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## Some Circular Designs (Page 3)

Most authorities construct curve-stitch parabolas using
s+1 distinct lines, where s is the number of equal-length segments into which each axis is divided. For example, if s = 3, we would have a figure like the following:

In this construction, a point on either axis is connected directly to a point on the other axis s segments away, counting toward the vertex and moving along the other axis once the vertex is reached. For s = 4, we have something like:

When I was taught how to construct curve-stitch parabolas in junior high school, I learned to use s+2 distinct lines, following the rule of spacing endpoints at a separation of s+1 segments, with the axes as the scaffolding on which the entire construction is built. This produces figures like the following, for s = 3:

Notice that this is actually the same construction as the earlier one, but using longer axes segmented into s+1 units and with the final segment on each axis lopped off.

Each construction method approximates parabolas, in that they result in tangent lines along a parabola that is never explicitly drawn. The first method approximates a parabola actually tangent to the axes at the endpoints. This is a useful property in some constructions. For example, if two curve-stitch parabolas share an axis and endpoint, and they are mirror images of one another, using the first (or standard) construction prevents a minor discontinuity where the parabolas meet:

The second method results in:

Nonetheless, the standard construction does not always seem to be the “right” one. In some combinations, standard parabolas produce very “spiky” elements, which one might prefer to avoid. Consider this juxtaposition of standard parabolas:

The alternate construction gives us the following instead:

I generally prefer the second method of construction, which gives many figures points that look like stars, rather than like antennas. Judge for yourself in the following pictures. (Click on a  design to see a larger image.)

##### Segments = 10, Wedges = 5, Levels = 2, Alternate Construction

Click on Next for more on the two possible constructions for curve-stitch parabolas.