def __init__(self, vmob, alpha, **kwargs):
digest_config(self, kwargs)
da = self.d_alpha
a1 = clip(alpha - da, 0, 1)
a2 = clip(alpha + da, 0, 1)
super().__init__(vmob.pfp(a1), vmob.pfp(a2), **kwargs)
self.scale(self.length / self.get_length())
def get_arc_center_and_angles(x0, y0, rx, ry, phi, large_arc_flag,
sweep_flag, x1, y1):
"""
The parameter functions of an ellipse rotated `phi` radians counterclockwise is (on `alpha`):
x = cx + rx * cos(alpha) * cos(phi) + ry * sin(alpha) * sin(phi),
y = cy + rx * cos(alpha) * sin(phi) - ry * sin(alpha) * cos(phi).
Now we have two points sitting on the ellipse: `(x0, y0)`, `(x1, y1)`, corresponding to 4 equations,
and we want to hunt for 4 variables: `cx`, `cy`, `alpha0` and `alpha_1`.
Let `d_alpha = alpha1 - alpha0`, then:
if `sweep_flag = 0` and `large_arc_flag = 1`, then `PI <= d_alpha < 2 * PI`;
if `sweep_flag = 0` and `large_arc_flag = 0`, then `0 < d_alpha <= PI`;
if `sweep_flag = 1` and `large_arc_flag = 0`, then `-PI <= d_alpha < 0`;
if `sweep_flag = 1` and `large_arc_flag = 1`, then `-2 * PI < d_alpha <= -PI`.
"""
xd = x1 - x0
yd = y1 - y0
if close_to_zero(xd) and close_to_zero(yd):
raise Exception(
"Cannot find arc center since the start point and the end point meet."
)
# Find `p = cos(alpha1) - cos(alpha0)`, `q = sin(alpha1) - sin(alpha0)`
eq0 = [rx * np.cos(phi), ry * np.sin(phi), xd]
eq1 = [rx * np.sin(phi), -ry * np.cos(phi), yd]
p, q = solve_2d_linear_equation(*zip(eq0, eq1))
# Find `s = (alpha1 - alpha0) / 2`, `t = (alpha1 + alpha0) / 2`
# If `sin(s) = 0`, this requires `p = q = 0`,
# implying `xd = yd = 0`, which is impossible.
sin_s = (p**2 + q**2)**0.5 / 2
if sweep_flag:
sin_s = -sin_s
sin_s = clip(sin_s, -1, 1)
s = np.arcsin(sin_s)
if large_arc_flag:
if not sweep_flag:
s = PI - s
else:
s = -PI - s
sin_t = -p / (2 * sin_s)
cos_t = q / (2 * sin_s)
cos_t = clip(cos_t, -1, 1)
t = np.arccos(cos_t)
if sin_t <= 0:
t = -t
# We can make sure `0 < abs(s) < PI`, `-PI <= t < PI`.
alpha0 = t - s
alpha_1 = t + s
cx = x0 - rx * np.cos(alpha0) * np.cos(phi) - ry * np.sin(
alpha0) * np.sin(phi)
cy = y0 - rx * np.cos(alpha0) * np.sin(phi) + ry * np.sin(
alpha0) * np.cos(phi)
return cx, cy, alpha0, alpha_1
def smooth(t, inflection=10.0):
error = sigmoid(-inflection / 2)
return clip(
(sigmoid(inflection * (t - 0.5)) - error) / (1 - 2 * error),
0,
1,
)
def get_sub_alpha(self, alpha, index, num_submobjects):
# TODO, make this more understanable, and/or combine
# its functionality with AnimationGroup's method
# build_animations_with_timings
lag_ratio = self.lag_ratio
full_length = (num_submobjects - 1) * lag_ratio + 1
value = alpha * full_length
lower = index * lag_ratio
return clip((value - lower), 0, 1)
def angle_between_vectors(v1: np.ndarray, v2: np.ndarray) -> float:
"""
Returns the angle between two 3D vectors.
This angle will always be btw 0 and pi
"""
n1 = get_norm(v1)
n2 = get_norm(v2)
if n1 == 0 or n2 == 0:
return 0
cos_angle = np.dot(v1, v2) / np.float64(n1 * n2)
return math.acos(clip(cos_angle, -1, 1))
def update(m, dt):
run_time = animation.get_run_time()
time_ratio = animation.total_time / run_time
if cycle:
alpha = time_ratio % 1
else:
alpha = clip(time_ratio, 0, 1)
if alpha >= 1:
animation.finish()
m.remove_updater(update)
return
animation.interpolate(alpha)
animation.update_mobjects(dt)
animation.total_time += dt
def interpolate(self, alpha: float) -> None:
# Note, if the run_time of AnimationGroup has been
# set to something other than its default, these
# times might not correspond to actual times,
# e.g. of the surrounding scene. Instead they'd
# be a rescaled version. But that's okay!
time = alpha * self.max_end_time
for anim, start_time, end_time in self.anims_with_timings:
anim_time = end_time - start_time
if anim_time == 0:
sub_alpha = 0
else:
sub_alpha = clip((time - start_time) / anim_time, 0, 1)
anim.interpolate(sub_alpha)
def rotate(self, angle: float, axis: np.ndarray = OUT, **kwargs):
curr_rot_T = self.get_inverse_camera_rotation_matrix()
added_rot_T = rotation_matrix_transpose(angle, axis)
new_rot_T = np.dot(curr_rot_T, added_rot_T)
Fz = new_rot_T[2]
phi = np.arccos(clip(Fz[2], -1, 1))
theta = angle_of_vector(Fz[:2]) + PI / 2
partial_rot_T = np.dot(
rotation_matrix_transpose(phi, RIGHT),
rotation_matrix_transpose(theta, OUT),
)
gamma = angle_of_vector(np.dot(partial_rot_T, new_rot_T.T)[:, 0])
self.set_euler_angles(theta, phi, gamma)
return self
def interpolate(self, alpha: float) -> None:
alpha = clip(alpha, 0, 1)
self.interpolate_mobject(self.rate_func(alpha))
def increment_phi(self, dphi):
phi = self.data["euler_angles"][1]
new_phi = clip(phi + dphi, 0, PI)
self.data["euler_angles"][1] = new_phi
self.refresh_rotation_matrix()
return self
def interpolate(self, alpha):
alpha = clip(alpha, 0, 1)
self.interpolate_mobject(self.rate_func(alpha))
def angle_between_vectors(v1, v2):
"""
Returns the angle between two 3D vectors.
This angle will always be btw 0 and pi
"""
return math.acos(clip(np.dot(normalize(v1), normalize(v2)), -1, 1))