Winding Numbers and Solid Angles

Abstract

Winding number is an important and familiar mathematical concept which is used in several medical and research areas such that in electro cardiology. In two dimensions, for any given polygon P we mean by winding number of P about a point p is how many time that this polygon travels around p. We established two methods for finding winding number for a given planar polygon P about an query point p in space, namely, direct method and projective method. In the first one we extend our works in two dimensions to three dimensions and in the second one base our approach on finding an orthogonal basis for a given plane in R 3 and project each vector in the space to the xy-plane. This thesis is mainly concerned with the winding number for a given polyhedron and solid angles. An analytical expression for solid angles including and explanation of triangular polyhedra with solid angles subtended by plane triangles are also introduced.