To the Graduate Council: I am submitting herewith a thesis written by ...

Chapter 4: Algorithm Overview 45These maximum and minimum directions {T 1 , T 2 } are the principal directions. Theadded benefit of choosing the principal directions as the basis set is that the curvaturesκ 1 = κ p (T 1 ) and κ 2 = κ p (T 2 ) associated with these directions lead to the followingrelationship for any normal curvature at p:κ ( Tpθ22) = κ1cos ( θ ) + κ2sin ( θ ),(4.1)where T θ = cos(θ)T 1 + sin(θ)T 2 and -π ≤ θ ≤ π is the angle to vector T 1 in the tangentplane. The maximum and minimum curvatures are known as the principal curvatures.The principal directions along with the principal curvatures completely specify thesurface curvature of S at p and thus describe the shape of S. Combinations of theprincipal curvatures lead to other common definitions of surface curvature. The mostcommonly used is the Gaussian curvature, and is the product of the principalcurvatures as shown in Equation 4.2.K p = κ 1 κ 2(4.2)This definition highlights that negative surface curvature that occurs at hyperbolicoccur where only one principal curvature is negative. The second definition ofcurvature is mean curvature. We specify mean curvature as the average of bothprincipal curvatures (Equation 4.3). Mean curvature gives insight to the degree offlatness of the surface.H p= ( κ 1 + κ 2 ) / 2(4.3)4.2.2 Curvature EstimationCurvature estimation is a challenging problem on digitized representations of curvesand surfaces. Consider a 2D function y=f(x).The curvature of the continuous functiony is mathematically defined as shown in Equation 4.4.κ =1+ 2d y2dxdydx23 2(4.4)Equation 4.4 assumes the rectangular coordinate system. If we parameterize y = f(x) inthe polar coordinate system the curvature equation can be rewritten as in Equation 4.5.κ =r2+ 2 r2θ− rr2 2 3( r + r )θθθ2;rθ∂ r∂ θ .(4.5)