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: Epicycloids

Epicycloid is a parametric geometric curve obtained by tracing the path of a chosen point on the circumference of a circle (outside of it). Variety of epicycloid curves find applications in mechanical and construction engineering, e.g. construction of gears, cams, valves, pendulum clocks, robotic actuators, and machine/structures design in general. Contribute also to making designs of arts and animations.

In this context, it is useful to know how to generate epicycloid curves. In this post I'm presenting some of the curves along with the Excel VBA macro for creating this kind of curves. You can try to use it for your own creations. Just copy it to one of the modules in your workbook sheets and experiment with different settings and parameters.

Here are some examples of charts with epicycloid curves, single and double plots. 

 

Creative Charts - Excel Art

These are some of my Excel creations: charts based on the equation of circle. Unlimited Excel...

 

Creating Surface Charts 3D

Surface charts in Excel deserve special attention. They allow to see your data in three dimensions and can be particularly useful in engineering applications (design and control of processes) as well as in the art of drawing and painting.

Typical surface chart shows how the dependent variable (z) behaves as a function of two independent variables (x & y). In other words, the chart illustrates outcome of any nonlinear function involving two variables. E.g. we can plot speed of some technological process (reaction) as a function of pressure and temperature.

In this post I'll show how you can prepare a template for creating surface charts based on equations including variety of math functions (algebraic, transcendental, periodic). Such template can be very useful in all kinds of design, as it allows you to visualize plenty of solutions in minutes.

Let's start with just one example of surface chart I've created. This is 3D plot of equation shown in the title of this chart:

 

How can you create similar charts based on your own equations?