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)