmin=2)
h0 = np.array(h0, ndmin=2)
h1 = np.array(h1, ndmin=2)
a1 = np.array(a1, ndmin=2)
# Tangent vectors at the start and end.
T0 = h0 - a0
T1 = a1 - h1
# Search for inflection points. If none are found, use the
# midpoint as a cut point.
# Based on http://www.caffeineowl.com/graphics/2d/vectorial/cubic-inflexion.html
has_infl = np.ones(len(a0), dtype=bool)
p = h0 - a0
q = h1 - 2 * h0 + a0
r = a1 - 3 * h1 + 3 * h0 - a0
def cross2d(v, w):
return v[:, 0] * w[:, 1] - v[:, 1] * w[:, 0]
a = cross2d(q, r)
b = cross2d(p, r)
c = cross2d(p, q)
disc = b * b - 4 * a * c
has_infl &= (disc > 0)
sqrt_disc = np.sqrt(abs(disc))
# print(a, b, c, sqrt_disc)
settings = np.seterr(all='ignore')
ti_min, ti_max = [
np.true_divide(
-b + sgn * sqrt_disc,
2 * a,
out=(-c / b),
where=(a != 0),
) for sgn in [-1, +1]
]
np.seterr(**settings)
ti_min_in_range = has_infl & (0 < ti_min) & (ti_min < 1)
ti_max_in_range = has_infl & (0 < ti_max) & (ti_max < 1)
# Choose a value of t which is starts as 0.5,
# but is updated to one of the inflection points
# if they lie between 0 and 1
t_mid = 0.5 * np.ones(len(a0))
t_mid[ti_min_in_range] = ti_min[ti_min_in_range]
t_mid[ti_max_in_range] = ti_max[ti_max_in_range]
m, n = a0.shape
t_mid = t_mid.repeat(n).reshape((m, n))
# Compute bezier point and tangent at the chosen value of t
mid = bezier([a0, h0, h1, a1])(t_mid)
Tm = bezier([h0 - a0, h1 - h0, a1 - h1])(t_mid)
# Intersection between tangent lines at end points
# and tangent in the middle
i0 = find_intersection(a0, T0, mid, Tm)
i1 = find_intersection(a1, T1, mid, Tm)
m, n = np.shape(a0)
result = np.zeros((6 * m, n))
result[0::6] = a0
result[1::6] = i0
result[2::6] = mid
result[3::6] = mid
result[4::6] = i1
result[5::6] = a1
return result
