【Manim】椭圆基础 (视频)

发表于 更新于 分类于 ，

这个是我用 Python 和 Manim 完成的数学笔记视频。做了很长一段时间，在这个过程中也算是积累了一些Python经验

内容为上课记的数学笔记，如有纰漏请在评论区指出~ 感谢观看！

源码展示

pages1

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
from manimlib import *
import numpy as np

class main(Scene):
    def construct(self):
        
        #BGcolor: Default(Black)

        # Prepare for clear the screen 
        self.save_state()

        # Screen Plane (For movements)
        #NumberP = NumberPlane()
        #self.add(NumberP)

        # Title
        title1 = Text("椭圆 · Part I · 定义", font="HarmonyOS Sans SC").scale(2)
        self.play(ShowCreation(title1))
        self.play(title1.animate.scale(0.5).to_corner(UL))

        # axesL&R 
        # Origin:L4U1 R4U1
        # withOUT lables 
        # Color:Default
        axesL = Axes(
            x_range=(-3,3,0.5), 
            y_range=(-2,2,0.5), 
            height=4,
            width=6 )
        axesR = axesL.copy()
        axesL.shift(UP+LEFT*4)
        axesR.shift(UP+RIGHT*4)
        self.play(
            ShowCreation(axesL),
            Write(axesL.get_axis_labels()))
        self.wait(0.5)
        
        # Focus Point:
        f = np.sqrt(7)/2
        dF1 = Dot(axesL.c2p(-f,0))
        dF2 = Dot(axesL.c2p(f,0))
        tF1 = Tex(r"F1").scale(0.5).next_to(dF1, direction=DOWN*1/2)
        tF2 = Tex(r"F2").scale(0.5).next_to(dF2, direction=DOWN*1/2)
        gF = VGroup(dF1, tF1, dF2, tF2)
        self.play(ShowCreation(gF))
        self.wait(0.5)

        # P点
        ellipseL = Ellipse(width=4,height=3).set_color(BLUE).move_to(axesL)
        ellipseR = Ellipse(width=3, height=4).set_color(GREEN).shift(UP+RIGHT*4)
        dP = Dot(color=ORANGE).move_to(ellipseL.pfp(0.3))
        tP = Tex(r"P").scale(0.5).next_to(dP, direction=DOWN*1/2)
        P = VGroup(dP, tP)
        self.play(ShowCreation(P))

        #PF
        PF1 = Line(start=dP, end=dF1)
        PF2 = Line(start=dP, end=dF2)
        PF = VGroup(PF1, PF2)
        PF.set_color(ORANGE)
        self.play(ShowCreation(PF))
        self.wait(0.5)

        # Definition Tex 
        tDefine = Tex(r"\left|{\rm PF}_1\right|+\left|{\rm PF}_2\right|=2a", color=BLUE).to_edge(UP).scale(0.75).shift(RIGHT*3)
        self.play(Write(tDefine))
        self.wait(1)
        
        # Ellipse
        self.play(ShowCreation(ellipseL))
        self.wait(0.5)

        #公式推导
        func_derivation = VGroup(
            Tex(r"\sqrt{{(x+c)}^2+y^2}+\sqrt{{(x-c)}^2+y^2}=2a"),                                    #0 l02
            Tex(r"\sqrt{{(x+c)}^2+y^2}=2a-\sqrt{{(x-c)}^2+y^2}"),                                    #1 l03
            Tex(r"\left(\sqrt{{(x+c)}^2+y^2}\ \right)^2=\left(2a-\sqrt{{(x-c)}^2+y^2}\right)^2"),    #2 l04
            Tex(r"a\sqrt{\left(x-c\right)^2+y^2}=a^2-cx"),                                           #3 l04(Transform)
            Tex(r"\left(a\sqrt{\left(x-c\right)^2+y^2}\right)^2=\left(a^2-cx\right)^2"),             #4 l05
            Tex(r"\left(a^2-c^2\right)x^2+a^2y^2=a^2(a^2-c^2)"),                                     #5 l05(Transform)
            Text(r"令", font="HarmonyOS Sans SC"),                                                   #6
            Tex(r"b=a^2-c^2")                                                                        #7
            #Tex(r"\frac{x^2}{a^2}+\frac{y^2}{b^2}=1")                                               #l07
        )
        func_derivation.match_x(tDefine)
        func_derivation.to_edge(UP)
        func_derivation.set_color(BLUE)
        func_derivation.scale(0.75)
        func_derivation[0].shift(DOWN*1)
        func_derivation[1].shift(DOWN*2)
        func_derivation[2].shift(DOWN*3)
        func_derivation[3].shift(DOWN*3)
        func_derivation[4].shift(DOWN*4)
        func_derivation[5].shift(DOWN*4)
        func_derivation[6].shift(DOWN*5)
        func_derivation[7].next_to(func_derivation[6])
        func_origin = Tex(r"\frac{x^2}{a^2}+\frac{y^2}{b^2}=1", isolate=["\frac{x^2}{a^2}+\frac{y^2}{b^2}=1"],color=WHITE).scale(0.75).match_x(tDefine).to_edge(UP).shift(DOWN*6)
        
        self.play(Write(func_derivation[0]))
        self.play(Write(func_derivation[1]))
        self.play(Write(func_derivation[2]))
        self.play(TransformMatchingTex(func_derivation[2], func_derivation[3], transform_mismatches=True))
        self.play(Write(func_derivation[4]))
        self.play(TransformMatchingTex(func_derivation[4], func_derivation[5], transform_mismatches=True))
        self.play(Write(func_derivation[6]))
        self.play(Write(func_derivation[7]))
        self.play(TransformFromCopy(func_derivation[4], func_origin))

        self.play(ShowCreationThenDestructionAround(func_origin))
        self.play(FadeOutToPoint(func_derivation, axesL.get_center()))

        # Standard Function
        funcL = Tex(r"\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)", isolate=["\frac{x^2}{a^2}+\frac{y^2}{b^2}=1"],color=BLUE).scale(0.75)
        funcL.move_to(func_origin)
        funcR = Tex(r"\frac{y^2}{a^2}+\frac{x^2}{b^2}=1(a>b>0)",color=GREEN).scale(0.75)
        funcR.move_to(axesR).shift(DOWN*2.5)

        self.play(TransformMatchingTex(func_origin, funcL))
        self.play(funcL.animate.move_to(axesL).shift(DOWN*2.5).set_color(BLUE))
        self.play(tDefine.animate.shift(LEFT*3).set_color(WHITE))
        self.play(ShowCreationThenDestructionAround(tDefine))
        self.wait(0.5)

        self.play(
            ShowCreation(axesR),
            Write(axesR.get_axis_labels()))
        gR = VGroup(ellipseR, funcR)
        self.play(ShowCreation(gR))
        self.wait(1.5)
        self.restore()
        self.wait(0.5)

pages2

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
from math import gamma
from re import A
from manimlib import *
import numpy as np

class main(Scene):
    def construct(self):
        #self.save_state()
        
        #BGcolor: Default(Black)

        # Screen Plane (For movements)
        #NumberP = NumberPlane()
        #self.add(NumberP)

        # Title
        title2 = Text("椭圆 · Part I · 基本性质", font="HarmonyOS Sans SC").scale(2)
        self.play(ShowCreation(title2))
        self.play(title2.animate.scale(0.5).to_corner(UL))
        
        # AxesL
        axesL = Axes(
            x_range=(-3,3,0.5), 
            y_range=(-2,2,0.5), 
            height=4,
            width=6 ).shift(UP*0.5+LEFT*4)
        self.play(
            ShowCreation(axesL),
            Write(axesL.get_axis_labels()))
        #self.wait(0.5)

        # Ellipse
        a = 2
        b = 1.5
        ellipseL = Ellipse(width=4,height=3).set_color(BLUE).move_to(axesL)
        self.play(ShowCreation(ellipseL))

        # Focus Point:
        f = np.sqrt(7)/2
        dF1 = Dot(axesL.c2p(-f,0)).set_color(ORANGE)
        dF2 = Dot(axesL.c2p(f,0)).set_color(ORANGE)
        tF1 = Tex(r"F1").set_color(ORANGE).scale(0.5).next_to(dF1, direction=DOWN*1/2)
        tF2 = Tex(r"F2").set_color(ORANGE).scale(0.5).next_to(dF2, direction=DOWN*1/2)
        gF = VGroup(dF1, tF1, dF2, tF2)
        self.play(ShowCreation(gF))
        
        # Standard Function
        funcL = Tex(r"\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)", isolate=["\frac{x^2}{a^2}+\frac{y^2}{b^2}=1"],color=BLUE).scale(0.75).move_to(axesL).shift(DOWN*2.5)
        funcL.move_to(axesL).shift(DOWN*3)
        self.play(Write(funcL))
        self.wait(0.5)

        # Centre Line
        line = Line(start=UP*4, end=DOWN*4)
        self.play(ShowCreation(line))

        # Sub title:
        stitle = Text("基本性质", font="HarmonyOS Sans SC").to_edge(UP).shift(RIGHT*3.5)
        self.play(TransformFromCopy(title2, stitle))
        self.wait(1)

        self.save_state()

        # 焦点焦距
        self.play(*[ApplyWave(mob) for mob in gF])
        焦点 = Text("F1,F2 (±c,0): 焦点", font="HarmonyOS Sans SC").set_color(ORANGE).scale(0.75).move_to(stitle).shift(DOWN+LEFT*1.5)
        self.play(Write(焦点))
        焦距线 = Line(start=dF1, end=dF2).set_color(ORANGE)
        焦距1 = Tex(r"\left|F_1F_2\right|").scale(0.75).next_to(焦点).set_color(ORANGE)
        焦距2 = Text(": 焦距", font="HarmonyOS Sans SC").scale(0.75).next_to(焦距1).set_color(ORANGE)
        self.play(ShowCreationThenDestruction(焦距线), run_time=1.5, rate_func=double_smooth)
        self.play(Write(焦距1), Write(焦距2))
        self.wait(1)

        # 左右顶点
        dA1 = Dot(axesL.c2p(-2,0)).set_color(GREEN)
        dA2 = Dot(axesL.c2p(2,0)).set_color(GREEN)
        tA1 = Tex(r"A_1").set_color(GREEN).scale(0.5).next_to(dA1, direction=DOWN*1/2)
        tA2 = Tex(r"A_2").set_color(GREEN).scale(0.5).next_to(dA2, direction=DOWN*1/2)
        gA = VGroup(dA1, tA1, dA2, tA2)
        self.play(ShowCreation(gA))
        左右顶点 = Text("A1,A2 (±a,0): 左右顶点", font="HarmonyOS Sans SC").set_color(GREEN).scale(0.75).move_to(焦点).shift(DOWN)
        self.play(Write(左右顶点))
        长轴 = Line(start=dA1, end=dA2).set_color(GREEN)
        长轴1 = Tex(r"\left|A_1A_2\right|").scale(0.75).next_to(左右顶点).set_color(GREEN)
        长轴2 = Text(": 长轴", font="HarmonyOS Sans SC").scale(0.75).next_to(长轴1).set_color(GREEN)
        self.play(ShowCreationThenDestruction(长轴), run_time=1.5, rate_func=double_smooth)
        self.play(Write(长轴1), Write(长轴2))
        self.wait(1)

        # 上下顶点
        dB1 = Dot(axesL.c2p(0,-1.5)).set_color(GOLD)
        dB2 = Dot(axesL.c2p(0,1.5)).set_color(GOLD)
        tB1 = Tex(r"B_1").set_color(GOLD).scale(0.5).next_to(dB1, direction=DOWN*1/2)
        tB2 = Tex(r"B_2").set_color(GOLD).scale(0.5).next_to(dB2, direction=DOWN*1/2)
        gB = VGroup(dB1, tB1, dB2, tB2)
        self.play(ShowCreation(gB))
        上下顶点 = Text("B1,B2 (±b,0): 上下顶点", font="HarmonyOS Sans SC").set_color(GOLD).scale(0.75).move_to(左右顶点).shift(DOWN)
        self.play(Write(上下顶点))
        短轴 = Line(start=dB1, end=dB2).set_color(GOLD)
        短轴1 = Tex(r"\left|B_1B_2\right|").scale(0.75).next_to(上下顶点).set_color(GOLD)
        短轴2 = Text(": 短轴", font="HarmonyOS Sans SC").scale(0.75).next_to(短轴1).set_color(GOLD)
        self.play(ShowCreationThenDestruction(短轴), run_time=1.5, rate_func=double_smooth)
        self.play(Write(短轴1), Write(短轴2))
        self.wait(2)

        self.restore()

        # 离心率部分
        # e_t:标题
        # e_d：定义公式
        e_t = Text("离心率", font="HarmonyOS Sans SC").scale(0.85).move_to(line, aligned_edge=LEFT).to_edge(UP).shift(DOWN*1.5+RIGHT*0.25)
        self.play(Write(e_t))
        e_d = Tex(r"e=\frac{c}{a}", isolate=["e="]).scale(0.75).next_to(e_t)
        self.play(Write(e_d))
        e_d2 = Tex(r"=\frac{\left|F_1F_2\right|}{\left|PF_1\right|+\left|PF_2\right|}").scale(0.75).next_to(e_d)#.move_to(stitle).shift(DOWN*2)
        self.play(TransformFromCopy(e_d, e_d2))

        # 焦点三角形
        M1 = Dot(color=YELLOW_D).move_to(ellipseL.pfp(0.3))
        M2 = Dot(color=YELLOW_D).move_to(ellipseL.pfp(0.58))
        CMF = Polygon(M1.get_center(), M2.get_center(), dF2.get_center()).set_color(YELLOW_D)
        CMF_t = Text("焦点三角形", font="HarmonyOS Sans SC").scale(0.85).move_to(line, aligned_edge=LEFT).shift(RIGHT*0.25).match_y(e_d).shift(DOWN)
        CMF_d = Tex(r"C_{\triangle ABF_2}=4a").scale(0.75).next_to(CMF_t)
        self.play(Write(CMF_t))
        self.play(ShowCreation(M1), ShowCreation(M2))
        self.play(ShowCreation(CMF))
        self.play(Write(CMF_d))
        self.wait(2)
        CMF_a = VGroup(M1, M2, CMF)
        self.play(*[FadeOut(mob) for mob in CMF_a])

        # 通经
        h = b*b/a
        dH1 = Dot(axesL.c2p(-f,h), color=YELLOW_D)
        dH2 = Dot(axesL.c2p(-f,-h), color=YELLOW_D)
        lH = Line(dH1, dH2).set_color(YELLOW_D)
        gH = VGroup(dH1, dH2, lH)
        h_t = Text("通径", font="HarmonyOS Sans SC").scale(0.85).move_to(line, aligned_edge=LEFT).shift(RIGHT*0.25).match_y(CMF_t).shift(DOWN)
        self.play(Write(h_t))
        self.play(ShowCreation(gH))
        h_d = Tex(r"=\frac{2b^2}{a}").scale(0.75).next_to(h_t)
        self.play(Write(h_d))
        self.wait(2)
        self.play(*[FadeOut(mob) for mob in gH])

        # 椭圆上一点到焦点性质
        P = Dot(color=YELLOW_D).move_to(ellipseL.pfp(0.15))
        P_l = Tex(r"P").scale(0.7).set_color(YELLOW_D).next_to(P)
        PF1 = Line(P, dF1).set_color(YELLOW_D)
        PF2 = Line(P, dF2).set_color(YELLOW_D)
        gP = VGroup(P, PF1, PF2, P_l)
        p_t = Text("椭圆上一点P", font="HarmonyOS Sans SC").scale(0.85).move_to(line, aligned_edge=LEFT).shift(RIGHT*0.25).match_y(h_t).shift(DOWN)
        self.play(ShowCreation(gP))
        self.play(Write(p_t))
        p_1 = Tex(r"\left|PF\right|\in\ \left[a-c,a+c\right]").scale(0.75).move_to(line, aligned_edge=LEFT).shift(RIGHT).match_y(p_t).shift(DOWN)
        self.play(Write(p_1))
        p_2 = Tex(r"\left|PF_1\right|\bullet\left|PF_2\right|=\left[b^2,a^2\right]").scale(0.75).move_to(line, aligned_edge=LEFT).shift(RIGHT).match_y(p_1).shift(DOWN)
        self.play(Write(p_2))
        self.wait(2)

        self.restore()
        
        # 面积
        self.play(ShowCreation(gP))
        a_1 = PF1.get_angle()
        a1=a_1
        a_2 = PF2.get_angle()
        a2 = a_2-a_1
        theta = Arc(start_angle=a1, angle=a2, arc_center=P.get_center(), radius=0.5)
        t_l = Tex(r"\theta").scale(0.7).set_color(GREEN).next_to(theta, DOWN)
        gt = VGroup(theta, t_l)
        self.play(ShowCreation(gt))
        S_d = Tex(r'S_{\triangle F_1PF_2}=b^2\bullet tan\frac{\theta}{2}', t2c={'\theta':GREEN}).scale(0.85).move_to(line, aligned_edge=LEFT).to_edge(UP).shift(DOWN*1.5+RIGHT*0.25)
        self.play(Write(S_d))
        func_derivation = VGroup(
            Tex(r"S_{\triangle F_1PF_2}=\frac{1}{2}\bullet\left|PF_1\right|\bullet\left|PF_2\right|\bullet\sin{\theta}", isolate=["\left|PF_1\right|\bullet\left|PF_2\right|"]).scale(0.6).match_x(stitle).to_edge(UP).shift(DOWN*2.5), #0
            Tex(r"\left|F_1F_2\right|^2=\left|PF_1\right|^2+\left|PF_2\right|^2-2\left|PF_1\right|\bullet\left|PF_2\right|\bullet\cos{\theta}", isolate=["\left|F_1F_2\right|^2=", "\left|PF_1\right|\bullet\left|PF_2\right|", "\bullet\cos{\theta}"]).scale(0.6).match_x(stitle).to_edge(UP).shift(DOWN*3.5),#1
            Tex(r"{\left|F_1F_2\right|^2=\left(\left|PF_1\right|+\left|PF_2\right|\right)}^2-2\left|PF_1\right|\bullet\left|PF_2\right|\bullet\left(1+\cos{\theta}\right)", isolate=["\left|F_1F_2\right|^2=", "\left|PF_1\right|\bullet\left|PF_2\right|", "\left(1+\cos{\theta}\right)"]).scale(0.5).match_x(stitle).to_edge(UP).shift(DOWN*4.5),#2
            Tex(r"{4c}^2={4a}^2-2\left|PF_1\right|\bullet\left|PF_2\right|\bullet\left(1+\cos{\theta}\right)", isolate=["\left|PF_1\right|\bullet\left|PF_2\right|", "\left(1+\cos{\theta}\right)"]).scale(0.6).match_x(stitle).to_edge(UP).shift(DOWN*5.5),
            Tex(r"\left|PF_1\right|\bullet\left|PF_2\right|=\frac{2b^2}{\left(1+\cos{\theta}\right)}").scale(0.6).match_x(stitle).to_edge(UP).shift(DOWN*6.5)
        )
        
        self.play(Write(func_derivation[0]))
        self.play(TransformFromCopy(func_derivation[0], func_derivation[1]))
        self.play(TransformFromCopy(func_derivation[1], func_derivation[2]))
        self.play(TransformFromCopy(func_derivation[2], func_derivation[3]))
        self.play(TransformFromCopy(func_derivation[3], func_derivation[4]))
        self.play(ShowCreationThenDestructionAround(func_derivation[4]))
        self.play(Indicate(S_d))
        self.wait(2)

        self.restore()
        self.add(P)
        PA = VGroup(
            Line(P, dA1),
            Line(P,dA2)
        )
        PA.set_color(YELLOW_D)
        PA_d = Tex(r"k_{PA_1}\bullet k_{PA_2}=-\frac{b^2}{a^2}").scale(0.8).move_to(line, aligned_edge=LEFT).to_edge(UP).shift(DOWN*1.5+RIGHT*0.25)
        self.play(ShowCreation(gA))
        self.play(ShowCreation(PA))
        
        self.play(Write(PA_d))
        self.wait(2)