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EulerAnglesAndGimbal
This commit is contained in:
Grant Sanderson
parent
33108aa9e4
commit
e8349f8d0d
1 changed files with 157 additions and 220 deletions
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@ -66,6 +66,86 @@ class RandyPrism(Cube):
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self.center()
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class Gimbal(VGroup):
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CONFIG = {
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"inner_r": 1.2,
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"outer_r": 2.6,
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}
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def __init__(self, alpha=0, beta=0, gamma=0, inner_mob=None, **kwargs):
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VGroup.__init__(self, **kwargs)
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r1, r2, r3, r4, r5, r6, r7 = np.linspace(
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self.inner_r, self.outer_r, 7
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)
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rings = VGroup(
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self.get_ring(r5, r6),
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self.get_ring(r3, r4),
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self.get_ring(r1, r2),
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)
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for i, p1, p2 in [(0, r6, r7), (1, r4, r5), (2, r2, r3)]:
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annulus = rings[i]
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lines = VGroup(
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Line(p1 * UP, p2 * UP),
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Line(p1 * DOWN, p2 * DOWN),
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)
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lines.set_stroke(RED)
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annulus.lines = lines
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annulus.add(lines)
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rings[1].lines.rotate(90 * DEGREES, about_point=ORIGIN)
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rings.rotate(90 * DEGREES, RIGHT, about_point=ORIGIN)
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rings.set_shade_in_3d(True)
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self.rings = rings
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self.add(rings)
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if inner_mob is not None:
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corners = [
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inner_mob.get_corner(v1 + v2)
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for v1 in [LEFT, RIGHT]
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for v2 in [IN, OUT]
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]
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lines = VGroup()
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for corner in corners:
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corner[1] = 0
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line = Line(
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corner, self.inner_r * normalize(corner),
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color=WHITE,
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stroke_width=1
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)
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lines.add(line)
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lines.set_shade_in_3d(True)
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rings[2].add(lines, inner_mob)
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# Rotations
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angles = [alpha, beta, gamma]
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for i, angle in zip(it.count(), angles):
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vect = rings[i].lines[0].get_vector()
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rings[i:].rotate(angle=angle, axis=vect)
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def get_ring(self, in_r, out_r, angle=TAU / 4):
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result = VGroup()
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for start_angle in np.arange(0, TAU, angle):
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start_angle += angle / 2
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sector = AnnularSector(
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inner_radius=in_r,
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outer_radius=out_r,
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angle=angle,
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start_angle=start_angle
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)
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sector.set_fill(LIGHT_GREY, 0.8)
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arcs = VGroup(*[
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Arc(
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angle=angle,
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start_angle=start_angle,
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radius=r
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)
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for r in [in_r, out_r]
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])
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arcs.set_stroke(BLACK, 1, opacity=0.5)
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sector.add(arcs)
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result.add(sector)
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return result
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# Scenes
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class Introduction(QuaternionHistory):
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@ -542,86 +622,34 @@ class RotationMatrix(ShowSeveralQuaternionRotations):
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self.add_fixed_orientation_mobjects(label)
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class EulerAnglesAndGimbal(SpecialThreeDScene):
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class EulerAnglesAndGimbal(ShowSeveralQuaternionRotations):
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CONFIG = {
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"cut_axes_at_radius": True,
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"inner_r": 1.2,
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"outer_r": 2.6,
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"use_lightweight_axes": True,
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"lightweight_axes_num_pieces": 10,
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}
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def construct(self):
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self.setup_position()
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self.setup_cube()
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self.setup_angle_trackers()
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self.setup_gimbal()
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self.add_axes()
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self.add_title()
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self.show_rotations()
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def add_axes(self):
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if self.use_lightweight_axes:
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self.axes = VGroup(*[
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VGroup(*Line(-v, v).scale(10).get_pieces(
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self.lightweight_axes_num_pieces
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))
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for v in [RIGHT, UP, OUT]
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])
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self.axes.set_stroke(WHITE, 1, opacity=0.5)
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self.axes.set_shade_in_3d(True)
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else:
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self.axes = self.get_axes()
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self.add(self.axes)
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def setup_position(self):
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self.add_axes()
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self.set_camera_orientation(
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theta=-140 * DEGREES,
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phi=70 * DEGREES,
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)
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self.begin_ambient_camera_rotation(rate=0.015)
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def setup_cube(self):
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cube = self.cube = RandyPrism()
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points = [
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cube[5].get_corner([i, 0, j])
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for i in (-1, 1)
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for j in (-1, 1)
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]
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lines = VGroup()
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for point in points:
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point[1] = 0
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proj_point = normalize(point) * self.inner_r
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line = Line(point, proj_point)
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line.set_stroke(WHITE, 1)
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lines.add(line)
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lines.set_shade_in_3d(True)
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cube.add(lines)
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self.add(cube)
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def setup_angle_trackers(self):
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self.alpha_tracker = ValueTracker(0)
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self.beta_tracker = ValueTracker(0)
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self.gamma_tracker = ValueTracker(0)
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def setup_gimbal(self):
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r1, r2, r3, r4, r5, r6, r7 = np.linspace(
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self.inner_r, self.outer_r, 7
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)
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gimbal = VGroup(
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self.get_ring(r5, r6),
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self.get_ring(r3, r4),
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self.get_ring(r1, r2),
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)
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for i, p1, p2 in [(0, r6, r7), (1, r4, r5), (2, r2, r3)]:
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annulus = gimbal[i]
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lines = VGroup(
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Line(p1 * UP, p2 * UP),
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Line(p1 * DOWN, p2 * DOWN),
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)
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lines.set_stroke(RED)
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annulus.lines = lines
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annulus.add(lines)
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gimbal[1].lines.rotate(90 * DEGREES, about_point=ORIGIN)
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gimbal.rotate(90 * DEGREES, RIGHT, about_point=ORIGIN)
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gimbal.set_shade_in_3d(True)
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gimbal = updating_mobject_from_func(self.get_gimbal)
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self.gimbal = gimbal
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self.add(gimbal)
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def add_title(self):
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title = TextMobject("Euler angles")
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@ -652,18 +680,22 @@ class EulerAnglesAndGimbal(SpecialThreeDScene):
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self.remove(gl_label)
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def show_rotations(self):
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cube = self.cube
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gimbal = self.gimbal
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self.add(cube, gimbal)
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alpha_tracker = self.alpha_tracker
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beta_tracker = self.beta_tracker
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gamma_tracker = self.gamma_tracker
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kwargs = {
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"about_point": ORIGIN,
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"run_time": 3,
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}
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angles = [-60 * DEGREES, 50 * DEGREES, 45 * DEGREES]
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for i, angle in zip(it.count(), angles):
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vect = gimbal[i].lines[0].get_vector()
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line = self.get_dotted_line(vect)
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trackers = [alpha_tracker, beta_tracker, gamma_tracker]
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in_rs = [0.6, 0.5, 0.6]
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for i in range(3):
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tracker = trackers[i]
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angle = angles[i]
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in_r = in_rs[i]
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ring = gimbal.rings[i]
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vect = ring.lines[0].get_vector()
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line = self.get_dotted_line(vect, in_r=in_r)
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angle_label = self.angle_labels[i]
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line.match_color(angle_label)
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self.play(
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@ -671,172 +703,77 @@ class EulerAnglesAndGimbal(SpecialThreeDScene):
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FadeInFromDown(angle_label)
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)
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self.play(
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Rotate(
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VGroup(cube, gimbal[i:]),
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angle=angle,
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axis=vect,
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**kwargs
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),
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tracker.set_value, angle,
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run_time=3
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)
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self.play(FadeOut(line))
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self.wait()
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self.wait(3)
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self.play(Write(self.gimbal_lock_label))
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self.play(
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Rotate(
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VGroup(cube, gimbal[1:]),
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angle=-angles[1],
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axis=gimbal[1].lines[0].get_vector(),
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**kwargs
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)
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alpha_tracker.set_value, 0,
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beta_tracker.set_value, 0,
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run_time=2
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)
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self.wait()
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self.play(
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Rotate(
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VGroup(cube, gimbal[0:]),
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angle=-angles[0],
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axis=gimbal[0].lines[0].get_vector(),
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**kwargs
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)
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alpha_tracker.set_value, 90 * DEGREES,
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gamma_tracker.set_value, -90 * DEGREES,
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run_time=3
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)
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self.play(
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FadeOut(self.gimbal_lock_label),
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*[ApplyMethod(t.set_value, 0) for t in trackers],
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run_time=3
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)
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self.play(
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alpha_tracker.set_value, 30 * DEGREES,
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beta_tracker.set_value, 120 * DEGREES,
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gamma_tracker.set_value, -50 * DEGREES,
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run_time=3
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)
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self.play(
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alpha_tracker.set_value, 120 * DEGREES,
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beta_tracker.set_value, -30 * DEGREES,
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gamma_tracker.set_value, 90 * DEGREES,
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run_time=4
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)
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self.play(
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beta_tracker.set_value, 150 * DEGREES,
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run_time=2
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)
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self.play(
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alpha_tracker.set_value, 0,
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beta_tracker.set_value, 0,
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gamma_tracker.set_value, 0,
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run_time=4
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)
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self.wait()
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for angle in [-120 * DEGREES, 120 * DEGREES]:
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self.play(
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Rotate(
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VGroup(cube, gimbal[2:]),
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angle=angle,
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axis=gimbal[2].lines[0].get_vector(),
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run_time=4,
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about_point=ORIGIN
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)
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)
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self.wait(4)
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#
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def get_dotted_line(self, vect):
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line = DashedLine(ORIGIN, 10 * normalize(vect))
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def get_gimbal(self):
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self.prism = RandyPrism()
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return Gimbal(
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alpha=self.alpha_tracker.get_value(),
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beta=self.beta_tracker.get_value(),
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gamma=self.gamma_tracker.get_value(),
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inner_mob=self.prism
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)
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def get_dotted_line(self, vect, in_r=0, out_r=10):
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line = VGroup(*it.chain(*[
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DashedLine(
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in_r * normalize(u * vect),
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out_r * normalize(u * vect),
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)
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for u in [-1, 1]
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]))
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line.sort_submobjects(get_norm)
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line.set_shade_in_3d(True)
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line.set_stroke(YELLOW, 5)
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line.center()
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return line
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def get_ring(self, in_r, out_r, angle=TAU / 4):
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result = VGroup()
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for start_angle in np.arange(0, TAU, angle):
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start_angle += angle / 2
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sector = AnnularSector(
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inner_radius=in_r,
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outer_radius=out_r,
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angle=angle,
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start_angle=start_angle
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)
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sector.set_fill(LIGHT_GREY, 0.8)
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arcs = VGroup(*[
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Arc(
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angle=angle,
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start_angle=start_angle,
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radius=r
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)
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for r in [in_r, out_r]
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])
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arcs.set_stroke(BLACK, 1, opacity=0.5)
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sector.add(arcs)
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result.add(sector)
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return result
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class QuaternionsDescribingRotation(EulerAnglesAndGimbal):
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CONFIG = {
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"use_lightweight_axes": True,
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"quaternions_and_imaginary_part_labels": [
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([1, 1, 0, 0], "{i}"),
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# ([1, 0, 1, 0], "{j}"),
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# ([0, 0, 0, 1], "{k}"),
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# ([1, 1, 1, 1], "\\left({{i} + {j} + {k} \\over \\sqrt{3}}\\right)"),
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],
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}
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class NewSceneName(Scene):
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def construct(self):
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self.setup_position()
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self.show_rotations()
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def show_rotations(self):
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for quat, ipl in self.quaternions_and_imaginary_part_labels:
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quat = normalize(quat)
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axis = quat[1:]
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angle = 2 * np.arccos(quat[0])
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label = self.get_label(angle, ipl)
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prism = RandyPrism()
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prism.scale(2)
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self.play(
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LaggedStart(FadeInFromDown, label),
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FadeIn(prism),
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)
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self.play(Rotate(
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prism,
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angle=angle, axis=axis,
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run_time=3,
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about_point=ORIGIN,
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))
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#
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def get_label(self, angle, imaginary_part_label):
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deg = int(angle / DEGREES)
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ipl = imaginary_part_label
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kwargs = {
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"tex_to_color_map": {
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"{i}": I_COLOR,
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"{j}": J_COLOR,
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"{k}": K_COLOR,
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}
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}
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p_label = TexMobject(
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"x{i} + y{j} + z{k}", **kwargs
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)
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arrow = TexMobject(
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"\\rightarrow"
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)
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q_label = TexMobject(
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"\\big(\\cos(%d^\\circ) + \\sin(%d^\\circ)%s \\big)" % (deg, deg, ipl),
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**kwargs
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)
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inner_p_label = TexMobject(
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"\\left(x{i} + y{j} + z{k} \\right)",
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**kwargs
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)
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q_inv_label = TexMobject(
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"\\big(\\cos(-%d^\\circ) + \\sin(-%d^\\circ)%s \\big)" % (deg, deg, ipl),
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**kwargs
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)
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equation = VGroup(
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p_label, arrow, q_label, inner_p_label, q_inv_label
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)
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equation.arrange_submobjects(RIGHT, buff=SMALL_BUFF)
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equation.set_width(FRAME_WIDTH - 1)
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equation.to_edge(UP)
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parts_text_colors = [
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(p_label, "\\text{3d point}", YELLOW),
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(q_label, "q", PINK),
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(inner_p_label, "\\text{3d point}", YELLOW),
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(q_inv_label, "q^{-1}", PINK),
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]
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braces = VGroup()
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for part, text, color in parts_text_colors:
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brace = Brace(part, DOWN, buff=SMALL_BUFF)
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label = brace.get_tex(text, buff=MED_SMALL_BUFF)
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label.set_color(color)
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brace.add(label)
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braces.add(brace)
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braces[-1][-1].shift(0.2 * UR)
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equation.add_to_back(BackgroundRectangle(equation))
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equation.braces = braces
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equation.add(*braces)
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self.add_fixed_in_frame_mobjects(equation)
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self.add_fixed_in_frame_mobjects(braces)
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return equation
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pass
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