Rendered and Animated Pictures

These pictures were rendered using POV-Ray 3.0 and animated with the GIF-Construction set. The source for the scene descriptions was entered manually.

One of my favorite math toys. Each quarter circle is connected to the previous with a rotating rigid joint. This is described by coordinate transformations exactly the way robot arms are described. The constraint of requiring the elements to close is a system of polynomials, and the solutions demonstrate some features of real algebraic geometry.
This particular one is one of 2 possible rigid configurations, the other one is the complete circle made from 4 elements. Closed configurations exist for all numbers of elements except 1, 2, 3, and 5. This can be proven easily using Grobner bases but it would be nice to have an 'understandable' proof.
rotating torus (33KB)

Another view of the object, this time, six quarter circles are configured so that the resulting object is non-rigid. The locus of orientations is topologically a circle, as the locus of orientations of the previous object is topologically a point. There are two configurations of 7 quarter circles both of which have circular locii. I am certain that there is only one locii of configurations for 8 or more elements, and in particular 8 elements have a toroidal locus.
Other configurations have locii that are real semi-algebraic manifolds, and they are very difficult to compute. There are even some surprises in this example, why not try it? Is it necessary that all such locii are orientable?
waving torus (126KB)

The bats are surfaces of revolution of a cubic spline except for the large end which is a superellipse. The points for the spline were taken from an actual size 30 bat.
three bats (15KB)

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One of my favorite math toys. Each quarter circle is connected to the previous with a rotating rigid joint. This is described by coordinate transformations exactly the way robot arms are described. The constraint of requiring the elements to close is a system of polynomials, and the solutions demonstrate some features of real algebraic geometry. This particular one is one of 2 possible rigid configurations, the other one is the complete circle made from 4 elements. Closed configurations exist for all numbers of elements except 1, 2, 3, and 5. This can be proven easily using Grobner bases but it would be nice to have an 'understandable' proof.
Another view of the object, this time, six quarter circles are configured so that the resulting object is non-rigid. The locus of orientations is topologically a circle, as the locus of orientations of the previous object is topologically a point. There are two configurations of 7 quarter circles both of which have circular locii. I am certain that there is only one locii of configurations for 8 or more elements, and in particular 8 elements have a toroidal locus. Other configurations have locii that are real semi-algebraic manifolds, and they are very difficult to compute. There are even some surprises in this example, why not try it? Is it necessary that all such locii are orientable?
The bats are surfaces of revolution of a cubic spline except for the large end which is a superellipse. The points for the spline were taken from an actual size 30 bat.