Drawing Super Circles

Everyone knows how to draw a circle or a square. An oval is just an elongated circle just as a rectangle is an elongated square. A diamond is also straightforward to draw. These shapes, along with many others are tied together in what I call the theory of super circles. This Instructable will show just how to do it.

Step 1: X Radius, Y Radius, and Poly Order

A super circle is defined by its parmeters; X radius, Y radius, and Polynomial order. Don't let the name scare you, the Poly Order is a number that tells you how puffy the circle is.

PolyDraw

To start. go to PolyDraw and set the # of Polys window to "1". You will get a screen with a single poly with; an X Radius of 4, a Y Radius of 4, and a Poly Order of 2" . Click Draw and you will get a PDF with a circle have a diameter of 8" (4" radius).

Now change the Y Radius value to 5 and hit draw. You now have a oval.

Change the Poly Order to 1 the new PDF will have a diamond.

Change the Poly Order to 100 and you will get something really close to a rectangle. (the Poly Order would have to be infinite for an actual rectangle.

Step 2: Draw a Specific Shaped Poly (Super Circle)

A Poly will have an X Radius, a Y Radius, and a D (diagonal) Radius. Unfortunately PloyDraw requires a Poly Order Value. The following program converts Poly Order to D radius or the reverse.

PolyAnalyze

Suppose you want a Poly with an X radius of 4", a Y radius of 4", and a D radius of 4.25" . Enter these values into PolyAnalyze and solve for the Poly Order. In this case it is 2.424.

Enter these values into PolyDraw and you get a PDF of this Poly. Print it out and measure it to show it has a diagonal diameter of 8.5". Polys with a Poly Order around 2.3 to 2.6 have a very Asian feel to them. Less than 2.3 an it looks mostly like a circle. Above 2.6 and they have a squarish feel to them.

PolyDraw allows you to draw several Polys at a time. The photo at the start had 9.

You are now ready to draw super circles. There are also tutorials available that will help you learn to exploit more PolyDraw features.

Comments

[TomV4]

The geometry and algebra at work here would be interesting to explore.

[dave.vaness.79]

The website has a tutorial on PolyAnalyze that has some theory in in. One thing that came out of my studies is that radiians are meant to be the length of an arc normalized to a radius of one. This makes it proportional to degrees. This only works for circles. It should be defined as the percntage of the closed loop normalized to 2Pi. This allows a point to move at a equaldistant spped on the curve. To not confuse other I call these linear radians of ladians.

[TomV4]

Right there are 2*Pi radians per 360 degrees. I wouldn't bother inventing mathematical terms. Mathematics is a highly defined subject, with terms that are universally accepted and understood for centuries, even millennia. "Ladians" is sort of funny though!