More Fun Ruled Sufaces

I chose to redo the topic of ruled surfaces as I find their concept especially interesting, and likewise, creating a ruled surface with code is surprisingly enjoyable. Last time, I did not feel as though my example was creative, as my ruled surface was only between two curves which were the same shape, and only set off by a phase. I had made the claim that the ruled surface can exist between two arbitrary curves, but this was not well-illustrated by my choice of highly similar curves.

As mentioned before, a ruled surface is created such that a point P on a surface S lies on exactly one distinct straight line connecting two arbitrary curves. The surface is constructed only from distinct straight lines, and no two lines intersect, and the curves the lines lie between do not self intersect either.

My example last time wasn't completely illustrative of the ability for surfaces to be between completely arbitrary non-self intersecting curves.

This time I chose two different asteroid-like curves. The classic example of the 4-point asteroid curve is shown below, along with its parameterization.

\[x(t) = 3 \ cos(t) + \cos(3t)\\ y(t) = 3\sin(t) - \sin(3t)\]

For my two curves, I chose to project the 6-point asteroid onto the 5-point asteroid, and also created a surface of the 6-point asteroid being projected onto a heavily distorted 5-point asteroid. Below I show the 2-D plots of the curves and their parameterizations.

\[x(t) = 5 \ cos(t) + \cos(5t)\\ y(t) = 5\sin(t) - \sin(5t)\]

\[x(t) = 4 \ cos(t+phase) + \cos(4(t+phase))\\ y(t) = 4\sin(t+phase) - \sin(4(t+phase))\]

I defined the phase shift to be 55 degrees in this assignment, and distorted the x(t) parameter for the 5-point curve by \(\displaystyle 1.5*phase\) and did not alter the y(t) parameter in order to create the distorted 5-point curve.

\[x'(t) = 4 \ cos(t+1.5*phase) + \cos(4(t+1.5*phase))\\ y'(t) = 4\sin(t+phase) - \sin(4(t+phase))\]

Below I give the OpenScad renderings of both objects, and their dimensions.

The above object will have a length and width of 80mm with a height of 20mm.

This object will also have a length and width of 80mm with a height of 20mm.

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