Special Plane Curves:
The Astroid

Eric Rane
Math 50C
9/28/97

The Astroid

History:

Roemer (1674) was the first to discover the set of cycloidal curves. He was searching for the best form for gear teeth, and in his search, he discovered the Asteroid. It wasn't until later, with Daniel Bernoulli (1725), did double generation of these curves take shape in the minds of mathmaticians. The astroid took many names including four-cusp-curve until the year 1838 when a book was found in Vienna which named the curve as "The Astroid". There is evidence of the curve in Leibniz's correspondence as early as 1715, where he writes the equation to be x^(2/3) + y^(2/3) = a^(2/3).

Astroid - Gerated using astroid.m (MatLab)

Parametric Equations:

The curve can be given by two equations show below:

x=4a*cos(t)^3   y=4a*sin(t)^3   0 <= t <= 2*pi

```
 x^(2/3) + y^(2/3) = a^(2/3)
```

To see this generation in action, download and execute astroid.m. Below are the instructions for astroid.m:

ASTROID(Z,BOOL)

Z=the angle step between tracings from 0 to 2pi ie. ASTROID(pi/2) renders four angles To see the circles that sweep out the astroid, put any number as the second parameter of astroid ie. ASTROID(Z,1)

Also, you can download a multi-layer 3d version of the astroid off of my web site at www.tidepool.com/boris. This requires that you are running windows '95 and have the OpenGL drivers installed on your machine. If you don't have the drivers, you can download them from microsoft at www.microsoft.com/kb/softlib/mslfiles/Opengl95.exe.

Make sure you start this rendering on a coffee break.

|
Matlab command line >>astroid(pi/25,1)

Generation of the Trancendental Parametric Equations:

The distance from A to B is 4a*cos(t)
The distance from C to P is ABcos(t) or 4acos(t)^2
The distance from the y-axis to P (the x component of the equation) is 4acos(t)^2cos(t) or 4a*cos(t)^3

References:

Lockwood, E.H. (1961) A Book of Curves. Great Britian: Cambridge University Press.

MacTutor History of Mathematics Archive. ``Astroid.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Astroid.html