What can we measure?

When characterizing a shape or changes in shape we must first ask, what can we measure about a shape? For example, for a region in ℝ^3 we may ask for its volume or its surface area. If the object at hand undergoes deformation due to forces acting on it we may need to formulate the laws governing the change in shape in terms of measurable quantities and their change over time. Usually such measurable quantities for a shape are defined with the help of integral calculus and often require some amount of smoothness on the object to be well defined. In this chapter we will take a more abstract approach to the question of measurable quantities which will allow us to define notions such as mean curvature integrals and the curvature tensor for piecewise linear meshes without having to worry about the meaning of second derivatives in settings in which they do not exist. In fact in this chapter we will give an account of a classical result due to Hadwiger, which shows that for a convex, compact set in ℝ^n there are only n + 1 unique measurements if we require that the measurements be invariant under Euclidian motions (and satisfy certain "sanity" conditions). We will see how these measurements are constructed in a very straightforward and elementary manner and that they can be read off from a characteristic polynomial due to Steiner. This polynomial describes the volume of a family of shapes which arise when we "grow" a given shape. As a practical tool arising from these consideration we will see that there is a well defined notion of the curvature tensor for piece-wise linear meshes and we will see very simple formulas for quantities needed in physical simulation with piecewise linear meshes. Much of the treatment here will initially be limited to convex bodies to keep things simple. This limitation that will be removed at the very end