Compass Only

The astroid by compass only art
     In mathematics, the Mohr–Mascheroni theorem states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone.  In other words, you don’t need the straightedge.  Remember, a straight line is determined by two points, and points can be located with a compass alone (no straightedge required); that’s what the Mohr-Mascheroni theorem means.  The theorem shows that any set of points that fully defines a constructed figure can be determined with compass alone, and the only reason to use a straightedge is for the aesthetics of seeing straight lines, which for the purposes of construction are functionally unnecessary.

     All of the constructions for this page are what I call “compass only” constructions; no straight lines or line segments--they are a bit more complex, but are fun to figure out and lend themselves very well to modifications that produce beautiful and interesting “art versions” of the animation.  The so-called art versions, however, will not necessarily be compass only.
     The animations presented below are in pairs.  The first of the pair is a basic “Compass Only” construction that generates the curve.  The second of the pair is an attempt to create an interesting dynamic animation with all of the underlying mathematical construction hidden, but based on the basic “compass only” construction.

 

The Hyperbola by Compass Only

The Hyperbola by Compass Only Art

The Cardioid by Compass Only

The Cardioid by Compass Only Art

An Ellipse by Compass Only

An Ellipse by Compass Only Art

A Bowditch Curve by Compass Only

A Bowditch Curve by Compass Only Art

The Astroid by Compass Only

The Astroid by Compass Only Art

The Bernoulli Lemniscate by Compass Only

The Bernoulli Lemniscate by Compass Only Art