Approximating Superellipses

Superellipses (Lamé curves) can be approximated in CorelDRAW using the B-Spline Tool (introduced in CorelDRAW X5).  The basic idea is that if the b-spline's control polyline (the line connecting the b-spline's control points) is an octagon with alternating side lengths, the resulting b-spline will approximate a superellipse.

The easiest way to accomplish this is to snap the control points to a grid:



The picture above shows a 6x6 grid (I used a graph paper object as a temporary guide, although the normal grid would work). The control points (blue dots) are placed on the verticies of an octagon (shown in blue).  The resulting b-spline, shaded in green, approximates a superellipse.

The roundness of the superellipse can be controlled by varying the lengths of the control octagon's sides:



Another Way
Here is an alternate pattern of control points that approximates a superellipse.  It is convenient because it can be created on a simple rectangle.  Just snap the control points to the verticies (nodes) and midpoints of the rectangle's edges.



Superellipses with n < 2
The above approach works well for superellipses with n > 2.  To approximate superellipses with 0 < n < 1 (e.g., astroids), create a 4-point star with the Star Tool, and place the b-spline's control points on its vertices. The shape ("sharpness") of the star dictates the shape of the resultant b-spline:



In this case, the outer vertices must be clamped (hold v while creating, or select them with the Shape Tool and clamp them in the property bar).

The case where 1 < n < 2 is harder to approximate simply.  Here is an example created on a 17x17 grid.  Note the extra control points added to adjust the curvature of the sides.
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