Shapes Based On Parametric Equations: Epitrochoids

Epitrochoids are curves generated by a point selected on a circle of smaller radius rolling around the outside of a fixed circle of larger radius, and that selected point can be chosen at some distance from the center of the smaller circle. These are then 3-parametric curves. They can be helpful in creating decorative designs and other art applications, and can serve also as pastime and recreation for children. Since 1965 there is available a toy (geometric drawing device) known as spirograph, combining mathematics and art. It allows producing physically numerous curve designs based exactly on epitrochoid and hypotrochoid parametric equations.

However, nowadays the epitrochoids (as well as hypotrochoids - see another post at https://draft.blogger.com/blog/post/edit/3323809043368251287/9189797010242374274?hl=en-GB) can be generated quite easily in Excel by using VBA macro like the one provided at the end of this post.

Here are some examples of charts showing epitrochoid curves generated with the macro. Some of them present combined double curves.

Shapes Based On Parametric Equations: Hypocycloids

Hypocycloid is a parametric curve traced by a fixed point on a circle that rolls on the interior of another larger circle. 

The curve - due to its parametric functionality - finds many applications, both theoretical and practical. They include mechanical and construction engineering, e.g. design of gears, cams, valves, robotics (motion planning), and machine/structures design in general. Contribute also to aesthetically pleasing art designs and animations.

The hypocycloid curves can be generated quite easily in Excel with VBA macro,  presented at the end of this post. Here are some examples of charts with generated hypocycloid curves, both single and combined double curves: