Differential Geometry of Surfaces Jordan Smith UC Berkeley CS284

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

Differential Geometry of a Curve Point p on the curve at u0 p C(u) p=C(u0)

Differential Geometry of a Curve Tangent T to the curve at u0 p Cu C(u)

Differential Geometry of a Curve Normal N and Binormal B to the curve at u0 B p Cu Cuu C(u) N

Differential Geometry of a Curve Curvature κ at u0 and the radius ρ osculating circle B p Cu Cuu C(u) N

Differential Geometry of a Curve Curvature at u0 is the component of -NT along T C(u0) C(u1) T N(u0) C(u) N(u1) NT

Computing the Curvature of a Curve

Computing the Curvature of a Curve

Computing the Curvature of a Curve

Computing the Curvature of a Curve

Computing the Curvature of a Curve

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 (u0,v0) p S(u,v)

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

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

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

First Fundamental Form IS Metric of the surface S

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

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

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

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

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 Su Tangent Plane of S

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

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

Change of Coordinates Ss St Sv b T s u t θ v p a Su A point T expressed in (u,v) and (s,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 Ss St Sv κT is a function of direction T b θ p a Su

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

Curvature E1 E2 φ p Ss St Su Sv a b θ Eigen analysis of IIŜ Eigenvalues = {κ1,κ2} Eigenvectors = {E1,E2} Eigendecompostion of IIŜ

Curvature E1 E2 φ p Ss St Su Sv a b θ α

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 E1 E2 φ p Ss St Su Sv a b θ

Weingarten Operator

Weingarten Operator If κ1≠ κ2 else umbilic (κ1= κ2), 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