Differential Geometry of Surfaces Jordan Smith UC Berkeley

Differential Geometry of Surfaces Jordan Smith UC Berkeley CS 284

Outline • Differential Geometry of a Curve • Differential Geometry of a Surface – I and II Fundamental Forms – Change of Coordinates (Tensor Calculus) – Curvature – Weingarten Operator – Bending Energy

Differential Geometry of a Curve C(u)

Differential Geometry of a Curve Point p on the curve at u 0 p C(u) p=C(u 0)

Differential Geometry of a Curve Tangent T to the curve at u 0 p C(u) Cu

Differential Geometry of a Curve Normal N and Binormal B to the curve at u 0 p C(u) N B Cu Cuu

Differential Geometry of a Curve Curvature κ at u 0 and the radius ρ osculating circle p C(u) N B Cu Cuu

Differential Geometry of a Curve Curvature at u 0 is the component of -NT along T C(u 0) C(u 1) C(u) N(u 0) NT N(u 1) T

Computing the Curvature of a Curve

Outline • Differential Geometry of a Curve • Differential Geometry of a Surface – I and II Fundamental Forms – Change of Coordinates (Tensor Calculus) – Curvature – Weingarten Operator – Bending Energy

Differential Geometry of a Surface S(u, v)

Differential Geometry of a Surface Point p on the surface at (u 0, v 0) p S(u, v)

Differential Geometry of a Surface Tangent Su in the u direction p S(u, v) Su

Differential Geometry of a Surface Tangent Sv in the v direction Sv p S(u, v) Su

Differential Geometry of a Surface Plane of tangents T Sv p S(u, v) T Su

First Fundamental Form IS • Metric of the surface S

Differential Geometry of a Surface Normal N N Sv p S(u, v) T Su

Differential Geometry of a Surface Normal section N Sv p S(u, v) T Su

Differential Geometry of a Surface Curvature N Sv p S(u, v) T Su

Differential Geometry of a Surface Curvature NT N Sv p S(u, v) T Su

Second Fundamental Form IIS

Outline • Differential Geometry of a Curve • Differential Geometry of a Surface – I and II Fundamental Forms – Change of Coordinates (Tensor Calculus) – Curvature – Weingarten Operator – Bending Energy

Change of Coordinates Sv p Tangent Plane of S Su

Change of Coordinates St b θ p a S u Construct an Orthonormal Basis Sv Ss

Change of Coordinates St b p First Fundamental Form θ a Su Sv Ss

Change of Coordinates St s b Sv θ Ss p a S u A point T expressed in (u, v) and (s, t) T u v t

Outline • Differential Geometry of a Curve • Differential Geometry of a Surface – I and II Fundamental Forms – Change of Coordinates (Tensor Calculus) – Curvature – Weingarten Operator – Bending Energy

Curvature St κT is a function of direction T b p θ a Su Sv Ss

Curvature St How do we analyze the κT function? p b θ a Su Sv Ss

Curvature Eigen analysis of IIŜ St b Eigenvalues = {κ 1, κ 2} E 2 Eigenvectors = {E 1, E 2} Sv φ p Eigendecompostion of IIŜ E 1 θ a Su Ss

Curvature St E 2 E 1 α p b Sv φ θ a Su Ss

Outline • Differential Geometry of a Curve • Differential Geometry of a Surface – I and II Fundamental Forms – Change of Coordinates (Tensor Calculus) – Curvature – Weingarten Operator – Bending Energy

Weingarten Operator St E 1 b E 2 Sv φ p θ a Su Ss

Weingarten Operator

Weingarten Operator If κ 1≠ κ 2 else umbilic (κ 1= κ 2), chose orthogonal directions

Outline • Differential Geometry of a Curve • Differential Geometry of a Surface – I and II Fundamental Forms – Change of Coordinates (Tensor Calculus) – Curvature – Weingarten Operator – Bending Energy

Bending Energy

Bending Energy Minimizing = Minimizing

Conclusion • Curvature of Curves and Surfaces • Computing Surface Curvature using the Weingarten Operator • Minimizing Bending Energy – Gauss-Bonnet Theorem