How to Put A Heart in A Desmos Graphing Calculator

Creating a heart shape in Desmos is a great way to visualize mathematical functions and explore parametric equations. This guide will walk you through the process step-by-step, from the basic equation to advanced customizations.

Introduction

Desmos is a powerful graphing calculator that allows users to create and explore mathematical functions visually. One of the most visually appealing shapes you can create is a heart. This guide will show you how to put a heart in Desmos using parametric equations.

Parametric equations allow you to define a curve using two separate functions for x and y coordinates, which is perfect for creating complex shapes like hearts. By combining trigonometric functions and polynomial terms, you can create a heart shape that looks realistic.

Basic Heart Equation

The most common parametric equation for a heart is:

x = 16 sin³(t) y = 13 cos(t) - 5 cos(2t) - 2 cos(3t) - cos(4t)

This equation uses trigonometric functions to create the characteristic shape of a heart. The x-coordinate is defined by a cubic sine function, while the y-coordinate combines several cosine terms to create the rounded and pointed parts of the heart.

Step-by-Step Implementation

Open the Desmos graphing calculator in your web browser.
Click on the "New Expression" button or press Enter to create a new expression.
Enter the x-coordinate equation: x = 16 sin³(t)
Press Enter and create another expression for the y-coordinate: y = 13 cos(t) - 5 cos(2t) - 2 cos(3t) - cos(4t)
Adjust the slider for t (theta) to see the heart shape being drawn.

The heart will appear in the graphing area, with the slider controlling the parameter t that traces out the curve. You can adjust the range of t to see the complete heart shape.

Customizing the Heart

Once you have the basic heart equation working, you can customize it in several ways to create different effects.

Changing the Size

To make the heart larger or smaller, you can multiply the entire equation by a scaling factor. For example:

x = 2 * 16 sin³(t) y = 2 * (13 cos(t) - 5 cos(2t) - 2 cos(3t) - cos(4t))

This will double the size of the heart. You can adjust the scaling factor to make the heart as large or small as you need.

Changing the Color

Desmos allows you to change the color of the heart by clicking on the expression and selecting a different color from the color palette.

Adding Animation

You can create an animated heart by adding a time-dependent term to the equation. For example:

x = 16 sin³(t) + 2 sin(5t) y = 13 cos(t) - 5 cos(2t) - 2 cos(3t) - cos(4t) + 2 cos(5t)

This will make the heart pulse or move in a more dynamic way.

Advanced Techniques

For more advanced users, there are several techniques you can use to create even more complex heart shapes.

Combining Multiple Hearts

You can create a scene with multiple hearts by defining additional parametric equations with different parameters. For example:

x1 = 16 sin³(t) y1 = 13 cos(t) - 5 cos(2t) - 2 cos(3t) - cos(4t) x2 = 16 sin³(t) + 5 y2 = 13 cos(t) - 5 cos(2t) - 2 cos(3t) - cos(4t)

This will create two hearts side by side.

Creating a Heartbeat Effect

You can simulate a heartbeat effect by making the heart pulse in size. One way to do this is to use a time-dependent scaling factor:

scale = 1 + 0.2 sin(3t) x = scale * 16 sin³(t) y = scale * (13 cos(t) - 5 cos(2t) - 2 cos(3t) - cos(4t))

This will make the heart gently pulse in size.

FAQ

Can I create a heart shape without using parametric equations?: Yes, you can create a heart shape using Cartesian equations, but parametric equations are generally more flexible and easier to work with for complex shapes.
How can I make the heart fill in?: In Desmos, you can fill in the area under a curve by selecting the expression and choosing the "Fill" option from the formatting menu.
Can I create a 3D heart in Desmos?: Desmos primarily works with 2D graphs, but you can create the illusion of a 3D heart by using shading and multiple layers.
How can I save my heart graph?: You can save your Desmos graph by clicking on the "Save" button in the top-right corner of the screen. This will create a link that you can share with others.
Can I export the heart graph as an image?: Yes, you can export your Desmos graph as an image by clicking on the "Download" button in the top-right corner of the screen and selecting the image format you prefer.