Graphing Calculator Drawing: Non-Euclidean Ellipses
Non-euclidean ellipses can be used to draw starry-eye sunglasses.
This post is part of the book Graphing Calculator Drawing Exercises. Suggested citation: Skycak, J. (2019). Graphing Calculator Drawing: Non-Euclidean Ellipses. In 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.
![image](https://justinmath.com/files/blog/graph-non-euclidean-circles.png)
Exercise. Change the non-Euclidean circles to non-Euclidean ellipses in the previous exercise to reproduce the graph shown below.
![image](https://justinmath.com/files/blog/graph-non-euclidean-ellipses.png)
Exercise. Shift the ellipses right and up to produce the graph below.
![image](https://justinmath.com/files/blog/graph-non-euclidean-ellipses-shifted.png)
Exercise. Create another set of ellipses, shifted right of the original set.
![image](https://justinmath.com/files/blog/graph-starry-eyes.png)
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.
![image](https://justinmath.com/files/blog/graph-starry-eyes-face.png)
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.
![image](https://justinmath.com/files/blog/graph-starry-eyes-face-hair.png)
Challenge. Try to make a narrower face with longer hair.
This post is part of the book Graphing Calculator Drawing Exercises. Suggested citation: Skycak, J. (2019). Graphing Calculator Drawing: Non-Euclidean Ellipses. In Graphing Calculator Drawing Exercises. https://justinmath.com/graphing-calculator-drawing-non-euclidean-ellipses/