  Extracting shape information from object bound- aries is a well studied
problem in vision, and has found tremen- dous use in applications like object
recognition. Conversely, studying the space of shapes represented by curves
satisfying certain constraints is also intriguing. In this paper, we model and
analyze the space of shapes represented by a 3D curve (space curve) formed by
connecting n pieces of quarter of a unit circle. Such a space curve is what we
call a Tangle, the name coming from a toy built on the same principle. We
provide two models for the shape space of n-link open and closed tangles, and
we show that tangles are a subset of trigonometric splines of a certain order.
We give algorithms for curve approximation using open/closed tangles, computing
geodesics on these shape spaces, and to find the deformation that takes one
given tangle to another given tangle, i.e., the Log map. The algorithms
provided yield tangles upto a small and acceptable tolerance, as shown by the
results given in the paper.
