A piecewise cubic PostScript trefoil

Fred Linton 1;a
                         1 Wesleyan Univ., United States of America
                         a flinton@wesleyan.edu

2010 Mathematics Subject Classification.  53-04, 97G70, 97N80, 97R60

Keywords.  
Trefoil, piecewise cubic parametrization, Bézier control point, PostScript language, rotation and reflection

pp. Geometry 148-149

Eschewing the traditional trigonometrical approach thereto, we offer here a piecewise
polynomial parametrization of the trefoil, a continuously differentiable stitching together of  
six rotated and reflected copies of one basic pattern-curve, part of the graph of a well-chosen
cubic polynomial.

Values for such a polynomial and its first derivative at the end points of a convenient
interval are easy to motivate, and it’s an elementary linear algebra exercise to confirm that
there is a unique cubic polynomial realizing those four values there.

It’s an easy calculation (cf. p. 393 of the Adobe Systems Inc. PostScript Language Refer-
ence Manual (the Red Book), 2nd edition, Addison-Wesley, Reading (MA), 1990, ISBN-13:
978-0-201-18127-2, or the equivalent) to determine two handy Bézier control points, which,
along with the start and end points, determine that cubic curve; and it then becomes an amus-
ing exercise in PostScript “curveto” boondogglery to plot that polynomial fragment, and to
assemble it and its rotated and reflected copies into the trefoil image desired, as slides will
illustrate.