Differential Geometry of Surfaces

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1 Differential Geometry of urfaces Jordan mith and Carlo équin C Diision, UC Berkeley Introduction These are notes on differential geometry of surfaces ased on reading Greiner et al. n. d.. Differential Geometry of urfaces Differential geometry of a D manifold or surface emedded in 3D is the study of the intrinsic properties of the surface as well as the effects of a gien parameterization on the surface. For the discussion elow, we will consider the local ehaior of a surface at a single point p with a gien local parameterization (u, ), see Figure. We would like to calculate the principle curature alues (κ, κ ), the maximum and minimum curature alues at p, as well as the local orthogonal arc length parameterization (s, t) which is aligned with the directions of maximum and minimum curature. N T fundamental form I. This measures the length of a tangent ector on a gien parametric asis y examining the quantity T T. T T ( ) + ( ) + ( ) () E + F + G (3) (4) (5) E F (6) F G U t I U (7) Equation 7 defines I. u I g ij u u E F F G (8) (9) (0) Figure : Tangent plane of 3 The First Fundamental Form With the gien parameterization, we can compute a pair of tangent ectors assuming the u and directions are distinct. These two ectors locally parameterize the tangent plane to at p. general tangent ector T can e constructed as a linear comination of scaled y infinitesimal coefficients U t. T + Consider rotating the direction of the tangent T around p y setting cos θ sin θ. In general, the magnitude and direction of T will ary ased on the skew and scale of the asis ectors, see Figure 3(a). The distortion of the local parameterization is descried y the metric tensor or the first (jordans sequin)@cs.erkeley.edu () If an orthonormal arc length parameterization (s, t) is chosen then there will e no local metric distortion and I will reduce to the identity matrix. 4 The econd Fundamental Form The unit normal N of a surface at p is the ector perpendicular to, i.e. the tangent plane of, at p. N can e calculated gien a general nondegenerate parametrization (u, ). N u () The curature of a surface at a point p is defined y the rate that N rotates in response to a unit tangential displacement as in Figure. normal section cure at p is constructed y intersecting with a plane normal to it, i.e a plane that contains N and a tangent direction T. The curature of this cure is the curature of in the direction T. The curature κ of a cure is the reciprocal of the radius ρ of the est fitting osculating circle, i.e. κ. The directional deriatie N T of N in the direction T is a linear comination ρ of the partial deriatie ectors N u N N N and is parallel to the tangent plane.

2 (uu ) + (u u) u N u + ( ) () N (u ) + (u ) N T N u + N N u N N u + ( ) (3) (4) N T Figure : Normal section of s with the tangent ector T, N T is suject to scaling y the metric of the parameter space. In an arc length parameter direction s, s N s is the curature of the normal section. Gien a general parameterization (u, ), we can compute T N T. This quantity defines the second fundamental form II which descries curature information distorted y the local metric. T N T ( N u) ( N + N u) ( N ) (5) N N (6) N u N N u N (7) U t II U (8) Equation 8 defines II. II can e simplified to a set of quanties which are easier to compute y sustituting the expressions in Equations,3 for N u and N respectiely. Two of the terms of in of the entries of the matrix in Equation anish due to the dot product of orthogonal ectors. Further simplification using properties of the ox product of ectors yields Equation. Equation 3 is deried y susituting N from Equation and shows that II can e computed simply y the dot product of the second partial deriaties of with N. Equation 3 also demonstrates that II is a symmetric matrix. T u II h ij N N u N u N N u N (u ) u u (u ) (u u) (u ) uu ( u ) ( ) u ( ) ( ) uu N u N N N L M M N (9) (0) () () (3) (4) Equations 3,4 can e sustituted for II to yield alternate formulas for T N T. T N T (u N) + ( N) + ( N) (5) L + M + N (6) u N N u N N L M M N (7) (8) II contains information aout the curature of, ut it will e distorted y the metric if the gien parameterization is not an arc length parameterization. It is still necessary to counter act this distortion in order to compute the curature of the. 5 Coordinate Transformations Our purpose in studying differential geometry and the fundamental forms has een to compute the principle curature alues (κ, κ ) and directions ( k, k ) of a parametric surface at a point p. Thus far we hae deried II which contains curature information ut can e distorted y the metric of the parameterization. I descries this metric information. We must comine II and I to find the arc length curature alues. The gien parameterization (u, ) defines a set of asis tangent ectors. We can construct an orthonormal asis s t due to an arc length parameterization (s, t) where s and are aligned, see Figure 3(a). point T can e expressed in either coordinate system, see Figure 3(). T s s t (9) The coordinates s measure the length of T as opposed to the coordinates. T T s + + (30) We would like to work in the s t, so we must find the transformations to and from. The quantities a,, and θ in Figure 3(a) are defined y.

3 t θ a (a) Constructing an orthonormal asis on top of the gien parametric asis s t θ a Figure 3: Effects of the metric s s u T () point T in (u, ) and (s, t) coordinates t T T I s ( I ) t s s ( t) ( ) t s s s (4) (43) (44) (45) (46) s + (47) It is now possile to transform our gien parameters to coordinates on an orthonormal asis where we can measure lengths. We can also transform coordinates on the orthonormal asis ack to coordinates on our gien parameterization. 6 Curature a 0 s cos θ sin θ t s t sin θ 0 u t a sin θ cos θ a u u s (3) (3) (33) (34) The first fundamental form I can e represented y these coordinate transformations. I u s t s t t 0 0 (35) (36) t (37) t (38) a a cos θ a cos θ (39) The curature at a point p on the surface in any tangential direction T is T N T. T N T N N (48) II (49) This function is parameterized y the gien parameterization. We would like to study the function that sweeps a unit length tangent around p, so it is more conenient to use arc length coordinates. We must transform coordinates so that setting the arc length coordinates s cos φ sin φ sweeps out the curature alues. T N T s ( II ) t s s s IIŜ E t E (50) (5) We can also derie transformations etween coordinates of the different sets of asis ectors y sustituting Equations 3,34 respectiely into Equation 9. a θ φ s s (40) s (4) To erify that this makes sense, consider T T again and transform the coordinates using Equations 40,4. Figure 4: The relationship etween the principle parameter directions (E, E ), the gien general parameter directions (, ), and the constructed orthonormal asis ( s, t)in the tangent plane to at p The curature function will hae maximum and minimum alues that occur at directions that are orthogonal to each other. The eigenalues of IIŜ are the principle curature alues (κ, κ ). The

4 eigenectors of IIŜ are the coordinates of the principle curature directions in the arc length coordinates. These coordinates will not e directly useful ecause we only know the ectors, so we would actually prefer the eigenectors in coordinates on this asis. IIŜ II ( ) t a sin θ sin θ 0 cos θ a L M M N a sin θ sin θ sin θ ( cos θl + am) sin θ cos θ 0 a (5) (53) sin θ ( cos θl + am) cos θl a cos θm + a N (54) oling for the eigenalues, we find the characteristic equation. The principle curature alues are the eigenalues. 0 ( a (a cos θ) ) λ ( L a cos θm + a N ) λ + ( LN M ) (55) 0 ( EG F ) λ (GL F M + EN) λ + ( LN M ) 7 The Weingarten Operator (56) n alternate way of computing the principle curature alues and directions is y using the Weingarten Operator W, also known as the shape operator. W is the inerse of the first fundamental form multiplied y the second fundamental form. I W I II (57) is the inerse of the first fundamental form. I ( t) ( ) t G F EG F F E (58) (59) We will proe that W has the same eigenalues as IIŜ, i.e. the same principle curature alues. We will also proe that the eigenectors of W are transformed ersions of the eigenectors of IIŜ to coordinates on the gien asis. eigenalue matrix as for IIŜ, so the eigenalues of W are the principle curature alues. ( )t V ( V t), so the columns of ( )t V are the eigenectors of W. This is a transformed ersion of V, the eigenectors of IIŜ oer the arc length asis. The transformation transforms the arc length coordinates to coordinates oer the gien asis, see Equation 4. s s V t t ( ) t ( V ) t s s t t u u (67) (68) (69) To sole for the principle curature alues and directions using the gien parameterization, we sustitute Equations 4,59 into Equation 57 and simplify to derie W written as the coefficients of the first and second fundamental forms. W I II (70) G F L M (7) EG F F E M N GL F M GM F N (7) EG F EM F L EN F M Then we use the representation of W in Equation 7 and compute the eigendecomposition. The eigenalues are the principle curatures, and the eigenectors are the coordinate coefficients oer the gien parametric asis ectors. GL F M + EN κ, λ, (EG F ) (GL F M + EN) 4 (EG F ) (LN M ) (EG F ) (73) κ M κ + κ GL F M + EN (EG F ) T race (W ) (74) κ G κ κ LN M Det (W ) (75) EG F References GREINER, G., LOO, J., ND WEELINK, W. Data Dependent Thin Plate Energy and its use in Interactie urface Modeling. IIŜ V ΛV (60) II IIŜ t (6) W I II (6) ( ) t II (63) ( ) t IIŜ t (64) ( ) t IIŜ t (65) W ( ) t V ΛV t (66) Equation 66 is the eigendecomposition of W. The diagonal matrix Λ has the eigenalues of W on its diagonal. It is the same

5 quare Matrices: Quick Reference. General Matrices We will e manipulating the matrices I and II, so it is important to reiew some fundamental properties. Matrix B in Equation - is a general matrix. The inerse, B, can e computed in closed form as shown in Equation -, if the determinant of B, Det (B) (ad c), does not equal to zero. α. Eigenalues B a α c d B α d ad c c a (-) (-) For a general n n square matrix, the eigenalues of are n special scalar alues λ i (i :, n) such that for special corresponding eigenectors i (i :, n), multiplication of with i is the same as scaling i y λ i, Equation -3. By sustituting i with itself multiplied y the identity matrix I in Equation -4, we can then comine terms in Equation -5. λ i i i λ i (I i) i (-3) (-4) λ ii i i 0 ( λ ii) i (-5) i is in the nullspace of the matrix ( λ ii), ut the ector i 0 always satisfies Equation -5. We are interested in nonzero eigenectors i only, so ( λ ii) must e singular so that its nullspace will e nonempty. ( λ ii) is singular if its determinant is zero. Equation -6 is the characteristic equation of, and the n roots of this equation define the eigenalues λ i (i :, n). 0 Det ( λ ii) (-6) For the case of the general matrix B, the characteristic equation, Equation -7 can e expanded analytically to yield an equation that is quadratic in λ i, Equation -. 0 Det (B λ ii) (-7) ( ) a λi 0 0 Det (-8) α c d 0 λ i 0 ( ) a α Det αλi 0 (-9) c d 0 αλ i 0 α a αλi (-0) c d αλ i 0 ( α λ i α (a + d) λ i (ad c) ) (-) α Equation - can e soled analytically using the quadratic equation to yield the eigenalues λ and λ in Equation -. λ, a + d ± (a d) + 4c α (-).3 Eigenectors For a general n n square matrix, once the eigenalues λ i (i :, n) hae een computed, the corresponding eigenectors i (i :, n) are computed y finding the nullspace of the matrix ( λ ii). The null space is computed for each λ i y sustituting its alue into the matrix ( λ ii) and then performing Gaussian elimination. The matrix will e singular, so at least one row will ecome all zeroes. It is then possile to sole for the components of i as parametric equations of a suset of free components. The parametric nature of the i s show that the eigenectors are not unique. In Equation -3, i can e replaced with itself multiplied y an aitrary scalar. This aritrary scaling of the eigenectors is the only degree of freedom if the eigenalues are unique, i.e. their algeraic multiplicities are all one. In this case, the nullspaces are all -dimensional suspaces, lines through the origin. If there are multiple roots to the characteristic equation, then the algeraic multiplicity m of some of the eigenalues will e greater than one. The m eigenectors corresponding to such an eigenalue will e a set of linearly independent ectors that span the m-dimensional suspace. The choice of these ectors has more degrees of freedom than the scaling in the -dimension case. If m, then the suspace is a plane through the origin, and there is a rotational degree of freedom for choosing the eigenectors as well as the scaling degree of freedom. In general, there are some matrices with eigenalues of algeraic multiplicity m > that do not hae a complete set of m associated eigenectors, i.e. the geometric multiplicity is less than the algeraic multiplicity. These matrices are defectie. For our purposes for the case of matrices, we will assume that the matrices are non-defectie. For the case of a non-defectie matrix B, we will try to sole for the eigenectors analytically y sustituting the eigenalues from Equation - into the matrix in Equation -0 and then performing Gaussian elimination. 0 a αλi α c d αλ i i a αλi i,x 0 c d αλ i i,y 0 a a+d± c d a+d± 0 (a d) c (a d) (a d) c i,x i,y (-3) (-4) i,x i,y (-5) i,x i,y (-6) (-7) Equation -7 shows that the matrix ( λ ii) is singular. We will let i,y then we can sole for i,y, Equation -8., (a d)± c (-8) In Equation -8, if c 0, then the eigenector is not well defined. We must handle the case where c 0 or 0 separately. In this case the eigenalues simplify to λ a α and λ d α, and gaussian elimination will produce a matrix with a single nonzero element. Hence, 0 t and 0 t, respectiely.

6 .4 Eigen Decomposition For a non-defectie n n square matrix, the geometric multiplicity equals the algeraic multiplicity for all of the eigenalues λ i (i :, n), so there is a corresponding set of n linearly independent eigenectors i (i :, n). It is then possile to diagonalize y transforming y the eigenectors and their inerse. Rememering Equation -9 that defines each eigenector, we can write all n equations as columns next to each other in Equation -0. i iλ i (-9)... n λ λ... nλ n (-0) We will define V as the matrix of column eigenectors. We can rewrite the right side of Equation -0 as the eigenector matrix V multiplied y a diagonal matrix Λ that has the n eigenalues along its diagonal, Equation -. fter right multiplying V, we derie the eigen decomposition of, Equation -4. V V V Λ V V V ΛV... n V ΛV λ λ λ n (-) (-) (-3) (-4) The eigen decomposition of, Equation -4, makes it ery efficient to raise to an integer exponent e. Consider first squaring, Equation -5. λ + λ λ λ a + d α ad c α T race () (-3) Det () (-3) We rememer that the determinant of the multiplication of two matrices is the product of their determinants. Det (B) Det () Det (B) (-33) We can then proe that the determinant of a matrix is equal to the product of its eigenalues. V ΛV Det () Det ( V ΛV ) (-34) (-35) Det (V ) Det (Λ) Det ( V ) (-36) Det ( V V Λ ) (-37) Det (Λ) (-38) λ i (-39) V ΛV V ΛV V ΛΛV V Λ V (-5) Equation -6 for to an integer exponent e is then deried y induction. This is ery efficient to compute ecause raising Λ to the exponent e can e computed y simply raising the n diagonal elements to e. e V Λ e V (-6) For the matrix B, the eigen decomposition is the following: B V Λ V (-7) a d c a d+ c V Λ a+d 0 α a+d+ 0 V c a d+ (a d) a d + 4c α c c (-8) (-9) (-30)