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ffd_visualization.py
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from manimlib.imports import *
CON_POINT_RANGE = 7
# B spline basis function
def B_0(u):
assert 0 <= u <= 1
return (1. - u) ** 3. / 6.
def B_1(u):
assert 0 <= u <= 1
return (3 * u ** 3 - 6 * u ** 2 + 4) / 6
def B_2(u):
assert 0 <= u <= 1
return (-3 * u ** 3 + 3 * u ** 2 + 3 * u + 1) / 6
def B_3(u):
assert 0 <= u <= 1
return u ** 3 / 6
def naive_transformation(pos_3d, mesh, delta, B, K):
pos = pos_3d[0:2]
pos = (pos - B) / K
pos_reg = pos / delta
pos_floor = np.floor(pos_reg)
uv = pos_reg - pos_floor
ij = pos_floor - 1.
B_00 = B_0(uv[0]) * B_0(uv[1])
B_01 = B_0(uv[0]) * B_1(uv[1])
B_02 = B_0(uv[0]) * B_2(uv[1])
B_03 = B_0(uv[0]) * B_3(uv[1])
B_10 = B_1(uv[0]) * B_0(uv[1])
B_11 = B_1(uv[0]) * B_1(uv[1])
B_12 = B_1(uv[0]) * B_2(uv[1])
B_13 = B_1(uv[0]) * B_3(uv[1])
B_20 = B_2(uv[0]) * B_0(uv[1])
B_21 = B_2(uv[0]) * B_1(uv[1])
B_22 = B_2(uv[0]) * B_2(uv[1])
B_23 = B_2(uv[0]) * B_3(uv[1])
B_30 = B_3(uv[0]) * B_0(uv[1])
B_31 = B_3(uv[0]) * B_1(uv[1])
B_32 = B_3(uv[0]) * B_2(uv[1])
B_33 = B_3(uv[0]) * B_3(uv[1])
B_all = np.array(((B_00, B_01, B_02, B_03),
(B_10, B_11, B_12, B_13),
(B_20, B_21, B_22, B_23),
(B_30, B_31, B_32, B_33)))
mesh_part = mesh[:, int(ij[0] + 1):int(ij[0] + 1) + 4, int(ij[1] + 1):int(ij[1] + 1) + 4]
tmp = B_all * mesh_part
output = np.zeros(3)
output[0:2] = [(tmp[0, :, :].sum() * K) + B, (tmp[1, :, :].sum() * K) + B]
return output
class FFDSquare(Scene):
def __init__(self, mesh, mesh_trans, delta, **scene_kwargs):
self.mesh = mesh
self.mesh_trans = mesh_trans
self.delta = delta
self.mesh_size = self.mesh.shape[1] - 3
self.K = CON_POINT_RANGE / (self.mesh_size + 1) / delta
self.B = CON_POINT_RANGE * (1 / (self.mesh_size + 1) - 0.5)
self.GEOMETRY_SIZE = self.K * delta * (self.mesh_size - 1) - 0.0001
self.AXIS_MIN = -0.5 * self.GEOMETRY_SIZE
self.CONFIG = {
"x_min": self.AXIS_MIN,
"x_max": -self.AXIS_MIN,
"y_min": self.AXIS_MIN,
"y_max": -self.AXIS_MIN,
"background_line_style": {
"stroke_color": "#FFFFFF",
},
"x_line_frequency": 0.5 * self.K * delta,
"y_line_frequency": 0.5 * self.K * delta,
}
super().__init__(**scene_kwargs)
def construct(self):
control_points = VGroup(*[Dot(point=[self.mesh[0, i, j] * self.K + self.B,
self.mesh[1, i, j] * self.K + self.B,
0])
for i in range(self.mesh_size + 2) for j in range(self.mesh_size + 2)])
control_points_trans = VGroup(*[Dot(point=[self.mesh_trans[0, i, j] * self.K + self.B,
self.mesh_trans[1, i, j] * self.K + self.B,
0])
for i in range(self.mesh_size + 2) for j in range(self.mesh_size + 2)])
control_lines = VGroup(
*[Line(np.append(self.mesh[:, i, j] * self.K + self.B, 0),
np.append(self.mesh[:, i + 1, j] * self.K + self.B, 0))
for i in range(self.mesh_size + 1) for j in range(self.mesh_size + 2)],
*[Line(np.append(self.mesh[:, i, j] * self.K + self.B, 0),
np.append(self.mesh[:, i, j + 1] * self.K + self.B, 0))
for i in range(self.mesh_size + 2) for j in range(self.mesh_size + 1)]
)
control_lines_trans = VGroup(
*[Line(np.append(self.mesh_trans[:, i, j] * self.K + self.B, 0),
np.append(self.mesh_trans[:, i + 1, j] * self.K + self.B, 0))
for i in range(self.mesh_size + 1) for j in range(self.mesh_size + 2)],
*[Line(np.append(self.mesh_trans[:, i, j] * self.K + self.B, 0),
np.append(self.mesh_trans[:, i, j + 1] * self.K + self.B, 0))
for i in range(self.mesh_size + 2) for j in range(self.mesh_size + 1)]
)
control_lines.set_color("#70c3ff")
control_lines_trans.set_color("#70c3ff")
grid = NumberPlane(**self.CONFIG)
square_num = 10.
square_side_length = self.GEOMETRY_SIZE / square_num
squares = VGroup(
*[Square(side_length=square_side_length, fill_opacity=1).shift(x * RIGHT + y * UP)
for x in np.arange(self.AXIS_MIN + 0.5 * square_side_length,
self.AXIS_MIN + self.GEOMETRY_SIZE - 0.4 * square_side_length,
square_side_length)
for y in np.arange(self.AXIS_MIN + 0.5 * square_side_length,
self.AXIS_MIN + self.GEOMETRY_SIZE - 0.4 * square_side_length,
square_side_length)])
squares.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
#
self.add(squares, grid, control_lines, control_points)
squares.save_state()
grid.save_state()
control_points.save_state()
control_lines.save_state()
grid.prepare_for_nonlinear_transform()
self.play(
Transform(control_points, control_points_trans),
Transform(control_lines, control_lines_trans),
ApplyPointwiseFunction(lambda p: naive_transformation(p, self.mesh_trans, self.delta, self.B, self.K),
squares),
ApplyPointwiseFunction(lambda p: naive_transformation(p, self.mesh_trans, self.delta, self.B, self.K),
grid),
run_time=1,
)
self.play(
Restore(grid, run_time=1),
Restore(squares, run_time=1),
Restore(control_points, run_time=1),
Restore(control_lines, run_time=1)
)
class FFDDots(Scene):
def __init__(self, mesh, mesh_trans, delta, **scene_kwargs):
self.mesh = mesh
self.mesh_trans = mesh_trans
self.delta = delta
self.mesh_size = self.mesh.shape[1] - 3
self.K = CON_POINT_RANGE / (self.mesh_size + 1) / delta
self.B = CON_POINT_RANGE * (1 / (self.mesh_size + 1) - 0.5)
GEOMETRY_SIZE = self.K * delta * (self.mesh_size - 1) - 0.0001
self.AXIS_MIN = -0.5 * GEOMETRY_SIZE
self.CONFIG = {
"x_min": self.AXIS_MIN,
"x_max": -self.AXIS_MIN,
"y_min": self.AXIS_MIN,
"y_max": -self.AXIS_MIN,
"background_line_style": {
"stroke_color": "#FFFFFF",
},
"x_line_frequency": 0.5 * self.K * delta,
"y_line_frequency": 0.5 * self.K * delta,
}
super().__init__(**scene_kwargs)
def construct(self):
control_lines = VGroup(
*[Line(np.append(self.mesh[:, i, j] * self.K + self.B, 0),
np.append(self.mesh[:, i + 1, j] * self.K + self.B, 0))
for i in range(self.mesh_size + 1) for j in range(self.mesh_size + 2)],
*[Line(np.append(self.mesh[:, i, j] * self.K + self.B, 0),
np.append(self.mesh[:, i, j + 1] * self.K + self.B, 0))
for i in range(self.mesh_size + 2) for j in range(self.mesh_size + 1)]
)
control_lines_trans = VGroup(
*[Line(np.append(self.mesh_trans[:, i, j] * self.K + self.B, 0),
np.append(self.mesh_trans[:, i + 1, j] * self.K + self.B, 0))
for i in range(self.mesh_size + 1) for j in range(self.mesh_size + 2)],
*[Line(np.append(self.mesh_trans[:, i, j] * self.K + self.B, 0),
np.append(self.mesh_trans[:, i, j + 1] * self.K + self.B, 0))
for i in range(self.mesh_size + 2) for j in range(self.mesh_size + 1)]
)
control_grid = VGroup(control_lines, control_lines_trans)
control_grid.set_color("#70c3ff")
grid = NumberPlane(**self.CONFIG)
dot_radius = 0.5 * 0.25 * self.delta * self.K
dots = VGroup(
*[Dot(radius=dot_radius, fill_opacity=1).shift(x * RIGHT + y * UP)
for x in np.arange(self.AXIS_MIN + 0.25 * self.delta * self.K,
-self.AXIS_MIN - 0.24 * self.delta * self.K,
0.5 * self.delta * self.K)
for y in np.arange(self.AXIS_MIN + 0.25 * self.delta * self.K,
-self.AXIS_MIN - 0.24 * self.delta * self.K,
0.5 * self.delta * self.K)])
dots.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
self.add(dots, grid, control_lines)
dots.save_state()
grid.save_state()
control_lines.save_state()
grid.prepare_for_nonlinear_transform()
self.play(
Transform(control_lines, control_lines_trans),
ApplyPointwiseFunction(lambda p: naive_transformation(p, self.mesh_trans, self.delta, self.B, self.K),
dots),
ApplyPointwiseFunction(lambda p: naive_transformation(p, self.mesh_trans, self.delta, self.B, self.K),
grid),
run_time=1,
)
self.play(Restore(grid, run_time=1),
Restore(dots, run_time=1),
Restore(control_lines, run_time=1))
class FFDVectorsWithGrid(Scene):
def __init__(self, mesh, mesh_trans, delta, **scene_kwargs):
self.mesh = mesh
self.mesh_trans = mesh_trans
self.delta = delta
self.mesh_size = self.mesh.shape[1] - 3
self.K = CON_POINT_RANGE / (self.mesh_size + 1) / delta
self.B = CON_POINT_RANGE * (1 / (self.mesh_size + 1) - 0.5)
GEOMETRY_SIZE = self.K * delta * (self.mesh_size - 1) - 0.0001
self.AXIS_MIN = -0.5 * GEOMETRY_SIZE
self.GEOMETRY_SIZE = self.K * delta * (self.mesh_size - 1) - 0.0001
self.AXIS_MIN = -0.5 * GEOMETRY_SIZE
self.CONFIG = {
"x_min": self.AXIS_MIN,
"x_max": -self.AXIS_MIN,
"y_min": self.AXIS_MIN,
"y_max": -self.AXIS_MIN,
"background_line_style": {
"stroke_color": "#FFFFFF",
},
"x_line_frequency": 0.5 * self.K * delta,
"y_line_frequency": 0.5 * self.K * delta,
"max_stroke_width_to_length_ratio": 10,
}
super().__init__(**scene_kwargs)
def construct(self):
control_line_width = 1.5
control_lines = VGroup(
*[Line(np.append(self.mesh[:, i, j] * self.K + self.B, 0),
np.append(self.mesh[:, i + 1, j] * self.K + self.B, 0), stroke_width=control_line_width)
for i in range(self.mesh_size + 1) for j in range(self.mesh_size + 2)],
*[Line(np.append(self.mesh[:, i, j] * self.K + self.B, 0),
np.append(self.mesh[:, i, j + 1] * self.K + self.B, 0), stroke_width=control_line_width)
for i in range(self.mesh_size + 2) for j in range(self.mesh_size + 1)]
)
control_lines_trans = VGroup(
*[Line(np.append(self.mesh_trans[:, i, j] * self.K + self.B, 0),
np.append(self.mesh_trans[:, i + 1, j] * self.K + self.B, 0), stroke_width=control_line_width)
for i in range(self.mesh_size + 1) for j in range(self.mesh_size + 2)],
*[Line(np.append(self.mesh_trans[:, i, j] * self.K + self.B, 0),
np.append(self.mesh_trans[:, i, j + 1] * self.K + self.B, 0), stroke_width=control_line_width)
for i in range(self.mesh_size + 2) for j in range(self.mesh_size + 1)]
)
control_grid = VGroup(control_lines, control_lines_trans)
control_grid.set_color("#70c3ff")
grid = NumberPlane(**self.CONFIG)
points = [x * RIGHT + y * UP
for x in np.arange(self.AXIS_MIN + 0.25 * self.delta * self.K,
-self.AXIS_MIN - 0.24 * self.delta * self.K,
0.5 * self.delta * self.K)
for y in np.arange(self.AXIS_MIN + 0.25 * self.delta * self.K,
-self.AXIS_MIN - 0.24 * self.delta * self.K,
0.5 * self.delta * self.K)
]
vectors = VGroup(*[Vector([0, 0, 0]).shift(point) for point in points])
scale_factor = 1
vectors_trans = VGroup(*[Vector(scale_factor *
(naive_transformation(point, self.mesh_trans, self.delta, self.B, self.K) - point),
**self.CONFIG).shift(point)
for point in points])
vectors_trans.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE)
self.add(vectors, grid, control_lines)
vectors.save_state()
grid.save_state()
control_lines.save_state()
grid.prepare_for_nonlinear_transform()
self.play(
Transform(control_lines, control_lines_trans),
Transform(vectors, vectors_trans),
ApplyPointwiseFunction(lambda p: naive_transformation(p, self.mesh_trans, self.delta, self.B, self.K),
grid),
run_time=1,
)
self.play(Restore(grid, run_time=1),
Restore(vectors, run_time=1),
Restore(control_lines, run_time=1))
class FFDVectors(Scene):
def __init__(self, mesh, mesh_trans, delta, **scene_kwargs):
self.mesh = mesh
self.mesh_trans = mesh_trans
self.delta = delta
self.mesh_size = self.mesh.shape[1] - 3
self.K = CON_POINT_RANGE / (self.mesh_size + 1) / delta
self.B = CON_POINT_RANGE * (1 / (self.mesh_size + 1) - 0.5)
GEOMETRY_SIZE = self.K * delta * (self.mesh_size - 1) - 0.0001
self.AXIS_MIN = -0.5 * GEOMETRY_SIZE
self.GEOMETRY_SIZE = self.K * delta * (self.mesh_size - 1) - 0.0001
self.AXIS_MIN = -0.5 * GEOMETRY_SIZE
self.CONFIG = {
"x_min": self.AXIS_MIN,
"x_max": -self.AXIS_MIN,
"y_min": self.AXIS_MIN,
"y_max": -self.AXIS_MIN,
"background_line_style": {
"stroke_color": "#FFFFFF",
},
"x_line_frequency": 0.5 * self.K * delta,
"y_line_frequency": 0.5 * self.K * delta,
"max_stroke_width_to_length_ratio": 10,
}
super().__init__(**scene_kwargs)
def construct(self):
control_points = VGroup(*[Dot(point=[self.mesh[0, i, j] * self.K + self.B,
self.mesh[1, i, j] * self.K + self.B,
0], radius=0.05 * self.delta * self.K)
for i in range(self.mesh_size + 2) for j in range(self.mesh_size + 2)])
control_points_trans = VGroup(*[Dot(point=[self.mesh_trans[0, i, j] * self.K + self.B,
self.mesh_trans[1, i, j] * self.K + self.B,
0], radius=0.05 * self.delta * self.K)
for i in range(self.mesh_size + 2) for j in range(self.mesh_size + 2)])
control_line_width = 1.5
control_lines = VGroup(
*[Line(np.append(self.mesh[:, i, j] * self.K + self.B, 0),
np.append(self.mesh[:, i + 1, j] * self.K + self.B, 0), stroke_width=control_line_width)
for i in range(self.mesh_size + 1) for j in range(self.mesh_size + 2)],
*[Line(np.append(self.mesh[:, i, j] * self.K + self.B, 0),
np.append(self.mesh[:, i, j + 1] * self.K + self.B, 0), stroke_width=control_line_width)
for i in range(self.mesh_size + 2) for j in range(self.mesh_size + 1)]
)
control_lines_trans = VGroup(
*[Line(np.append(self.mesh_trans[:, i, j] * self.K + self.B, 0),
np.append(self.mesh_trans[:, i + 1, j] * self.K + self.B, 0), stroke_width=control_line_width)
for i in range(self.mesh_size + 1) for j in range(self.mesh_size + 2)],
*[Line(np.append(self.mesh_trans[:, i, j] * self.K + self.B, 0),
np.append(self.mesh_trans[:, i, j + 1] * self.K + self.B, 0), stroke_width=control_line_width)
for i in range(self.mesh_size + 2) for j in range(self.mesh_size + 1)]
)
control_grid = VGroup(control_lines, control_lines_trans)
control_grid.set_color("#70c3ff")
points = [x * RIGHT + y * UP
for x in
np.arange(self.AXIS_MIN + 0.25 * self.delta * self.K,
-self.AXIS_MIN - 0.24 * self.delta * self.K,
0.5 * self.delta * self.K)
for y in
np.arange(self.AXIS_MIN + 0.25 * self.delta * self.K,
-self.AXIS_MIN - 0.24 * self.delta * self.K,
0.5 * self.delta * self.K)
]
vectors = VGroup(*[Vector([0, 0, 0]).shift(point) for point in points])
scale_factor = 2
vectors_trans = VGroup(*[Vector(scale_factor *
(naive_transformation(point, self.mesh_trans, self.delta, self.B, self.K) - point),
**self.CONFIG).shift(point)
for point in points])
vectors_trans.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE)
self.add(vectors, control_lines, control_points)
vectors.save_state()
control_lines.save_state()
control_points.save_state()
self.play(
Transform(control_lines, control_lines_trans),
Transform(control_points, control_points_trans),
Transform(vectors, vectors_trans),
run_time=1,
)
self.play(Restore(vectors, run_time=1),
Restore(control_points, run_time=1),
Restore(control_lines, run_time=1))