class ZoomInOnFordCircles(Scene):
CONFIG = {
"q_max" : 100,
"label_height_factor" : 0.5,
"axes_center": 2.5*DOWN,
"axes_config": {
"x_min": -1,
"x_max": 2,
"number_line_config": {
"unit_size" : 1,
"color": LIGHT_GREY,
"include_tip" : False,
"tick_size": 0,
"numbers_with_elongated_ticks" : [],
"exclude_zero_from_default_numbers": False,
},
},
"circle_config": {
"stroke_width" : 1,
"stroke_color" : BLUE,
},
}
def construct(self):
self.setup_axes()
self.setup_circles_and_labels()
self.add_remark()
self.show_different_levels_of_zooming()
def setup_axes(self):
axes = Axes(**self.axes_config)
axes.center()
axes.shift(self.axes_center)
axes.x_axis.set_stroke(width = 1)
axes.y_axis.set_stroke(width = 0)
self.add_foreground_mobjects(axes)
self.axes = axes
def setup_circles_and_labels(self):
circles = VGroup()
labels = VGroup()
for q in range(1, self.q_max+1):
for p in get_coprime_numers_by_denom(q):
circle = self.generate_circle_by_fraction(p, q)
circle.add_updater(
lambda m: m.set_stroke(width = get_stroke_width_by_height(m.get_height()))
)
# label = TexMobject("\\dfrac{%d}{%d}" % (p, q))
label = AssembledFraction(p, q)
label.set_height(1)
label.set_height(circle.get_height() * self.label_height_factor)
label.move_to(circle.get_center())
circles.add(circle)
labels.add(label)
self.add(circles, labels)
self.circles = circles
self.labels = labels
def add_remark(self):
nl_text = TextMobject("数轴")
nl_arrow = Arrow(ORIGIN, UP).match_height(nl_text)
nl_remark = VGroup(nl_arrow, nl_text)
nl_remark.scale(0.8)
nl_remark.set_color(LIGHT_GREY)
nl_remark.arrange_submobjects(RIGHT, buff = 0.1)
nl_remark.next_to(self.axes.coords_to_point(0, 0), DOWN, buff = 0.1)
nl_remark.to_edge(LEFT, buff = 0.15)
frac_remark = TextMobject("圆内分数为圆心横坐标")
frac_remark.scale(0.6)
frac_remark.to_corner(DL, buff = 0.15)
farey_sum_remark = TexMobject(
"\\text{Farey Sum: }", "\\dfrac{a}{b} \\oplus \\dfrac{c}{d}", "=", "\\dfrac{a+c}{b+d}"
)
farey_sum_remark[1].set_color(YELLOW)
farey_sum_remark[-1].set_color(PINK)
farey_sum_remark.to_corner(DR, buff = 0.15)
self.add(nl_remark, frac_remark, farey_sum_remark)
def show_different_levels_of_zooming(self):
self.zoom_in_on(1/2., 6)
self.wait()
self.acl = VGroup(self.axes, self.circles, self.labels)
self.acl.save_state()
# First Zoom
self.play_farey_sum_animation(0, 1, 1, 1)
self.wait()
# Second Zoom
self.play_zooming_animation(1/np.sqrt(2), 9, run_time = 2)
self.play_farey_sum_animation(2, 3, 3, 4)
self.wait()
# Third Zoom
self.play_zooming_animation(0.73, 5, run_time = 2)
self.play_farey_sum_animation(5, 7, 8, 11)
self.wait()
# Fourth Zoom
self.play_zooming_animation(0.74, 5, run_time = 2)
self.play_farey_sum_animation(11, 15, 14, 19)
self.play_farey_sum_animation(14, 19, 17, 23)
self.wait()
# Reset
self.play(Restore(self.acl), lag_ratio = 0, run_time = 4)
self.wait()
def generate_circle_by_fraction(self, p, q):
radius = 1./(2 * q**2)
center = self.axes.coords_to_point(p/q, radius)
circle = Circle(radius = radius, **self.circle_config)
circle.rotate(-PI/2.)
circle.move_to(center)
return circle
def zoom_in_on(self, center, scaling_factor, update_circles = True):
VGroup(self.axes, self.circles, self.labels).scale(
scaling_factor, about_point = self.axes.coords_to_point(center, 0)
)
if update_circles:
for circle in self.circles:
circle.update(0)
def play_zooming_animation(self, center, scaling_factor, **kwargs):
self.play(
ApplyMethod(
VGroup(self.axes, self.circles, self.labels).scale, scaling_factor,
{"about_point" : self.axes.coords_to_point(center, 0)}
),
lag_ratio = 0, **kwargs,
)
def get_farey_sum_key_mobjects(self, p1, q1, p2, q2):
assert (q1 + q2 <= self.q_max)
c1 = self.get_circle_by_fraction(p1, q1).deepcopy()
c2 = self.get_circle_by_fraction(p2, q2).deepcopy()
c3 = self.get_circle_by_fraction(p1+p2, q1+q2).deepcopy()
l1 = self.get_label_by_fraction(p1, q1).deepcopy()
l2 = self.get_label_by_fraction(p2, q2).deepcopy()
l3 = self.get_label_by_fraction(p1+p2, q1+q2).deepcopy()
for c, color in zip([c1, c2, c3], [YELLOW, YELLOW, PINK]):
c.clear_updaters()
c.set_stroke(width = 5)
c.set_color(color)
return c1, c2, c3, l1, l2, l3
def play_farey_sum_animation(self, p1, q1, p2, q2):
c1, c2, c3, l1, l2, l3 = self.get_farey_sum_key_mobjects(p1, q1, p2, q2)
l3.set_color(PINK)
self.wait()
self.play(
ShowCreation(c1), ApplyMethod(l1.set_color, YELLOW),
ShowCreation(c2), ApplyMethod(l2.set_color, YELLOW),
)
self.play(
ReplacementTransform(l1.numer.deepcopy(), l3.numer),
ReplacementTransform(l1.line.deepcopy(), l3.line),
ReplacementTransform(l1.denom.deepcopy(), l3.denom),
ReplacementTransform(l2.numer.deepcopy(), l3.numer),
ReplacementTransform(l2.line.deepcopy(), l3.line),
ReplacementTransform(l2.denom.deepcopy(), l3.denom),
ReplacementTransform(c1.deepcopy(), c3),
ReplacementTransform(c2.deepcopy(), c3),
)
self.wait()
self.play(FadeOut(VGroup(c1, c2, c3, l1, l2, l3)))
def get_circle_by_fraction(self, p, q, thres = 1e-6):
x = p/q
for circle in self.circles:
cx = self.axes.point_to_coords(circle.get_center())[0]
if np.abs(x - cx) < thres:
return circle
return VMobject()
def get_label_by_fraction(self, p, q, thres = 1e-6):
x = p/q
for label in self.labels:
lx = self.axes.point_to_coords(label.get_center())[0]
if np.abs(x - lx) < thres:
return label
return VMobject()
