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Up to AngularMomentumArgument
This commit is contained in:
Grant Sanderson
@ -1,8 +1,12 @@
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from __future__ import absolute_import
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from big_ol_pile_of_manim_imports import *
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from active_projects.div_curl import VectorField
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from active_projects.div_curl import get_force_field_func
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COBALT = "#0047AB"
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class Orbiting(ContinualAnimation):
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CONFIG = {
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"rate": 0.3,
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@ -577,12 +581,14 @@ class TheMotionOfPlanets(Scene):
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class AskAboutEllipses(TheMotionOfPlanets):
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CONFIG = {
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"camera_config": {"background_opacity": 1},
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"sun_height": 0.5,
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"sun_center": ORIGIN,
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"animate_sun": True,
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"a": 3.5,
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"b": 2.0,
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"ellipse_color": WHITE,
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"ellipse_stroke_width": 1,
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"comet_height": 0.2,
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}
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def construct(self):
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@ -600,17 +606,20 @@ class AskAboutEllipses(TheMotionOfPlanets):
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self.title = title
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def add_sun(self):
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sun = ImageMobject("sun", height=0.5)
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sun = ImageMobject("sun", height=self.sun_height)
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sun.move_to(self.sun_center)
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self.sun = sun
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self.add(sun)
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if self.animate_sun:
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self.add(SunAnimation(sun))
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sun_animation = SunAnimation(sun)
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self.add(sun_animation)
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self.add_foreground_mobjects(
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sun_animation.mobject
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)
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def add_orbit(self):
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sun = self.sun
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comet = ImageMobject("comet")
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comet.scale_to_fit_height(0.2)
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comet = self.get_comet()
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ellipse = self.get_ellipse()
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orbit = Orbiting(comet, sun, ellipse)
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@ -798,6 +807,11 @@ class AskAboutEllipses(TheMotionOfPlanets):
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self.wait(6)
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# Helpers
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def get_comet(self):
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comet = ImageMobject("comet")
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comet.scale_to_fit_height(self.comet_height)
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return comet
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def get_ellipse(self):
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a = self.a
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b = self.b
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@ -809,7 +823,7 @@ class AskAboutEllipses(TheMotionOfPlanets):
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)
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ellipse.stretch(fdiv(b, a), dim=1)
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ellipse.move_to(
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self.sun.get_center() + c * LEFT / 2
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self.sun.get_center() + c * LEFT,
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)
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self.focus_points = [
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self.sun.get_center(),
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@ -1848,75 +1862,75 @@ class EndOfGeometryProofiness(GeometryProofLand):
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class KeplersSecondLaw(AskAboutEllipses):
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CONFIG = {
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"sun_center": 3 * RIGHT + DOWN,
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"animate_sun": False,
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"sun_center": 4 * RIGHT + 0.75 * DOWN,
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"animate_sun": True,
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"a": 5.0,
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"b": 2.5,
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"b": 3.0,
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"ellipse_stroke_width": 2,
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"area_color": COBALT,
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"area_opacity": 0.75,
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"arc_color": YELLOW,
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"arc_stroke_width": 3,
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"n_sample_sweeps": 5,
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"fade_sample_areas": True,
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}
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def construct(self):
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self.add_title()
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self.add_sun()
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self.add_foreground_mobjects(self.sun)
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self.add_orbit()
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self.add_foreground_mobjects(self.comet)
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self.show_several_sweeps()
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self.contrast_close_to_far()
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def add_title(self):
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title = TextMobject("Kepler's 2nd law")
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title = TextMobject("Kepler's 2nd law:")
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title.scale(1.0)
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title.to_edge(UP)
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self.add(title)
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self.title = title
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subtitle = TextMobject(
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"Orbits sweep a constant area per unit time"
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)
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subtitle.next_to(title, DOWN, buff=0.2)
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subtitle.set_color(BLUE)
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self.add(subtitle)
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def show_several_sweeps(self):
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n_sweeps = 3
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shown_areas = VGroup()
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for x in range(n_sweeps):
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for x in range(self.n_sample_sweeps):
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self.wait()
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area = self.show_area_sweep()
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shown_areas.add(area)
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self.wait(2)
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self.play(FadeOut(shown_areas))
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self.wait()
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if self.fade_sample_areas:
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self.play(FadeOut(shown_areas))
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def contrast_close_to_far(self):
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orbit = self.orbit
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sun_point = self.sun.get_center()
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start_prop = 0.8
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start_prop = 0.9
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self.wait_until_proportion(start_prop)
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area = self.show_area_sweep()
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end_prop = max(0.9, orbit.proportion)
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self.show_area_sweep()
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end_prop = orbit.proportion
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arc = self.get_arc(start_prop, end_prop)
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radius = Line(sun_point, arc.points[0])
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radius.set_color(PINK)
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radius.set_color(WHITE)
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radius_words = self.get_radius_words(
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radius, "Short"
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)
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radius_words = self.get_radius_words(radius, "Short")
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radius_words.next_to(radius.get_center(), LEFT, SMALL_BUFF)
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arc_words = TextMobject("Long arc")
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angle = 9 * DEGREES
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arc_words.rotate(angle)
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arc_words.scale(0.1)
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vect = rotate_vector(RIGHT, angle)
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arc_words.next_to(vect, vect)
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arc_words.rotate(90 * DEGREES)
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arc_words.scale(0.5)
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arc_words.next_to(RIGHT, RIGHT)
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arc_words.apply_complex_function(np.exp)
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arc_words.scale(2)
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arc_words.scale(0.8)
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arc_words.next_to(
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arc.point_from_proportion(0.5),
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rotate_vector(vect, 90 * DEGREES),
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buff=-MED_SMALL_BUFF,
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arc, RIGHT, buff=-SMALL_BUFF
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)
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arc_words.match_color(arc)
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@ -1934,14 +1948,30 @@ class KeplersSecondLaw(AskAboutEllipses):
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# Show narrow arc
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# (Code repetition...uck)
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start_prop = 0.4
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start_prop = 0.475
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self.wait_until_proportion(start_prop)
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area = self.show_area_sweep()
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end_prop = max(0.45, orbit.proportion)
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arc = self.get_arc(start_prop, end_prop)
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radius = Line(sun_point, arc.points[0])
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radius.set_color(PINK)
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radius_words = self.get_radius_words(radius, "Long")
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self.show_area_sweep()
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end_prop = orbit.proportion
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short_arc = self.get_arc(start_prop, end_prop)
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long_radius = Line(sun_point, short_arc.points[0])
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long_radius.set_color(WHITE)
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long_radius_words = self.get_radius_words(long_radius, "Long")
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short_arc_words = TextMobject("Short arc")
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short_arc_words.scale(0.5)
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short_arc_words.rotate(90 * DEGREES)
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short_arc_words.next_to(short_arc, LEFT, SMALL_BUFF)
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short_arc_words.match_color(short_arc)
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self.play(
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ShowCreation(long_radius),
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Write(long_radius_words),
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)
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self.play(
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ShowCreation(short_arc),
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Write(short_arc_words)
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)
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self.wait(15)
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# Helpers
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def show_area_sweep(self, time=1.0):
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@ -1986,7 +2016,6 @@ class KeplersSecondLaw(AskAboutEllipses):
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return result
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def get_arc(self, prop1, prop2):
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sun_point = self.sun.get_center()
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ellipse = self.get_ellipse()
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prop1 = prop1 % 1.0
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prop2 = prop2 % 1.0
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@ -2019,18 +2048,371 @@ class KeplersSecondLaw(AskAboutEllipses):
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if self.skip_animations:
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self.orbit.proportion = prop
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else:
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while self.orbit.proportion < prop:
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self.wait(0.2)
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while (self.orbit.proportion % 1) < prop:
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self.wait(self.frame_duration)
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def get_radius_words(self, radius, adjective):
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radius_words = TextMobject(
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"%s radius" % adjective,
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)
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radius_words.scale_to_fit_width(
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0.8 * radius.get_length()
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)
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min_width = 0.8 * radius.get_length()
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if radius_words.get_width() > min_width:
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radius_words.scale_to_fit_width(min_width)
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radius_words.match_color(radius)
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radius_words.next_to(ORIGIN, DOWN, SMALL_BUFF)
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radius_words.rotate(radius.get_angle(), about_point=ORIGIN)
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radius_words.next_to(ORIGIN, UP, SMALL_BUFF)
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angle = radius.get_angle()
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angle = ((angle + PI) % TAU) - PI
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if np.abs(angle) > PI / 2:
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angle += PI
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radius_words.rotate(angle, about_point=ORIGIN)
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radius_words.shift(radius.get_center())
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return radius_words
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class AngularMomentumArgument(KeplersSecondLaw):
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CONFIG = {
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"animate_sun": False,
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"sun_center": 4 * RIGHT + DOWN,
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"comet_start_point": 4 * LEFT,
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"comet_end_point": 5 * LEFT + DOWN,
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"comet_height": 0.3,
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}
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def construct(self):
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self.add_sun()
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self.show_small_sweep()
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self.show_sweep_dimensions()
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self.show_conservation_of_angular_momentum()
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def show_small_sweep(self):
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sun_center = self.sun_center
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comet_start = self.comet_start_point
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comet_end = self.comet_end_point
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triangle = Polygon(
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sun_center, comet_start, comet_end,
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fill_opacity=1,
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fill_color=COBALT,
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stroke_width=0,
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)
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triangle.save_state()
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alt_triangle = Polygon(
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sun_center,
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interpolate(comet_start, comet_end, 0.9),
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comet_end
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)
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alt_triangle.match_style(triangle)
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comet = self.get_comet()
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comet.move_to(comet_start)
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velocity_vector = Arrow(
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comet_start, comet_end,
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color=WHITE,
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buff=0
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)
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velocity_vector_label = TexMobject("\\vec{\\textbf{v}}")
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velocity_vector_label.next_to(
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velocity_vector.get_center(), UL,
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buff=SMALL_BUFF
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)
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small_time_label = TextMobject(
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"Small", "time", "$\\Delta t$",
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)
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small_time_label.to_edge(UP)
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small = small_time_label.get_part_by_tex("Small")
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small_rect = SurroundingRectangle(small)
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self.add_foreground_mobjects(comet)
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self.play(
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ShowCreation(
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triangle,
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rate_func=lambda t: interpolate(1.0 / 3, 2.0 / 3, t)
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),
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MaintainPositionRelativeTo(
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velocity_vector, comet
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),
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MaintainPositionRelativeTo(
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velocity_vector_label,
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velocity_vector,
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),
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ApplyMethod(
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comet.move_to, comet_end,
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rate_func=None,
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),
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run_time=2,
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)
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self.play(Write(small_time_label), run_time=2)
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self.wait()
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self.play(
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Transform(triangle, alt_triangle),
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ShowCreation(small_rect),
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small.set_color, YELLOW,
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)
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self.wait()
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self.play(
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Restore(triangle),
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FadeOut(small_rect),
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small.set_color, WHITE,
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)
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self.wait()
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self.triangle = triangle
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self.comet = comet
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self.delta_t = small_time_label.get_part_by_tex(
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"$\\Delta t$"
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)
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self.velocity_vector = velocity_vector
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self.small_time_label = small_time_label
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def show_sweep_dimensions(self):
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triangle = self.triangle
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# velocity_vector = self.velocity_vector
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delta_t = self.delta_t
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comet = self.comet
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triangle_points = triangle.get_anchors()[:3]
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top = triangle_points[1]
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area_label = TexMobject(
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"\\text{Area}", "=", "\\frac{1}{2}",
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"\\text{Base}", "\\times", "\\text{Height}",
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)
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area_label.set_color_by_tex_to_color_map({
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"Base": PINK,
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"Height": YELLOW,
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})
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area_label.to_edge(UP)
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equals = area_label.get_part_by_tex("=")
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area_expression = TexMobject(
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"=", "\\frac{1}{2}", "R", "\\times",
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"\\vec{\\textbf{v}}_\\perp",
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"\\Delta t",
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)
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area_expression.set_color_by_tex_to_color_map({
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"R": PINK,
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"textbf{v}": YELLOW,
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})
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area_expression.next_to(area_label, DOWN)
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area_expression.align_to(equals, LEFT)
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self.R_v_perp = VGroup(*area_expression[-4:-1])
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self.R_v_perp_rect = SurroundingRectangle(
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self.R_v_perp,
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stroke_color=BLUE,
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fill_color=BLACK,
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fill_opacity=1,
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)
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base = Line(triangle_points[2], triangle_points[0])
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base.set_stroke(PINK, 3)
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base_point = line_intersection(
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base.get_start_and_end(),
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[top, top + DOWN]
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)
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height = Line(top, base_point)
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height.set_stroke(YELLOW, 3)
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radius_label = TextMobject("Radius")
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radius_label.next_to(base, DOWN, SMALL_BUFF)
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radius_label.match_color(base)
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R_term = area_expression.get_part_by_tex("R")
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R_term.save_state()
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R_term.move_to(radius_label[0])
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R_term.set_fill(opacity=0.5)
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v_perp = Arrow(*height.get_start_and_end(), buff=0)
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v_perp.set_color(YELLOW)
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v_perp.shift(comet.get_center() - v_perp.get_start())
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v_perp_label = TexMobject(
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"\\vec{\\textbf{v}}_\\perp"
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)
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v_perp_label.set_color(YELLOW)
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v_perp_label.next_to(v_perp, RIGHT, buff=SMALL_BUFF)
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v_perp_delta_t = VGroup(v_perp_label.copy(), delta_t.copy())
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v_perp_delta_t.generate_target()
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v_perp_delta_t.target.arrange_submobjects(RIGHT, buff=SMALL_BUFF)
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v_perp_delta_t.target.next_to(height, RIGHT, SMALL_BUFF)
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self.small_time_label.add(v_perp_delta_t[1])
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self.play(
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FadeInFromDown(area_label),
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self.small_time_label.scale, 0.5,
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self.small_time_label.to_corner, UL,
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)
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self.wait()
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self.play(
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ShowCreation(base),
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Write(radius_label),
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)
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self.wait()
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self.play(ShowCreation(height))
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self.wait()
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self.play(
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GrowArrow(v_perp),
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Write(v_perp_label, run_time=1),
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)
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self.wait()
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self.play(MoveToTarget(v_perp_delta_t))
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self.wait()
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self.play(*[
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ReplacementTransform(
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area_label.get_part_by_tex(tex).copy(),
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area_expression.get_part_by_tex(tex),
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)
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for tex in "=", "\\frac{1}{2}", "\\times"
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])
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self.play(Restore(R_term))
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self.play(ReplacementTransform(
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v_perp_delta_t.copy(),
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VGroup(*area_expression[-2:])
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))
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self.wait()
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def show_conservation_of_angular_momentum(self):
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R_v_perp = self.R_v_perp
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R_v_perp_rect = self.R_v_perp_rect
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sun_center = self.sun_center
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comet = self.comet
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comet.save_state()
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vector_field = VectorField(
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get_force_field_func((sun_center, -1))
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)
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vector_field.set_fill(opacity=0.8)
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vector_field.sort_submobjects(
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lambda p: -np.linalg.norm(p - sun_center)
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)
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|
||||
stays_constant = TextMobject("Stays constant")
|
||||
stays_constant.next_to(
|
||||
R_v_perp_rect, DR, buff=MED_LARGE_BUFF
|
||||
)
|
||||
stays_constant.match_color(R_v_perp_rect)
|
||||
stays_constant_arrow = Arrow(
|
||||
stays_constant.get_left(),
|
||||
R_v_perp_rect.get_bottom(),
|
||||
color=R_v_perp_rect.get_color()
|
||||
)
|
||||
|
||||
sun_dot = Dot(sun_center, fill_opacity=0.25)
|
||||
big_dot = Dot(fill_opacity=0, radius=FRAME_WIDTH)
|
||||
|
||||
R_v_perp.save_state()
|
||||
R_v_perp.generate_target()
|
||||
R_v_perp.target.to_edge(LEFT, buff=MED_LARGE_BUFF)
|
||||
lp, rp = parens = TexMobject("()")
|
||||
lp.next_to(R_v_perp.target, LEFT)
|
||||
rp.next_to(R_v_perp.target, RIGHT)
|
||||
|
||||
self.play(Transform(
|
||||
big_dot, sun_dot,
|
||||
run_time=1,
|
||||
remover=True
|
||||
))
|
||||
self.wait()
|
||||
self.play(
|
||||
DrawBorderThenFill(R_v_perp_rect),
|
||||
Animation(R_v_perp),
|
||||
Write(stays_constant, run_time=1),
|
||||
GrowArrow(stays_constant_arrow),
|
||||
)
|
||||
self.wait()
|
||||
foreground = VGroup(*self.get_mobjects())
|
||||
self.play(
|
||||
LaggedStart(GrowArrow, vector_field),
|
||||
Animation(foreground)
|
||||
)
|
||||
for x in range(3):
|
||||
self.play(
|
||||
LaggedStart(
|
||||
ApplyFunction, vector_field,
|
||||
lambda mob: (lambda m: m.scale(1.1).set_fill(opacity=1), mob),
|
||||
rate_func=there_and_back,
|
||||
run_time=1
|
||||
),
|
||||
Animation(foreground)
|
||||
)
|
||||
self.play(
|
||||
FadeIn(parens),
|
||||
MoveToTarget(R_v_perp),
|
||||
)
|
||||
self.play(
|
||||
comet.scale, 2,
|
||||
comet.next_to, parens, RIGHT, {"buff": SMALL_BUFF}
|
||||
)
|
||||
self.wait()
|
||||
self.play(
|
||||
FadeOut(parens),
|
||||
R_v_perp.restore,
|
||||
comet.restore,
|
||||
)
|
||||
self.wait(3)
|
||||
|
||||
|
||||
class KeplersSecondLawImage(KeplersSecondLaw):
|
||||
CONFIG = {
|
||||
"animate_sun": False,
|
||||
"n_sample_sweeps": 8,
|
||||
"fade_sample_areas": False,
|
||||
}
|
||||
|
||||
def construct(self):
|
||||
self.add_sun()
|
||||
self.add_foreground_mobjects(self.sun)
|
||||
self.add_orbit()
|
||||
self.add_foreground_mobjects(self.comet)
|
||||
self.show_several_sweeps()
|
||||
|
||||
|
||||
class HistoryOfAngularMomentum(TeacherStudentsScene):
|
||||
CONFIG = {
|
||||
"camera_config": {"fill_opacity": 1}
|
||||
}
|
||||
|
||||
def construct(self):
|
||||
am = VGroup(TextMobject("Angular momentum"))
|
||||
k2l = TextMobject("Kepler's 2nd law")
|
||||
arrow = Arrow(ORIGIN, RIGHT)
|
||||
|
||||
group = VGroup(am, arrow, k2l)
|
||||
group.arrange_submobjects(RIGHT)
|
||||
group.next_to(self.hold_up_spot, UL)
|
||||
|
||||
k2l_image = ImageMobject("Kepler2ndLaw")
|
||||
k2l_image.match_width(k2l)
|
||||
k2l_image.next_to(k2l, UP)
|
||||
k2l.add(k2l_image)
|
||||
|
||||
angular_momentum_formula = TexMobject(
|
||||
"R", "\\times", "m", "\\vec{\\textbf{v}}_\\perp",
|
||||
)
|
||||
angular_momentum_formula.set_color_by_tex_to_color_map({
|
||||
"R": PINK,
|
||||
"v": YELLOW,
|
||||
})
|
||||
angular_momentum_formula.next_to(am, UP)
|
||||
am.add(angular_momentum_formula)
|
||||
|
||||
self.play(
|
||||
self.teacher.change, "raise_right_hand",
|
||||
FadeInFromDown(group),
|
||||
self.get_student_changes(*3 * ["pondering"])
|
||||
)
|
||||
self.wait()
|
||||
self.play(
|
||||
am.next_to, arrow, RIGHT,
|
||||
{"index_of_submobject_to_align": 0},
|
||||
k2l.next_to, arrow, LEFT,
|
||||
{"index_of_submobject_to_align": 0},
|
||||
path_arc=90 * DEGREES,
|
||||
run_time=2
|
||||
)
|
||||
self.wait(3)
|
||||
|
||||
|
||||
class FeynmanRecountingNewton(Scene):
|
||||
def construct(self):
|
||||
pass
|
||||
|
||||
Reference in New Issue
Block a user