Differential characterizations of curves of constant breadth and spherical curves

This thesis consists of nine sections. In the first section, basic definitions and theorems in 3-dimensional Euclidean and Minkowski spaces are given. Lucas polynomials are introduced. Then sum formula, recurrence relation, Binet formula and some identities of the Lucas polynomials are given. In the second section, the curves of constant breadth according to Bishop frame in 3-dimensional Euclidean space, and the timelike and spacelike curves of constant breadth according to Bishop frame in 3-dimensional Minkowski space are studied. Then differential equations and differential equation systems characterizing these curves are given. In the third section, the non-unit speed spherical curves according to Frenet frame in Euclidean 3-space, and the non-unit speed Lorentzian spherical timelike and spacelike curves and the non-unit speed hyperbolic spherical spacelike curves are studied. Then differential equation characterizations of these curves are given. In the fourth section, for solving linear differential equations with variable coefficients, general linear differential equation systems and systems of linear differential equation in normal form, Taylor collocation and Lucas collocation methods based on residual error function are given. In the fifth section, approximate solutions of the differential equation systems characterizing curves of constant breadth according to Bishop frame in 3-dimensional Euclidean space are given by using Taylor and Lucas collocation methods. In the sixth section, approximate solutions of the differential equation systems characterizing timelike and spacelike curves of constant breadth according to Bishop frame in 3-dimensional Minkowski space are given by using Taylor and Lucas collocation methods. In the seventh section, approximate solutions of the differential equations characterizing non-unit speed spherical curves according to Frenet frame in Euclidean 3-space are given by using Taylor and Lucas collocation methods. In the eighth section, approximate solutions of the differential equations characterizing non-unit speed Lorentzian spherical timelike and spacelike curves according to Frenet frame in Minkowski 3-space are given by using Taylor and Lucas collocation methods. In the ninth section, approximate solutions of the differential equations characterizing non-unit speed hyperbolic spherical spacelike curves according to Frenet frame in Minkowski 3-space are given by using Taylor and Lucas collocation methods.