diff options
Diffstat (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Geometric_Proof.py')
| -rw-r--r-- | FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Geometric_Proof.py | 89 |
1 files changed, 89 insertions, 0 deletions
diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Geometric_Proof.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Geometric_Proof.py new file mode 100644 index 0000000..2c1d668 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Geometric_Proof.py @@ -0,0 +1,89 @@ +from manimlib.imports import* + +#---- visualization of geometric proof of Lagrange multiplier +class firstScene(ThreeDScene): + def construct(self): + axes = ThreeDAxes().scale(0.7).rotate(math.radians(180)) + label_x = TextMobject("$x$").shift(4*RIGHT).fade(0.4) #---- x axis + label_y = TextMobject("$y$").shift(3.2*DOWN+0.2*RIGHT).rotate(math.radians(180)).fade(0.4) #---- y axis + + surface = ParametricSurface( + lambda u, v: np.array([ + 1*np.sin(u)*np.cos(v), + 1*np.sin(u)*np.sin(v), + -1*np.sin(u)*np.sin(u)+2 + ]),u_min=0,u_max=PI/2,v_min=0,v_max=2*PI).set_color(GREEN).scale(1).shift([-1.5,-1.5,0]) + + d = Dot([-2,-2.55,0],color = '#800000') + a_df = Arrow(color = '#00FFFF').rotate(-2).shift(3.2*DOWN+2.3*LEFT) #---- f parallel to g + a_dg = Arrow(color = '#FF00FF').scale(0.8).shift(3.2*DOWN+2.3*LEFT).rotate(-2) #---- f parallel to g + + b_dg = Arrow(color = '#00FFFF').rotate(1.1).shift(0.82*LEFT+0.15*UP) #---- f parallel to g + b_df = Arrow(color = '#FF00FF').scale(0.6).rotate(-2).shift(1.43*LEFT+1.1*DOWN) #---- f parallel to g + + + qd = Dot(color = '#800000').shift(1.2*LEFT+0.6*DOWN) + + l1 = Line([-1,-3.1,0],[-4,-3.1,0],color = PINK).rotate(-0.3).fade(0.6) + l2 = Line([-0.9,-2.9,0],[-4,-2.9,0],color = PINK).rotate(-0.3).fade(0.6) + l3= Line([-0.8,-2.7,0],[-4,-2.7,0],color = PINK).rotate(-0.3).fade(0.6) + l4= Line([-0.7,-2.45,0],[-4,-2.45,0],color = PINK).rotate(-0.3).fade(0.6) + l5= Line([-0.6,-2.2,0],[-4,-2.25,0],color = PINK).rotate(-0.3).fade(0.6) + l6 = Line([-0.5,-2,0],[-4,-2,0],color = PINK).rotate(-0.3).fade(0.6) + l7 = Line([-0.4,-1.8,0],[-4,-1.8,0],color = PINK).rotate(-0.3).fade(0.6) + l8 = Line([-0.3,-1.6,0],[-4,-1.6,0],color = PINK).rotate(-0.3).fade(0.6) + l9= Line([-0.2,-1.4,0],[-4,-1.4,0],color = PINK).rotate(-0.3).fade(0.6) + l10= Line([-0.1,-1.2,0],[-4,-1.2,0],color = PINK).rotate(-0.3).fade(0.6) + l11 = Line([-0,-1,0],[-4,-1,0],color = PINK).rotate(-0.3).fade(0.6) + l12 = Line([-0,-0.8,0],[-4,-0.8,0],color = PINK).rotate(-0.3).fade(0.6) + l13= Line([-0,-0.55,0],[-4,-0.55,0],color = PINK).rotate(-0.3).fade(0.6) + l14= Line([-0,-0.35,0],[-4,-0.35,0],color = PINK).rotate(-0.3).fade(0.6) + l15= Line([-0.,-0.15,0],[-4,-0.15,0],color = PINK).rotate(-0.3).fade(0.6) + + rel_text = TextMobject("$\\nabla f = \\lambda \\nabla g$",color = TEAL).shift([3,3.2,0]).scale(0.5) + + f_text = TextMobject("$\\nabla f$",color = '#800000').shift([1,1,0]).scale(0.5) + g_text = TextMobject("$\\nabla g$").shift([1.2,-0.8,0]).scale(0.5) + + p_text= TextMobject("$P$").shift([1.8,2.6,0]).scale(0.5) + + l1_text = TextMobject("$w=$ 17").rotate(math.radians(180)).scale(0.4).shift(2.7*DOWN+4.36*LEFT) + l2_text = TextMobject("$w=$ 16").rotate(math.radians(180)).scale(0.4).shift(2.46*DOWN+4.36*LEFT) + l3_text = TextMobject("$w=$ 15").rotate(math.radians(180)).scale(0.4).shift(2.2*DOWN+4.36*LEFT) + l4_text = TextMobject("$w=$ 14").rotate(math.radians(180)).scale(0.4).shift(1.97*DOWN+4.36*LEFT) + l5_text = TextMobject("$w=$ 13").rotate(math.radians(180)).scale(0.4).shift(1.74*DOWN+4.36*LEFT) + l6_text = TextMobject("$w=$ 12").rotate(math.radians(180)).scale(0.4).shift(1.5*DOWN+4.36*LEFT) + l7_text = TextMobject("$w=$ 11").rotate(math.radians(180)).scale(0.4).shift(1.26*DOWN+4.36*LEFT) + l8_text = TextMobject("$w=$ 10").rotate(math.radians(180)).scale(0.4).shift(1.05*DOWN+4.36*LEFT) + l9_text = TextMobject("$w=$ 9").rotate(math.radians(180)).scale(0.4).shift(0.8*DOWN+4.32*LEFT) + l10_text = TextMobject("$w=$ 8").rotate(math.radians(180)).scale(0.4).shift(0.6*DOWN+4.32*LEFT) + l11_text = TextMobject("$w=$ 7").rotate(math.radians(180)).scale(0.4).shift(0.4*DOWN+4.32*LEFT) + l12_text = TextMobject("$w=$ 6").rotate(math.radians(180)).scale(0.4).shift(0.2*DOWN+4.32*LEFT) + l13_text = TextMobject("$w=$ 5").rotate(math.radians(180)).scale(0.4).shift(-0.02*DOWN+4.32*LEFT) + l14_text = TextMobject("$w=$ 4").rotate(math.radians(180)).scale(0.4).shift(-0.23*DOWN+4.32*LEFT) + l15_text = TextMobject("$w=$ 3").rotate(math.radians(180)).scale(0.4).shift(-0.44*DOWN+4.32*LEFT) + + level_Curve = VGroup(l1,l1_text,l2,l2_text,l3,l3_text,l4,l4_text,l5,l5_text,l6,l6_text,l7,l7_text,l8,l8_text,l9,l9_text,l10,l10_text,l11,l11_text,l12,l12_text,l13,l13_text,l14,l14_text,l15,l15_text) + + self.set_camera_orientation(phi=0 * DEGREES, theta = 90*DEGREES) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.wait(1) + self.add(surface) + self.wait(1) + self.play(ShowCreation(level_Curve)) + self.wait(1) + self.play(ShowCreation(a_df),ShowCreation(a_dg),Write(d)) + self.wait(1) + self.add_fixed_in_frame_mobjects(rel_text) + self.add_fixed_in_frame_mobjects(p_text) + self.wait(1) + self.play(Write(qd)) + self.wait(1) + self.play(ShowCreation(b_df)) + self.add_fixed_in_frame_mobjects(f_text) + self.wait(1) + self.play(ShowCreation(b_dg)) + self.add_fixed_in_frame_mobjects(g_text) + self.wait(1) |