Hearts Equations In Math – Mattia Giuri's bizarre blog

In this article, I’ll show you five different equations you can use to plot your own heart.

Let’s start with the simplest:

$(x^2+y^2-1)^3 -x^2y^3 = 0$

I think this is the most classic of the five I’m going to show you.

If you prefer one which is a little bit thinner, there is this one:

$(y-\frac{3}{4} x^{\frac{2}{3}})^2 = 1 - x^2$

On the other hand, there is also one that is fatter:

$x^2 + 2(\frac{2}{5}x^{\frac{2}{5}}-y)^2 - 1 = 0$

Python Implementation For Plotting Two Parametric Hearts

To plot the hearts, I’m using the numpy and matplotlib libraries of Python.

The first parametric equation is:

$\begin{cases} x = +- (-t^2+40t+1200)\sin{\frac{t}{90}} \\ y = (-t^2+40t+1200)\cos{\frac{t}{60}} \end{cases}$

Where

$0 \leq t \leq 60$

Its code implementation is:

import numpy as np
import matplotlib.pyplot as plt


def heart_plotter(precision):
    t = np.linspace(0, 60, precision)
    X = 1/100*(-t**2 + 40*t + 1200)*np.sin(t/90)
    Y = 1/100*(-t**2 + 40*t + 1200)*np.cos(t/60)
    Y_1 = 1/100*(-t**2 + 40*t + 1200)*np.cos(t/60)
    X_1 = -1/100*(-t**2 + 40*t + 1200)*np.sin(t/90)

    X = np.concatenate((X, np.flip(X_1)))
    Y = np.concatenate((Y, np.flip(Y_1)))

    return X, Y


x, y = heart_plotter(1000)
plt.plot(x, y, c='r')
plt.show()

A very classic shape. As you can see it’s very easy to perform calculations with numpy arrays and its builtin functions.

Warning: if you don’t reverse the order of the arrays before concatenating (np.flip), the line x=0 would appear in the plot connecting the two ‘cusps’ of the heart, because the plot function of matplotlib would think there’s a gap in the graph of the equation.

The second parametric equation is the one I prefer:

$\begin{cases} x = \sin{t}\cos{t}\log |t| \\ y = \sqrt{|t|} \cos{t} \end{cases}$

Where

$-1 \leq t< 0, 0<t \leq 1$

Its code implementation is:

import numpy as np
import matplotlib.pyplot as plt


def heart_plotter(precision):
    t = np.linspace(-1, 1, 2*precision+1)
    X = np.sin(t)*np.cos(t)*np.log(np.abs(t))
    Y = np.sqrt(np.abs(t))*np.cos(t)

    return X, Y


x, y = heart_plotter(1000)
plt.plot(x, y, c='r')
plt.show()

I really like the shape of this heart, it seems like it’s constantly expanding.

I’m using 2*precision + 1 because, this way, there is that little gap near 0. If precision was even, there would be a line joining the two borders, which is not very aesthetically pleasing.