CS Tutorial 5 - Differential Geometry I - Surfaces

Transcription

1 Numerical Geometry of Images Tutorial 5 Differential Geometry II Surfaces c 2009

2 Parameterized surfaces A parameterized surface X : U R 2 R 3 a differentiable map 1 X from an open set U R 2 to R 3. Explicitly, X (U) is written as X (u, v) = (x(u, v), y(u, v), z(u, v)). X (U) R 3 is called the trace of X. 1 X (u, v) is called differentiable if x(u, v), y(u, v) and z(u, v) are differentiable functions w.r.t. the parameters u, v.

3 Parameterizations of a sphere - examples 2 (a) (b) (c) Figure: (a) Sphere in R 3. (b) Six overlapping parameterization of a kind X (x, y) = (x, y, ) 1 x 2 y 2. (c) X (θ, φ) = (sin θ cos φ, sin θ sin φ, cos θ). 2 Differential geometry of curves and surfaces, by Manfredo P. do Carmo

4 Differential of X Let X : U R n R m be a differential map. For each p U we will define the differential of X at p by dx p : U R n R m Let w R n and let α : ( ɛ, ɛ) U be a differential curve such that α(0) = p, α (0) = w. An image of α in R m s given by Then: β = X α( ɛ, ɛ) R m dx p (w) = β (0)

5 Differential of X : illustration ε v w X( p) z ( ) dx w p ε 0 α p u X X α = β y x

6 Calculation of the differential in R 3 Let X : U R 2 R 3 be a differential map. α(t) = (u(t), v(t)) β(t) = X α(t) = (x (u(t), v(t)), y (u(t), v(t)), z (u(t), v(t))) β (0) = x u y u z u x v y v z v ( du dt dv dt ) = dx p (w) Note: dx p (w) depends only on w, and not on the choice of α(t).

7 Exercise: df p calculation Calculate df p for F (u, v) = ( u 2 v 2, 2uv ), (u, v) R 2. Solution: df p = x = u 2 v 2, y = 2uv ( x u y u x v y v df (1,1) (2, 3) = df p ( 1 1 ) ( ) 2x 2y = 2y 2x ) = ( 2, 10)

8 Regular parameterization, Tangent space, Normal We will say that a differential parameterization X (U) is regular if X u (u, v) and X v (u, v) are linearly independent for all (u, v) U. Tangent plane (space) at p: a plane T p containing all the tangents to curves passing through the point p S. The tangent plane is spanned by the vectors X u (u, v), X v (u, v). Surface normal: a unit vector orthogonal to the tangent plane at the point p, defined by N(p) = X u X v X u X v (p)

9 Surfaces in R 3 - examples

10 Exercise: tangent space, normal Calculate the equation of the tangent plane of a surface which is a graph of a differentiable function z = f (x, y), at the point (x 0, y 0 ). Solution: X (x, y) = (x, y, f (x, y)) X x (x, y) = (1, 0, f x (x, y)) X y (x, y) = (0, 1, f y (x, y)) N = X u X v X u X v = ( f x, f y, 1) fx 2 + fy Equation of a plane passing through (x 0, y 0 ) and perpendicular to N(x 0, y 0 ) is: f x (x 0, y 0 )(x x 0 ) + f y (x 0, y 0 )(y y 0 ) (z f (x 0, y 0 )) = 0

11 The first fundamental form In order to measure local distances on the regular surface S, we assign an inner product on the tangent plane T p of S at every point p. The first fundamental form: a quadratic form I p : T p (S) T p (S) R, given by I p (w 1, w 2 ) = w 1, w 2 p. <, > p is a standard inner product of R 3 ; p stands for the tangent plane T p of the surface S at the point p. The form I p (w 1, w 2 ) is bilinear, symmetric and positive-definite.

12 The first fundamental form in terms of X u, X v Let α 1 (t) = X (u 1 (t), v 1 (t)) and α 2 (t) = X (u 2 (t), v 2 (t)) be two differential curves such that α 1 (0) = α 2 (0) = p, α 1(0) = w 1, α 2(0) = w 2. The first fundamental form is given by: I p (w 1, w 2 ) = α 1(0), α 2(0) = X u u 1 + X v v 1, X u u 2 + X v v 2 ( ) u T ( ) ( ) = 1 Xu, X u X u, X v u 2 v 1 X v, X u X v, X v v 2 ( ) u T ( ) ( ) = 1 E F u 2 v 1 F G v 2

13 The first fundamental form - alternative notation ( E F F G ) ( g11 g = 12 g 21 g 22 ) = ( Xu, X u X u, X v X v, X u X v, X v ) In coordinate notation: I p (w 1, w 2 ) = 2 2 g ij w 1 i w j 2 = g ij w 1 i w j 2 i=1 j=1 where we use the Einstein summation convention, according to which identical sub- and super-scripts are summed up; w k = (u k, v k ), k = 1, 2.

14 Exercise: calculation of I p Calculate the first fundamental form of a unit sphere parameterized by X (θ, φ) = (sin θ cos φ, sin θ sin φ, cos θ) Solution: X θ = (cos θ cos φ, cos θ sin φ, sin θ) X φ = (sin θ sin φ, sin θ cos φ, 0) E = X θ, X θ = cos 2 θ cos 2 φ + cos 2 θ sin 2 φ + sin 2 θ = 1 F = X θ, X φ = cos θ cos φ sin θ sin φ + cos θ sin φ sin θ cos φ = 0 G = X φ, X φ = sin 2 θ sin 2 φ + sin 2 θ cos 2 φ = sin 2 θ

15 Length calculation The length of a parameterized curve β : [0, T ] S is given by s(t) = t 0 β ( t) d t = t 0 I p (β, β )d t Given that α(t) = (u(t), v(t)), β(t) = X α(t), s(t) = t 0 E(u ) 2 + 2Fu v + G(v ) 2 d t Also, local element of the arclength is given by ds 2 = Edu 2 + 2Fdudv + Gdv 2

16 Area calculation Area of a bounded region R of a regular surface S = X (U) is given by A(R) = X u X v dudv, Q = X 1 (R) Q From the equality X u X v 2 + X u, X v 2 = X u 2 X v 2 it follows that X u X v = EG F 2 Note that EG F 2 > 0, by definition (prove).

17 Exercise: arclength and area measured on a sphere Calculate ds 2 and da = X u X v dθdφ of a unit sphere parameterized by X (θ, φ) = (sin θ cos φ, sin θ sin φ, cos θ) Solution: the first fundamental form of a sphere is given by ( ) ( ) E F 1 0 = F G 0 sin 2 θ Therefore: ds 2 = Edθ 2 + 2Fdθdφ + Gdφ 2 = dθ 2 + sin 2 θdφ 2 da = EG F 2 dθdφ = sin 2 θdθdφ

18 Orientation of surfaces We saw that a surface normal at point p S is defined by N(p) = X u X v X u X v (p) We will say that a regular surface S is orientable if and only if there exist a differentiable field of unit normal vectors N : S R 3 on S. Orientable surfaces: sphere, paper roll. Non-orientable surface: Möbius strip.