Eric Turner
September 11, 2012
CS284


1)

See attached screenshots

2)

We get a prismatic curve, with no "funny points", only when there are no inflection points.  As the curvature goes to zero, the local Frenet Frame becomes ill-defined.

Setting minimizeTorsion to true will result in a smooth curve (in 2D) and almost smooth in 3D, with a potential discontinuity at the start/end of the curve.  Increasing the z-scale increases this angular discontinuity.

Since angles are periodic with 360, we can take advantage of any symmetry in the structure to correct this rotation only to the nearest symmetry orientation.  So for a 5-pointed star, the rotational symmetry of the cross-section is 60 degrees.  Thus, we only need to correct the discontinuity by up to 60/2 = 30 degrees.

3)

As shown in the example with B-splines, we can modify the final twist of a sweep, and the extent of that twist will be distributed evenly throughout the curve.

If we plot the bezier curve with no modifications, the cubic components do not match up.  Their current orientation denotes a minimum torsion alignment for the individual cubic curves, which do not necessarily align separate curve segments.

Thus, we can determine the angular mismatch at each of the endpoints, and twist each individual bezier curve so these ends match.

Since any twisting is distributed along the length of a curve, rather than twisting to meet at the halfway angle, I would try to meet at the weighted average of the angles, where the weighting is inversely proportional to the curve length (so short curves need to twist less than long curves).  The curve length can be approximated by the circumference of the control polygon for each bezier curve.

However, SLIDE doesn't have a mechanism to read off the final angular offset, so the above calculations must be done manually.  This was accomplished by creating additional sliders for the azimuth and twist of each curve.  The manual twisting of each segment was performed with the above in mind, so that larger segments got more of the twist.

4)

The joints look very bad under the frenet frame, since each joint represents a dramatic discontinuity in the curvature.

The joints are very nicely mitered under minimum torsion conditions (with the exception of the first/last joint).  Adjusting the uniform twist of -19 degrees over the whole polyline allows this final joint to be matched as well (for a five-pointed star cross-section).