How to create animations of elliptic curves in manim?

Question

I'm playing with manim and I'd like to create an animation with an elliptic curve. This is my code, in the file functions.py:

from manim import *

class EllipticCurve(Scene):

    def construct(self):
        basic_ec = FunctionGraph(
           lambda x: x**1.5 - x**0.5 + 19**0.5
        )

        self.play(Create(basic_ec))

When I execute this command manim -pql functions.py EllipticCurve, I get the following error:

ValueError: array must not contain infs or NaNs

I believe the the method FunctionGraph expects a function, instead of a curve, but how can I animate and plot an elliptic curve? Is there any other method? Am I missing something?

`lambda x: x**1.5 - x**0.5 + 19**0.5` is not an elliptic curve. — President James K. Polk, May 12 '22 at 21:50

score 0 · Answer 1 · answered May 11 '22 at 22:38

0

You are passing the function to FunctionGraph correctly, the problem is that if you don't explicitly specify a x_range for the plot, Manim will choose the range [-config.frame_x_radius, config.frame_x_radius] (that is, it spans over the entire width of the frame; by default from -7.11 to +7.11).

Plugging in negative values in your function is problematic, and so Manim complains. Either pass x_range=[0, 7] to FunctionGraph, or check out ImplicitFunction (which to me seems more useful for working with elliptic curves).

And one final hint: It can be a bit tricky to get sane scaling for FunctionGraph, you might want to consider creating an Axes mobject and then use the corresponding Axes.plot or Axes.plot_implicit methods.

answered May 11 '22 at 22:38

Benjamin Hackl

1,011
2
8

Thank you for the answer. I’d need to plot negative values too. Is there any way to plot the EC with its Weierstrass form: `y^2 = x^3 - x + 19` ? – Hyperion May 12 '22 at 12:34
1

@Alejandro Yes, this is what the linked [`ImplicitFunction`](https://docs.manim.community/en/stable/reference/manim.mobject.graphing.functions.ImplicitFunction.html?highlight=ImplicitFunction#manim.mobject.graphing.functions.ImplicitFunction) does. – Benjamin Hackl May 12 '22 at 13:20

score 0 · Answer 2 · answered Jun 18 '22 at 13:21

The following worked for me:

#!/usr/bin/env python
"""
# python -m manim --quality l simple_ec.py EllipticCurve1 -p
"""
from manim import *

class EllipticCurve(Scene):
    def construct(self):
        ax = Axes(x_range=[-3.2, 6.2, 1], y_range=[-21, 21, 10]
                  , x_length=14         , y_length=7.5
                  , color=BLUE
                  , x_axis_config={"numbers_to_include": range(-3, 6 + 1, 1),
                                   "font_size": 24,}
                  , y_axis_config={"numbers_to_include": range(-20, 20 + 10, 10),
                                   "font_size": 24,}
                  , tips=False)
        
        a = ax.plot_implicit_curve(lambda x, y: -y**2 + x**3 - x + 19
                                   , color=YELLOW)

        plane = NumberPlane(x_range=[-3.2, 6.2, 1], y_range=[-21, 21, 10]
                            , x_length=14         , y_length=7.5
                            , color=GRAY)

        self.add(ax, a, plane)
        self.wait(5)

It uses an implicit plot, the method plot_implicit_curve of the axes object ax, and is producing:

Note: The wanted elliptic curve was extracted from the comments. It is the curve y² = x³ - x + 19 . (Of course, we cannot isolate y by extracting the square root "termwise" from the R.H.S. - whatever the signs of the terms may be...) The curve (seen over the rationals) has no torsion points, its rank is one, and a generator is (2, 5).

How to create animations of elliptic curves in manim?

2 Answers2