# Graphing Calculator Drawing: Non-Euclidean Ellipses

*Non-euclidean ellipses can be used to draw starry-eye sunglasses.*

*This post is a chapter in the book Graphing Calculator Drawing Exercises.* *Suggested citation:* Skycak, J. (2019). Graphing Calculator Drawing: Non-Euclidean Ellipses. *Graphing Calculator Drawing Exercises.* https://justinmath.com/graphing-calculator-drawing-non-euclidean-ellipses/

** Setup.** Navigate to https://www.desmos.com/calculator. Be sure to sign in so that you can save your graph.

** Demonstration - Non-Euclidean Circles.** Observe the graph as you type each of the following inputs. In general, the graph of the unit circle is given by $\vert x\vert^n+ \vert y \vert^n=1.$ For $n=2,$ this makes a Euclidean circle, i.e. all those points whose distance from the origin is 1, where distance is measured by the Euclidean metric $\sqrt{x^2+y^2}.$ For other values of $n,$ these equations make non-Euclidean circles, i.e. all the points whose “distance” from the origin is 1, where distance is measured by the metric $\sqrt[n]{\vert x \vert^n+ \vert y\vert^n}.$

** Demonstration - Non-Euclidean Ellipses.** Observe the graph as you type each of the following inputs. In general, the graph of $\left\vert \frac{x-a}{A} \right\vert^n + \left\vert \frac{y-b}{B} \right\vert^n = 1$ makes an ellipse with horizontal radius $A$ and vertical radius $B$ centered at the point $(a,b).$ When $n \neq 2,$ this is a non-Euclidean ellipse.

** Exercise.** Reproduce the graph shown below using non-Euclidean circles.

** Exercise.** Change the non-Euclidean circles to non-Euclidean ellipses in the previous exercise to reproduce the graph shown below.

** Exercise.** Shift the ellipses right and up to produce the graph below.

** Exercise.** Create another set of ellipses, shifted right of the original set.

** Exercise.** Add some details to form a face. The head can be made using a non-Euclidean ellipse, the frame of the glasses can be made using a parabola and two lines, and the smile can be made using a parabola with sine shading.

** Exercise.** Lastly, add some hair on the head. You can do this by duplicating the biggest ellipse that outlines the face, restricting the range, and shading via sine.

** Challenge.** Try to make a narrower face with longer hair.

*This post is a chapter in the book Graphing Calculator Drawing Exercises.* *Suggested citation:* Skycak, J. (2019). Graphing Calculator Drawing: Non-Euclidean Ellipses. *Graphing Calculator Drawing Exercises.* https://justinmath.com/graphing-calculator-drawing-non-euclidean-ellipses/