The goal of this handout is to discuss models of Hyperbolic and Euclidean Geometry, and the consistency of Hyperbolic Geometry. 1. Some concepts from Euclidean Geometry We will use circles in Euclidean Geometry to build up models for Hyperbolic Geometry. So we will review some basic facts about circles. (Mostly without proofs). In this section assume that we are working in Euclidean Geometry. D...

A manifold M is like a curve in 2D and a surface in 3D. It is a set of points with topology (or neighbor structures), and locally looks like R. Example: our earth. Locally it is planar. Rigidly, for a patch U ⊂ M , we have a local coordinate system xU : U 7→ R as i = 1 · n local coordinates. Typically M can be covered by several such patches. (eg, A sphere can be covered by 2 patches but not on...

This paper introduces a preprocessed incenter subdivision scheme to make the limit curves interpolate G1 or G2 Hermite data. The key idea is to add a preprocessed step, changing the compute rule of new inserted points and their corresponding tangent vectors in the first subdivision step of the incenter subdivision scheme[4]. Moreover, we prove that the limit curves generated by our algorithm in...

We derive a lower bound for energies of harmonic maps of convex polyhedra in R 3 to the unit sphere S, with tangent boundary conditions on the faces. We also establish that C maps, satisfying tangent boundary conditions, are dense with respect to the Sobolev norm, in the space of continuous tangent maps of finite energy. Mathematics Subject Classifications (2000). 58E20, 35J55.

We analyze characteristic curves of vector fields and report on locations where they have cusps in their spatial projection, i.e., isolated points on the curve with abruptly turning tangent direction. Cusps appear in places where a projection of the corresponding tangent curve vector field exhibits critical points. We show that such cusps are only possible for streak and path lines, whereas the...

A new paradigm for designing smooth surfaces is described. A finite set of points with weights specifies a closed surface in space referred to as skin. It consists of one or more components, each tangent continuous and free of self-intersections and intersections with other components. The skin varies continuously with the weights and locations of the points, and the variation includes the poss...

A theorem of Bôcher and Grace states that the critical points of a cubic polynomial are the foci of an ellipse tangent to the sides of the triangle joining the zeros. A more general result of Siebert and others states that the critical points of a polynomial of degree N are the algebraic foci of a curve of class N − 1 which is tangent to the lines joining pairs of zeroes. We prove the analogous...