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from manimlib.imports import*
import math as m
class Minima(ThreeDScene):
def construct(self):
heading = TextMobject("Nondegenerate Hessian Matrix",color = BLUE)
axes = ThreeDAxes()
label_x = TextMobject("$x$").shift([5.5,-0.3,0]) #---- x axis
label_y = TextMobject("$y$").shift([-0.3,5.5,0]).rotate(-4.5) #---- y axis
h_text = TextMobject("Case 1: $\\frac{\\partial^2 f}{\\partial x^2}>0$ and $\\frac{\\partial^2 f}{\\partial y^2}>0$").scale(1)
#---- determiniant of Hessian Matrix
hessian_surface = ParametricSurface(
lambda u, v: np.array([
u,
v,
-0.5*m.exp(-u**2-v**2)
]),u_min = -PI, u_max = PI, v_min = -PI, v_max =PI).set_color(TEAL).shift([0,0,0]).scale(1).fade(0.2)
det_text= TextMobject("$det \\hspace{1mm} H = (\\frac{\\partial^2 f}{\\partial x^2})(\\frac{\\partial^2 f}{\\partial y^2})-(\\frac{\\partial^2 f}{\\partial x \\partial y})^2 $").to_corner(UL).scale(0.7)
#---- function f(x,y)
f_surface = ParametricSurface(
lambda u, v: np.array([
u,
v,
u**2+v**2
]),u_min = -1.3, u_max = 1.3, v_min = -1.3, v_max = 1.3).set_color(TEAL).shift([0,0,-0.5])
f_text= TextMobject("surface of the function").to_corner(UL).scale(0.8)
d = Dot(color = "#800000").shift([0,0,-0.52]) #---- critical point
self.set_camera_orientation(phi = 75*DEGREES, theta = 40*DEGREES)
self.add_fixed_in_frame_mobjects(heading)
self.wait(1)
self.play(FadeOut(heading))
self.wait(1)
self.add_fixed_in_frame_mobjects(h_text)
self.wait(1)
self.play(FadeOut(h_text))
self.wait(1)
self.add(axes)
self.add(label_x)
self.add(label_y)
self.play(Write(hessian_surface))
self.wait(1)
self.add_fixed_in_frame_mobjects(det_text)
self.move_camera(phi = 90*DEGREES, theta= 60*DEGREES)
self.play(Write(d))
self.wait(1)
self.play(FadeOut(det_text),ReplacementTransform(hessian_surface,f_surface))
self.wait(1)
self.add_fixed_in_frame_mobjects(f_text)
self.wait(1)
self.play(FadeOut(f_text),FadeOut(f_surface),FadeOut(axes),FadeOut(label_x),FadeOut(label_y),FadeOut(d))
class Maxima(ThreeDScene):
def construct(self):
axes = ThreeDAxes()
label_x = TextMobject("$x$").shift([5.5,-0.3,0]) #---- x axis
label_y = TextMobject("$y$").shift([-0.3,5.5,0]).rotate(-4.5) #---- y axis
h_text = TextMobject("Case 2: $\\frac{\\partial^2 f}{\\partial x^2}<0$ and $\\frac{\\partial^2 f}{\\partial y^2}<0$").scale(1)
#---- determiniant of Hessian Matrix
hessian_surface = ParametricSurface(
lambda u, v: np.array([
u,
v,
0.5*m.exp(-u**2-v**2)
]),u_min = -PI, u_max = PI, v_min = -PI, v_max =PI).set_color(TEAL).shift([0,0,0]).scale(1).fade(0.2)
det_text= TextMobject("$det \\hspace{1mm} H = (\\frac{\\partial^2 f}{\\partial x^2})(\\frac{\\partial^2 f}{\\partial y^2})-(\\frac{\\partial^2 f}{\\partial x \\partial y})^2 $").to_corner(UL).scale(0.7)
#---- function g(x,y)
g_surface = ParametricSurface(
lambda u, v: np.array([
u,
v,
-u**2-v**2
]),u_min = -1.3, u_max = 1.3, v_min = -1.3, v_max = 1.3).set_color(TEAL).shift([0,0,0.5])
g_text= TextMobject("surface of the function").to_corner(UL).scale(0.8)
d = Dot(color = "#800000").shift([0,0,0.5]) #---- critical point
self.set_camera_orientation(phi = 75*DEGREES, theta = 40*DEGREES)
self.add_fixed_in_frame_mobjects(h_text)
self.wait(1)
self.play(FadeOut(h_text))
self.wait(1)
self.add(axes)
self.add(label_x)
self.add(label_y)
self.play(Write(hessian_surface))
self.wait(1)
self.add_fixed_in_frame_mobjects(det_text)
self.play(Write(d))
self.wait(1)
self.play(FadeOut(det_text),ReplacementTransform(hessian_surface,g_surface))
self.wait(1)
self.add_fixed_in_frame_mobjects(g_text)
self.wait(1)
self.play(FadeOut(g_text),FadeOut(g_surface),FadeOut(axes),FadeOut(label_x),FadeOut(label_y),FadeOut(d))
class SaddlePoint(ThreeDScene):
def construct(self):
axes = ThreeDAxes()
label_x = TextMobject("$x$").shift([5.5,-0.3,0]) #---- x axis
label_y = TextMobject("$y$").shift([-0.3,5.5,0]).rotate(-4.5) #---- y axis
h_text = TextMobject("Case 3: $\\frac{\\partial^2 f}{\\partial x^2}$ and $\\frac{\\partial^2 f}{\\partial y^2}$ have opposite signs").scale(1)
#---- determiniant of Hessian Matrix
hessian_surface = ParametricSurface(
lambda u, v: np.array([
u,
v,
m.exp(0.5*u**2-0.5*v**2)
]),u_min = -1.2, u_max = 1.2, v_min = -2.5, v_max = 2.5).set_color(TEAL).shift([0,0,-1]).scale(1).fade(0.2)
det_text= TextMobject("$det \\hspace{1mm} H = (\\frac{\\partial^2 f}{\\partial x^2})(\\frac{\\partial^2 f}{\\partial y^2})-(\\frac{\\partial^2 f}{\\partial x \\partial y})^2 $").to_corner(UL).scale(0.7)
#---- function p(x,y)
p_surface = ParametricSurface(
lambda u, v: np.array([
u,
v,
u**2-v**2
]),u_min = -1, u_max = 1, v_min = -1, v_max =1).set_color(TEAL).shift([0,0,0]).scale(2)
p_text= TextMobject("surface of the function").to_corner(UL).scale(0.8)
d = Dot(color = "#800000").shift([0,0,0]) #---- critical point
self.set_camera_orientation(phi = 80*DEGREES, theta = 60*DEGREES)
self.add_fixed_in_frame_mobjects(h_text)
self.wait(1)
self.play(FadeOut(h_text))
self.wait(1)
self.add(axes)
self.add(label_x)
self.add(label_y)
self.wait(1)
self.play(Write(hessian_surface))
self.play(Write(d))
self.wait(1)
self.add_fixed_in_frame_mobjects(det_text)
self.wait(2)
self.play(FadeOut(det_text),ReplacementTransform(hessian_surface,p_surface))
self.add_fixed_in_frame_mobjects(p_text)
self.wait(2)
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