# Graphing Calculator Drawing: Euclidean Ellipses

*Euclidean ellipses can be combined with sine wave shading to form three-dimensional shells.*

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

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

** Demonstration - Circles.** Observe the graph as you type each of the following inputs. In general, the graph of $(x-a)^2+(y-b)^2=r^2$ makes a circle with radius $r$ centered at the point $(a,b).$

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

** Demonstration - Shading.** Observe the graph as you type each of the following inputs. In general, for large $v,$ the graph of $\left( \frac{x-a}{A} \right)^2 + \left( \frac{y+A\sin(vx)-b}{B} \right)^2 = 1$ shades vertically around the graph of $\left( \frac{x-a}{A} \right)^2 + \left( \frac{y-b}{B} \right)^2 = 1$ with thickness $2A.$

** Exercise.** Use an absolute value function together with an ellipse to draw a cone.

** Exercise.** Stack ellipses vertically on the cone.

** Exercise.** Thicken the ellipses to form cylindrical shells.

** Challenge.** Try stacking cylindrical shells on the peaks of $y=\sin x.$

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