Parametric curves

Pretty math

Parametric curves are curve where the x and y positions are defined by functions eg. $(\cos(3t),\sin(2t))$. I have used them for years to make interesting patterns and animations with desmos and python. I use them in laser cutting, rendering, puzzle box, the sebsite logo, and CNC.

Desmos dots

The first cool thing I did with parametric functions was in desmos, an excellent graphing calculator. I defined a parametric curve as a path and them made a lot of points follow that path all at slightly faster speeds. This is super simple but the emergent behaviors are beautiful to watch. You can play with my graph here.

Rendering

Python + rendering engine > graphing calculator

When I started teaching myself python I quickly realized I could do so much more than with a graphing calculator, which lead to me finding a rendering engine to use with python. I did some of the same 2D things, but was also able to expand into 3D parametric curves and model waves in 2D planes and 3D volumes. Look at some of these renderings on my rendering page.

Every ball simply moves in a circle, and the outer balls circle faster than the inner balls.

Python DXF

Laser cutting is excellent because anything that you can make into a DXF file you can use. I wanted to laser cut parametric curves so I figured out a way to go from an equation to a cuttable DXF file using python and a module called ezdxf. The first version worked by approximating parametric curves using a lot of really small straight lines, then I improved this to make a fitted curve through a bunch of points on the parametric curve.

To create these curves I define an equation with constants and then vary the constants until I find something cool. For example: $$ (X,Y) \\ X=\sin(t)+\frac{\sin(at)}{2}+ \frac{\sin(bt)}{4} \\ Y=\cos(t)+\frac{\cos(at)}{2}+ \frac{\cos(bt)}{4}\\ $$ But this is slow. So I simply added a few lines to my python DXF program to loop through 40 values of a and b between 0 and 20. This created 1600 curves at a time! Then, I just had to peruse them and find my favorites.

Coasters

These coasters have all been laser cut with parametric designs I created.