We live in a three-dimensional world, using three dimensional objects in our daily life; some of these objects are convex, some are not. From the viewpoint of mathematics, we are interested in studying properties that are not changed under certain operation on the geometric objects. The familiar concepts of length for a curve, the area for a region, and the volume for a solid body, have the properties unchanged under rigid motions of the Euclidean space. _i ⁿ...[ Read more ]

We live in a three-dimensional world, using three dimensional objects in our daily life; some of these objects are convex, some are not. From the viewpoint of mathematics, we are interested in studying properties that are not changed under certain operation on the geometric objects. The familiar concepts of length for a curve, the area for a region, and the volume for a solid body, have the properties unchanged under rigid motions of the Euclidean space.

In this thesis we study length, area, and volume for general geometric objects of the three dimensional Euclidean space. We start with the classical Buffon needle problem in Chapter 1, giving both analytic and probabilistic interpretations. In Chapter 2 we introduce intrinsic invariant measures μ_i (1 ≤ i ≤ n) in Rⁿ for parallelotopes; these intrinsic invariant measures can be extended to convex sets and their finite unions. In Chapter 3 we study the length for any convex regions and mixed area for any pair of convex regions. In Chapter 4 we consider the group of motions of the two-dimensional Euclidean space, giving the kinematic density of the rigid motions and other interesting formulas. Finally, in Chapter 5 we consider the kinematic density of the rigid motions for the three-dimensional Euclidean space. We derive kinematic formula and generalize Buffon needle problem to higher dimensional case. We also generalize the kinematic formula for unbounded case.