How to Draw a Butt on a Graphing Calculator

TR Smith is a product designer and former teacher who uses math in her work every day.

Butts come in many shapes and sizes and there are just as many ways to draw a butt with equations on a graphing calculator. The most commonly used graphing calculators in high school and college are the TI 84 Plus and Hewlitt-Packard 50, but you can enter these equations on any device or online graphing calculator to create spectacular math butts.

Special thanks to my niece and nephew for requesting and helping with this short tutorial on graphing calculator art.

Basic Butts with Polar Coordinates

Polar coordinates are ideal for drawing many kinds of closed curves such as hearts and butterflies because a single equation will often trace out the entire figure. In fact, the equations to make butt graphs with polar coordinates are similar to the equations to make hearts and butterflies since you need to make two symmetric lobes in all three cases. One family of equations to draw a butt on a graphing calculator is

r(θ) = A + B * |cos(θ)| ^ (1/N)

where A, B, and N are positive constants, and N is greater than 1. The smaller the ratio A:B, the less crack there is. The larger the value of B, the larger the graph is. The larger the value of N, the closer together the cheeks are, while smaller values of N yield wider butts. Some examples are shown in the gallery below.

Click thumbnail to view full-size

Cartesian Butts

Drawing a butt in Cartesian coordinates (rectangular coordinates) on a graphing calculator using functions of the form y = f(x), is more cumbersome because it requires you to piece together many partial arcs of different curves. The simplest butt graph can be made by graphing two ellipses side by side. This requires four equations to get the top and bottom halves of each cheek.

A better way to graph a butt equation is to use a single implicit function of the form f(x, y) = 0. Implicitly defined functions can trace out closed curves with a single pass if you construct the function correctly. A simple example is

y^2 + x^2 - 10sqrt|x| = 4

The equation above can also be expressed as the pair y = sqrt( 4 - x^2 + 10sqrt|x| ) and y = -sqrt( 4 - x^2 + 10sqrt|x| ). Generally, you can create perfectly serviceable butts with implicit equations of the form

A*|y|^n + B*|y|^m + C*|x|^p + D*|x|^q = K

where A, B, C, D, n, m, p, q, and K are some constants whose variations will change the shape of the butt. Here are some more examples with this family of functions.

Click thumbnail to view full-size

Miscellaneous Butt Equations

Here are some other equations that you can enter on a graphing calculator to create butt-shaped curves.

The drawing below is comprised of six different functions of the form y=f(x).

The butt graph below is generated using a double periodic implicit function. Since there are only sine and cosine terms, the butts fill up the plane infinitely.

These graphs were created using Desmos Online Calculator. Other images from Pixabay public domain stock.

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• Rochelle Frank 22 months ago from California Gold Country

Truly amazing.

Even the non-math inclined among us have some knowledge of butts and cookies. I can see on the related hub subjects that hearts and butterflies have already been described mathematically.

• bryn 22 months ago

this is great! i didn't think that equations to draw a butt would be so similar to drawing a heart. what is the difference that makes one have horizontal reflective symmetry and the other a point?

• Author

TR Smith 22 months ago from Eastern Europe

Thanks for the comments. You're right that the equations for butts and hearts are very similar, especially in polar coordinates. The heart equations often longer periods. Taking the absolute value of a trig function halves the period.

• Alan 14 months ago from Tasmania

Regardless of the Butts, I am still waiting for the Ifs....lol!

calculus-geometry, you have a wonderful brain. I must revisit your hubs, please give me time.

• Dirk 4 months ago

how can i apply this to real world math problems?

• Author

TR Smith 4 months ago from Eastern Europe

I suppose the most salient application of this turorial is...drawing a butt on a graphing calculator.

• Alan 4 months ago from Tasmania

We often find ourselves looking at butts on giraffing escalators.