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In this video Grant performs a non linear transform. The transform from cartesian to polar is also non-linear. We find the determinant of the jacobian matrix to be 1/r. I could replicate the transform that Grant did, but how to animate this cartesian to polar using manim. In other words, what would be f1 and f2. I tried (x^2 + y^2)^0.5 and arctan(y/x)

def construct(self):
    def arctan(x, y):
        if x and y == 0:
            return 0
        else:
            return np.arctan(y / x)

    grid = NumberPlane()
    grid.prepare_for_nonlinear_transform()
    self.play(
        grid.apply_function,
        lambda p: p + np.array([
            (p[0]**2 + p[1]**2)**0.5,
            arctan(p[0], p[1]),
            0,
        ]),
        run_time=3,
    )
Aravindh Vasu
  • 159
  • 7

1 Answers1

1
class AV(Scene):
    def construct(self):
        def polar2c(p):
            return np.array([
                p[0]*np.cos(p[1]),
                p[0]*np.sin(p[1]),
                0
                ])

        grid = NumberPlane(
            x_line_frequency=PI/4,
            y_line_frequency=PI/4,
            x_min=-PI,
            x_max=PI,
            y_min=-PI,
            y_max=PI
            )
        func = FunctionGraph(lambda x: 0.5*np.sin(5*x)+2,x_min=-PI,x_max=PI)
        grid.add(func)
        self.add(grid)
        grid.faded_lines[4:9].fade(1)
        grid.faded_lines[12:].fade(1)
        grid.background_lines[4:9].fade(1)
        grid.background_lines[12:].fade(1)
        self.play(Rotating(func,radians=PI,axis=UR,about_point=ORIGIN,run_time=2,rate_func=smooth))
        grid.generate_target()
        grid.target.prepare_for_nonlinear_transform()
        grid.target.apply_function(lambda p: polar2c(p))

        self.play(
            MoveToTarget(grid,run_time=4)
        )
        self.wait(3)
TheoremOfBeethoven
  • 1,864
  • 6
  • 15
  • Why we're rotating the `func` then apply the transformation on the grid? Isn't $r=\sqrt{(x^2+y^2)},\theta =\tan^{-1}\frac{y}{x}$? What if we want to apply linear transformation? @TheoremOfBeethoven. And thanks for your YouTube videos. I am learning from you a lot. – falamiw Apr 11 '21 at 19:58