1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
|
from manimlib.imports import *
class MaximaMinima(ThreeDScene):
def construct(self):
axes = ThreeDAxes()
paraboloid = ParametricSurface(
lambda u, v: np.array([
3.5*np.sin(u)*np.cos(v),
3.5*np.sin(u)*np.sin(v),
3.5*3.5*np.sin(u)*np.sin(u)*(1+2*np.sin(v)*np.sin(v))*np.exp(1 - 3.5*3.5*np.sin(u)*np.sin(u) )
]),u_min=0,u_max=PI,v_min=0,v_max=2*PI, color = BLUE_C, fill_color = BLUE_C, fill_opacity = 0.1,
resolution=(15, 32)).scale(1)
paraboloid_copy1 = paraboloid.copy()
paraboloid_copy2 = paraboloid.copy()
paraboloid_x = ParametricSurface(
lambda u, v: np.array([
3.5*np.sin(u)*np.cos(v),
3.5*np.sin(u)*np.sin(v),
3.5*3.5*np.sin(u)*np.sin(u)*(1+2*np.sin(v)*np.sin(v))*np.exp(1 - 3.5*3.5*np.sin(u)*np.sin(u) )
]),u_min=0,u_max=PI,v_min=PI,v_max=2*PI, color = BLUE_C, fill_color = BLUE_C, fill_opacity = 0.1,
resolution=(15, 32)).scale(1)
paraboloid_y = ParametricSurface(
lambda u, v: np.array([
3.5*np.sin(u)*np.cos(v),
3.5*np.sin(u)*np.sin(v),
3.5*3.5*np.sin(u)*np.sin(u)*(1+2*np.sin(v)*np.sin(v))*np.exp(1 - 3.5*3.5*np.sin(u)*np.sin(u) )
]),u_min=0,u_max=PI,v_min=PI/2,v_max=3*PI/2, color = BLUE_C, fill_color = BLUE_C, fill_opacity = 0.1,
resolution=(15, 32)).scale(1)
parabola_x_out =ParametricFunction(
lambda u : np.array([
u,
0,
(u*u )*np.exp(1-u*u)
]),color=RED_E,t_min=-3.5,t_max=3.5,
)
parabola_y_out =ParametricFunction(
lambda u : np.array([
0,
u,
(3*u*u)*np.exp(1-u*u)
]),color=PINK,t_min=-3.5,t_max=3.5,
)
plane1 = Polygon(np.array([-3.5,0,-3]),np.array([3.5,0,-3]),np.array([3.5,0,3]),np.array([-3.5,0,3]),np.array([-3.5,0,-3]), color = RED_C, fill_color = RED_C, fill_opacity = 0.2)
plane_text_x = TextMobject(r"$y = 0$", color = RED_C).move_to(2*UP + 4.5*RIGHT)
plane2 = Polygon(np.array([0,-3.5,-3]),np.array([0,3.5,-3]),np.array([0,3.5,3]),np.array([0,-3.5,3]),np.array([0,-3.5,-3]), color = PINK, fill_color = PINK, fill_opacity = 0.2)
plane_text_y = TextMobject(r"$x = 0$", color = PINK).move_to(2*UP + 4.5*RIGHT)
surface_eqn = TextMobject("Surface", r"$z = (x^2 + 3y^2)e^{(1 - x^2 - y^2)}$", color = YELLOW_C).scale(0.6).move_to(np.array([3.5*LEFT +3.5*UP]))
surface_eqn[0].set_color(BLUE_C)
self.set_camera_orientation(phi=60 * DEGREES, theta = 45*DEGREES)
self.add(axes)
axis = TextMobject(r"X",r"Y",r"Z")
axis[0].move_to(6*RIGHT)
axis[1].move_to(6*UP)
axis[2].move_to(np.array([0,0,3.7]))
self.add_fixed_orientation_mobjects(axis[2])
self.add_fixed_orientation_mobjects(axis[0])
self.add_fixed_orientation_mobjects(axis[1])
self.play(ShowCreation(paraboloid))
#self.move_camera(phi=60 * DEGREES, theta = 45*DEGREES,run_time=3)
plane_x = Polygon(np.array([-3.5,2,-3]),np.array([3.5,2,-3]),np.array([3.5,2,3]),np.array([-3.5,2,3]),np.array([-3.5,2,-3]), color = YELLOW_C, fill_color = YELLOW_A, fill_opacity = 0.2)
plane_y = Polygon(np.array([2,-3.5,-3]),np.array([2,3.5,-3]),np.array([2,3.5,3]),np.array([2,-3.5,3]),np.array([2,-3.5,-3]), color = GREEN_C, fill_color = GREEN_A, fill_opacity = 0.2)
text_x = TextMobject(r"$x$", "is fixed on this" ,"plane").scale(0.7).to_corner(UL)
text_y = TextMobject(r"$y$", "is fixed on this" ,"plane").scale(0.7).to_corner(UR)
text_x[0].set_color(RED_C)
text_y[0].set_color(PINK)
text_x[1].set_color(BLUE_C)
text_y[1].set_color(BLUE_C)
text_x[2].set_color(GREEN_C)
text_y[2].set_color(YELLOW_C)
self.add_fixed_in_frame_mobjects(text_x, text_y)
for i in range(2,-4,-1):
parabola_x =ParametricFunction(lambda u : np.array([u,i,(u*u + 3*i*i)*np.exp(1- u*u - i*i)]),color=RED_C,t_min=-3.5,t_max=3.5,)
parabola_y =ParametricFunction(lambda u : np.array([i,u,(i*i + 3*u*u)*np.exp(1- u*u - i*i)]),color=PINK,t_min=-3.5,t_max=3.5,)
if(i==2):
self.play(ShowCreation(plane_x), ShowCreation(plane_y))
parabola_copy_x = parabola_x.copy()
parabola_copy_y = parabola_y.copy()
self.play(ShowCreation(parabola_copy_x), ShowCreation(parabola_copy_y))
self.wait()
self.play(FadeOut(parabola_copy_x), FadeOut(parabola_copy_y))
else:
self.play(ApplyMethod(plane_x.move_to, np.array([0,i,0])),ReplacementTransform(parabola_copy_x, parabola_x),ApplyMethod(plane_y.move_to, np.array([i,0,0])),ReplacementTransform(parabola_copy_y, parabola_y))
self.play(FadeOut(parabola_x), FadeOut(parabola_y))
self.wait()
parabola_copy_x = parabola_x.copy()
parabola_copy_y = parabola_y.copy()
self.play(FadeOut(plane_x), FadeOut(plane_y), FadeOut(text_x), FadeOut(text_y))
self.add_fixed_in_frame_mobjects(surface_eqn)
self.move_camera(phi=80 * DEGREES, theta = 95*DEGREES)
self.play(ShowCreation(plane1))
self.add_fixed_in_frame_mobjects(plane_text_x)
self.wait()
self.play(ReplacementTransform(paraboloid, paraboloid_x))
self.play(FadeOut(plane1), FadeOut(plane_text_x))
line_x = Line(np.array([-3.5,0,0]), np.array([3.5,0,0]), color = YELLOW_E)
self.play(ShowCreation(parabola_x_out), ShowCreation(line_x))
slope_text_x = TexMobject("Slope =", "{\\partial", "f", "\\over", "\\partial", "x}").scale(0.6).move_to(2*UP + 3.5*RIGHT)
slope_text_x[0].set_color(ORANGE)
slope_text_x.set_color_by_tex("\\partial",GREEN_E)
slope_text_x.set_color_by_tex("f",RED_E)
slope_text_x[5].set_color(YELLOW_E)
self.add_fixed_in_frame_mobjects(slope_text_x)
dot_x = Dot().rotate(PI/2).set_color(YELLOW_E)
alpha_x = ValueTracker(0)
vector_x = self.get_tangent_vector(alpha_x.get_value(),parabola_x_out,scale=1.5)
dot_x.add_updater(lambda m: m.move_to(vector_x.get_center()))
self.play(
ShowCreation(parabola_x_out),
GrowFromCenter(dot_x),
GrowArrow(vector_x)
)
vector_x.add_updater(
lambda m: m.become(
self.get_tangent_vector(alpha_x.get_value()%1,parabola_x_out,scale=1.5)
)
)
self.add(vector_x,dot_x)
self.play(alpha_x.increment_value, 1, run_time=10, rate_func=linear)
self.wait(2)
self.play(FadeOut(parabola_x_out), FadeOut(line_x), FadeOut(vector_x), FadeOut(dot_x), FadeOut(slope_text_x))
self.move_camera(phi=80* DEGREES,theta= 5*DEGREES)
self.play(ReplacementTransform(paraboloid_x, paraboloid_copy1))
self.wait()
self.play(ShowCreation(plane2))
self.add_fixed_in_frame_mobjects(plane_text_y)
self.wait()
self.play(ReplacementTransform(paraboloid_copy1, paraboloid_y))
self.play(FadeOut(plane2), FadeOut(plane_text_y))
line_y = Line(np.array([0,-3.5,0]), np.array([0,3.5,0]), color = GREEN_E)
self.play(ShowCreation(parabola_y_out), ShowCreation(line_y))
slope_text_y = TexMobject("Slope =", "{\\partial", "f", "\\over", "\\partial", "y}").scale(0.6).move_to(2*UP + 3.5*RIGHT)
slope_text_y[0].set_color(ORANGE)
slope_text_y.set_color_by_tex("\\partial",YELLOW_E)
slope_text_y.set_color_by_tex("f",PINK)
slope_text_y[5].set_color(GREEN_E)
self.add_fixed_in_frame_mobjects(slope_text_y)
dot_y = Dot().rotate(PI/2).set_color(GREEN_E)
alpha_y = ValueTracker(0)
vector_y = self.get_tangent_vector(alpha_y.get_value(),parabola_y_out,scale=1.5)
dot_y.add_updater(lambda m: m.move_to(vector_y.get_center()))
self.play(
ShowCreation(parabola_y_out),
GrowFromCenter(dot_y),
GrowArrow(vector_y)
)
vector_y.add_updater(
lambda m: m.become(
self.get_tangent_vector(alpha_y.get_value()%1,parabola_y_out,scale=1.5)
)
)
self.add(vector_y,dot_y)
self.play(alpha_y.increment_value, 1, run_time=10, rate_func=linear)
self.wait(2)
self.play(FadeOut(parabola_y_out), FadeOut(line_y), FadeOut(vector_y), FadeOut(dot_y), FadeOut(slope_text_y))
self.move_camera(phi=60* DEGREES,theta= 45*DEGREES)
self.play(ReplacementTransform(paraboloid_y, paraboloid_copy2))
self.wait()
def get_tangent_vector(self, proportion, curve, dx=0.001, scale=1):
coord_i = curve.point_from_proportion(proportion)
coord_f = curve.point_from_proportion(proportion + dx)
reference_line = Line(coord_i,coord_f)
unit_vector = reference_line.get_unit_vector() * scale
vector = Line(coord_i - unit_vector , coord_i + unit_vector, color = ORANGE, buff=0)
return vector
|
