Project: PlotEquation

Feb 19

Click to see gallery of full-sized images…

Introduction

It was my sophomore year in high school and it was my first engineering class I’ve ever taken. Mr. Niebergall, my engineering teacher, had mentioned to me that he is looking for a solution in bridging the gap between teaching engineering and teaching math, and that it would be a really nice addition to the CAD software Rhinoceros3D if it could generate math equations (much like Desmos) that students can then laser-cut and 3D-print. With my trigonometry-level math education (at the time) and a passion for programming, I discovered that making plugins in Rhino is really easy and well within my capabilities to provide a solution for my teacher. Within a week, I had a simple 2D cartesian equation grapher working (curves in the form y=f(x)), but I quickly realized that extending the plugin’s capabilities was really easy to do. By the next day, the plugin was able to graph 2D spherical equations as well (curves in the form r=f(Θ)), but this cracked open the door of opportunity even further as I realized extending these curves to 3D surfaces was really easy to do in Rhino. By the end of the next week, I had a fully functional 2D and 3D grapher (much like Geogebra) that could plot cartesian, spherical, cylindrical, and parametric curves and surfaces, and was able to 3D print my first mathematical object: a small 3D paraboloid. But I didn’t just print it out for fun: I wanted to see how strong it was and how much weight it could hold up. 473 pounds later I had run out of space to add any more weights from the weight room, and I realized the potential for this plugin was limitless. Flash forward 2 years later during my senior year of high school, and this plugin evolved to be much more than a simple equation grapher: it could now apply stereographic projections, generate various fractal objects, and more - far exceeding my engineering teacher’s original expectations. And thanks to this plugin, I was able to learn C# as well as gain a much more advanced understanding of mathematical concepts (such as quaternion algebra) well before I would be formally introduced to them.

x = (2 + sin(7u + 5v))cos(u)sin(v), y = (2 + sin(7u + 5v))sin(u)sin(v), z = (2 + sin(7u + 5v))cos(v)-min.PNG

Features

Surfaces

Cartesian Expressions: x=f(x,y); y=f(x,z); z=f(x,y)

Spherical Expressions: r=f(Θ,φ); Θ=f(r,φ); φ=f(r,Θ)

Cylindrical Expressions: r=f(Θ,z); Θ=f(r,z); z=f(r,Θ)

Parametric Expressions:
Cartesian: x=f(u,v), y=f(u,v), z=f(u,v)
Spherical: r=f(u,v), Θ=f(u,v), φ=f(u,v)
Cylindrical: r=f(u,v), Θ=f(u,v), z=f(u,v)

Implicit Expressions: 0 = f(x,y,z,r,Θ,φ)

Curves

Cartesian Expressions: x=f(y); y=f(x)

Polar Expressions: r=f(Θ); Θ=f(r)

Parametric Expressions (3D optional):
Cartesian: x=f(t), y=f(t), z=f(t)
Spherical: r=f(t), Θ=f(t), φ=f(t)
Cylindrical: r=f(t), Θ=f(t), z=f(t)

Implicit Expressions:
Cartesian: 0=f(x,y)
Polar: 0=f(r,Θ)

Miscellaneous

Stereographic Projections

Inverse Stereographic Projections

“4D” Expressions:
Cartesian: x=f(w,x,y); y=f(w,x,z); z=f(w,x,y)
Spherical: r=f(s,Θ,φ); Θ=f(s,r,φ); φ=f(s,r,Θ)
Cylindrical: r=f(s,Θ,z); Θ=f(s,r,z); z=f(s,r,Θ)

“4D” Parametric Expressions:
Cartesian: x=f(w,x,y); y=f(w,x,z); z=f(w,x,y)
Spherical: r=f(s,Θ,φ); Θ=f(s,r,φ); φ=f(s,r,Θ)
Cylindrical: r=f(s,Θ,z); Θ=f(s,r,z); z=f(s,r,Θ)