Differential Geometry of Surfaces Jordan Smith and Carlo Sequin´ ∗ CS Division, UC Berkeley 1 Introduction fundamental form IS. This measures the length of a tangent vector on a given parametric basis by examining the quantity T T . These are notes on differential geometry of surfaces based on read- · ing [Greiner et al. n. d.]. 2 2 T T = (Su Su) @u + 2 (Su Sv) @u@v + (Sv Sv) @v (2) · · · · 2 Differential Geometry of Surfaces = E@u2 + 2F @u@v + G@v2 (3) Su @u Differential geometry of a 2D manifold or surface embedded in 3D = @u @v Su Sv (4) Sv · @v is the study of the intrinsic properties of the surface as well as the ef- fects of a given parameterization on the surface. For the discussion Su Su Su Sv @u = @u @v · · (5) below, we will consider the local behavior of a surface S at a single Su Sv Sv Sv @v point p with a given local parameterization (u; v), see Figure 1. We · · E F @u would like to calculate the principle curvature values (κ1; κ2), the @u @v (6) = F G @v maximum and minimum curvature values at p, as well as the local t orthogonal arc length parameterization (s; t) which is aligned with = U ISU (7) the directions of maximum and minimum curvature. Equation 7 defines IS. N Su S IS = gij = Su Sv (8) v Sv · Su Su Su Sv T = · · (9) Su Sv Sv Sv S · · u E F (10) = F G If an orthonormal arc length parameterization (s; t) is chosen then there will be no local metric distortion and IS will reduce to the identity matrix. Figure 1: Tangent plane of S 4 The Second Fundamental Form 3 The First Fundamental Form The unit normal N of a surface S at p is the vector perpendicular With the given parameterization, we can compute a pair of tangent to S, i.e. the tangent plane of S, at p. N can be calculated given a @S @S general nondegenerate parametrization (u; v). vectors Su Sv = @u @v assuming the u and v direc- tions are distinct. These two vectors locally parameterize the tan- gent plane to S at p.A general tangent vector T can be constructed Su Sv as a linear combination of Su Sv scaled by infinitesimal co- N = × (11) Su Sv @u @v t k × k efficients U = . The curvature of a surface S at a point p is defined by the rate @u T @uS @vS Su Sv (1) that N rotates in response to a unit tangential displacement as in = u + v = @v Figure 2. A normal section curve at p is constructed by intersecting Consider rotating the direction of the tangent T around p by set- S with a plane normal to it, i.e a plane that contains N and a tan- ting @u @v = cos θ sin θ . In general, the magnitude gent direction T . The curvature of this curve is the curvature of S in and direction of T will vary based on the skew and scale of the the direction T . The curvature κ of a curve is the reciprocal of the radius ρ of the best fitting osculating circle, i.e. κ = 1 . The direc- basisvectors Su Sv , see Figure 3(a). The distortion of the ρ local parameterization is described by the metric tensor or the first tional derivative NT of N in the direction T is a linear combination @N @N of the partial derivative vectors Nu Nv = @u @v ∗e-mail: (jordansjsequin)@cs.berkeley.edu and is parallel to the tangent plane. −1 Su (Suu Sv) + (Su Svu) @ Su Sv IIS = hij = Nu Nv (19) Nu = × × + (Su Sv) k × k Sv · − − Su Sv × @u k × k (12) Su Nu Su Nv = − · − · (20) −1 Sv Nu Sv Nv (Suv Sv) + (Su Svv) @ Su Sv − · − · Nv = × × + (Su Sv) k × k Su·(Suu×Sv) Su·(Suv×Sv) Su Sv × @u u v u v = − kS ×S k − kS ×S k (21) k × k (13) Sv·(Su×Svu) Sv·(Su×Svv ) " − kSu×Sv k − kSu×Sv k # @u Suu·(Su×Sv) Suv ·(Su×Sv) N = @uN + @vN = Nu Nv (14) T u v @v = kSu×Svk kSu×Svk (22) Svu·(Su×Sv ) Svv ·(Su×Sv) " kSu×Svk kSu×Svk # Suu N Suv N = · · (23) Suv N Svv N · · L M (24) N T = M N N Equations [23,24] can be substituted for IIS to yield alternate formulas for T NT . − · Sv 2 2 T NT = (Suu N) @u + 2 (Suv N) @u@v + (Svv N) @v T − · · · · (25) Su = L@u2 + 2M@u@v + N@v2 (26) Suu N Suv N @u = @u @v · · Svu N Svv N @v · · (27) L M @u = @u @v (28) M N @v Figure 2: Normal section of S IIS contains information about the curvature of S, but it will As with the tangent vector T , NT is subject to scaling by the be distorted by the metric if the given parameterization is not an metric of the parameter space. In an arc length parameter direc- arc length parameterization. It is still necessary to counter act this tion s, Ss Ns is the curvature of the normal section. Given a distortion in order to compute the curvature of the S. − · general parameterization (u; v), we can compute T NT . This − · quantity defines the second fundamental form IIS which describes curvature information distorted by the local metric. 5 Coordinate Transformations Our purpose in studying differential geometry and the fundamental forms has been to compute the principle curvature values (κ1; κ2) 2 T NT = (Su Nu) @u (Su Nv + Sv Nu) @u@v and directions (Sk1; Sk2) of a parametric surface S at a point p. − · − · − · · 2 Thus far we have derived IIS which contains curvature informa- (Sv Nv ) @v (15) − · tion but can be distorted by the metric of the parameterization. IS Su @u describes this metric information. We must combine IIS and IS to = @u @v − Nu Nv find the arc length curvature values. Sv · @v − The given parameterization u; v defines a set of basis tan- (16) ( ) gent vectors Su Sv . We can construct an orthonormal basis Su Nu Su Nv @u = @u @v − · − · Ss St due to an arc length parameterization (s; t) where Ss Sv Nu Sv Nv @v − · − · and Su are aligned, see Figure 3(a). A point T can be expressed in (17) either coordinate system, see Figure 3(b). t = U IISU (18) S S T = @u @v u = @s @t s (29) Sv St Equation 18 defines IIS. IIS can be simplified to a set of quanties which are easier to compute by substituting the expres- sions in Equations [12,13] for Nu and Nv respectively. Two of the The coordinates @s @t measure the length of T as op- terms of in of the entries of the matrix in Equation 21 vanish due to posed to the coordinates @u @v . the dot product of orthogonal vectors. Further simplification using properties of the box product of vectors yields Equation 22. Equa- T T =@s2 + @t2 = @u2 + @v2 (30) tion 23 is derived by subsituting N from Equation 11 and shows · 6 We would like to work in the Ss St , so we must find the that IIS can be computed simply by the dot product of the second transformations to and from S S . The quantities a, b, and partial derivatives of S with N. Equation 23 also demonstrates that u v II is a symmetric matrix. θ in Figure 3(a) are defined by S S . S u v St Su @u T T = @u @v Su Sv (42) · Sv · @v @u Sv @u @v I (43) = S @v s u T S b t −1 −1 t @s v = @s @t A I A (44) S @t S v t θ b t @s = @s @t A−1 AAt A−1 (45) Ss θ @t a a S S S u u s @s @s @t (46) = @t (a) Constructing an or- (b) A point T in (u; v) and thonormal basis on top of (s; t) coordinates = @ s2 + @t2 (47) the given parametric basis It is now possible to transform our given parameters to coordi- nates on an orthonormal basis where we can measure lengths. We Figure 3: Effects of the metric can also transform coordinates on the orthonormal basis back to coordinates on our given parameterization. 6 Curvature The curvature at a point p on the surface S in any tangential direc- Su a 0 Ss tion T is T N . = (31) T Sv b cos θ b sin θ St − · S = A s (32) Su @u S T NT = @u @v Nu Nv t − · Sv · − − @v S 1 b sin θ 0 S (48) s = u (33) St ab sin θ b cos θ a Sv @u − = @u @v IIS (49) S @v = A−1 u (34) Sv This function is parameterized by the given parameterization. We would like to study the function that sweeps a unit length tan- gent around p, so it is more convenient to use arc length coordinates. The first fundamental form IS can be represented by these coor- dinate transformations. We must transform coordinates so that setting the arc length coordi- nates @s @t = cos φ sin φ sweeps out the curvature values. Su IS = Su Sv (35) −1 −1 t @s Sv · T N = @s @t A II A (50) T S @t − · Ss t = A Ss St A (36) @s St · = @s @t II (51) S^ @t 1 0 t = A A (37) 0 1 S = AAt (38) t E1 a2 ab cos θ = 2 (39) S ab cos θ b E v 2 b We can also derive transformations between coordinates of the φ different sets of basis vectors by substituting Equations [32,34] re- θ S spectively into Equation 29.