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The
Astroid
by Peter Spiegel
Roemer was the first to investigate the set of cycloidal curves in 1674. In his search for the best form for gear teeth, he discovered the astroid.
The astroid is a 4 cusped Hypocycloid. It is also known as cubocycloid, paracycle and tetracuspoid.
The astroid can be defined as the locus of a point on a circle with radius r
rolling on the inside of a circle of radius 4*r
To visualize this I created the following sketch with The Geometer’s Sketchpad
astroid1
Alternatively it can bedefined as the envelope of a line segment of constant length, which’s endpoints slide along the x and y axis. To visualize this the analogy of a ladder which is pushed up a wall, can be used. Following is a Matlab plot and a GSP sketch.
astroidlinesegment
A set of ellipses who’s major and minor axis always add up to a constant number also can be used to generate an astroid.
Again I created a Matlab plot and a Sketchpad sketch to demonstrate this.
astroidEllipses
To find a parameterization of the astroid, vectors come in quite handy.
This plot is to accompany the following calculations, to find the Cartesian parameterization in terms of theta, where theta is the angle to the center of the small circle if looked at in polar coordinates.
S1 and S2 are the arc lengths of the large circle from the x-axis to where it touches the small circle and of the small circle from where it touches the large circle to the point P. Given that there is no slippage these two arc lengths must be equal.
O is the origin and C is the center of the smaller circle.
Phi is the angle between the horizontal line and the vector from C to P.
To see what this looks like as the angle theta goes from 0 to pi/2 I provided the following set of plots
An interesting property of the astroid is that the evolute, which is the trace of the center of the osculating circles, is another astroid twice the size. This is shown in the following sketch.
astroidosculating
References
http://www.xahlee.org/SpecialPlaneCurves_dir/Astroid_dir/astroid.html
http://mathworld.wolfram.com/Astroid.html