Topics. CS Advanced Computer Graphics. Differential Geometry Basics. James F. O Brien. Vector and Tensor Fields.

Transcription

1 CS 94-3 Advanced Computer Graphics Differential Geometry Basics James F. O Brien Associate Professor U.C. Berkeley Topics Vector and Tensor Fields Divergence, curl, etc. Parametric Curves Tangents, curvature, and etc. Parametric Surfaces Normals, tangents, curvature, etc. Implicit Surfaces Normals, curvature, etc.

2 Vectors A vectors defines a magnitude and direction v ˆv v/ v Not just a list of numbers Particular numbers are an artifact of the coordinate system we chose Not all coordinate systems are orthonormal v [v x,v y,v z ] Nearly everything that is useful can be defined w/o coordinate system Vectors transform like vectors v A v No set location (e.g. no root) But may be functions of location v v(u) v v(x, y) 3 3 Tensors Tensors transform like tensors e.g. T A T A T Tensors used to define oriented quantities Independent of coordinate system Specific realization will depend on coordinate system Cartesian tensors -- orthonormal coordinate system General tensors -- non-orthonormal coordinate system Tensors have rank Not related to dimension of space Rank 0 scalars Rank vectors Rank matrices Rank 3 don t work well in matrix-vector notation T A T A 4 4

3 Tensors Examples a b a T b a b Cos( ab) a b T P (A a) (A b) T A (a b T ) A T A P A T R x x T + y y T + z z T v v T + v v T S v v Note the way inner and otter products behave Summation Notation Notation due to Einstein a a i A A ij Makes life much easier Takes a while to get used to Useful in other contexts as well Free index c A b c i A ij b j Appears on both sides Unique in each term Implied for all Dummy index Appears exactly twice in each term Implied sum over A RAR T A ij A kl R ik R jl Different for general tensors s a b s a i b i A a b T A ij a i b j c T b T A T c i b j A ij c A b c i b j A ij 6 6

4 Summation Notation Two special symbols Delta ij Permutation ai ij kij ijk aj kab ia jb ib ja if i j 0 if i j ijk ij if i, j, k are even permutation of,, 3 if i, j, k are odd permutation of,, 3 0 else If you re slumming in ij 0 if i, j are, if i, j are, else Scalar Fields Scalar as function of some spatial variable(s) e.g.: f (x, y) f (x) Sin(x)Sin(y) Density Plot Height-field Plot

5 Vector Fields Vector as function of some spatial variable(s) e.g.: v(x, y) v(x) [Sin(x), Cos(y)] v(x, y) v(x) [, Sin(y), 0] 9 9 Differential Operators on Fields Derivatives of field w.r.t. spatial coordinates Coordinates implicit given field parameterization Linear operators on the field Not tied to any particular coordinate system Basic operators Gradient Divergence Curl Laplacian e i x i i [ x, y, z ] i i xi All expressed with (a.k.a. Nabla or del) 0 0

6 Gradient Often applied to scalar fields Gives direction of steepest accent Also has meaning for higher rank fields Elevates rank by one e.g. velocity gradient of a Newtonian fluid gives the strain rate gradf(x) f(x) f(x) x, f(x) x, f(x) x 3, f(x) x + y f(x) [x, y] Divergence For a vector field it describes the net expansion or contraction Lowers rank by one Divergence of vector field is a scalar An inner product of derivatives with the field div v(x) v(x) T v(x) v x(x) x + v y(x) x + v z(x) x 3 [Sin(x), Cos(y)] Cos(x) + Sin(y)

7 Curl For a vector field it describes the net rotation Cross product of derivatives with the field Scaler in D, vector in 3D curl v(x) v(x) [Cos(y), 0] Sin(y) cos (x) sin (y) + sin (x) sin (y) 4 4 Laplacian Divergence of Gradient Scalar second derivative operator Difference between a point and its surround Often used for smoothing of some sort cos (x) sin (y) xx + yy + zz cos (x) cos (y)

8 Notation Examples v(x) f(x) v i i f s(x) v(x) s i v i c(x) v(x) c i ijk j v k a(x) (v(x) )b(x) a i v j j b i 5 5 Fun Facts ( v) 0 ( s) 0 Both are obvious in tensor notation Helmholtz-Hodge decomposition Scalar and vector potentials Smooth, differentiable vector field a s + v + h s v h irrotational or curl-free part solenoidal or divergence-free part harmonic part 6 6

9 Directional Derivative df dx x f Add a picture or something Parametric Curves Curve is a geometric entity Set of points in space In neighborhood of any point it is isomorphic to a line Generator function: x x(t) A vector valued function (careful with vector ) A scalar function for each dimension of embedding space A particular parameterization is arbitrary and not unique Parameterization is not intrinsic [cos( ), sin( )] u u +, u u + 8 8

10 Derivatives Given function for curve we can take derivatives w.r.t. the parameter: ẋ dx dt The derivatives have names based on physical analogs Velocity Acceleration Jerk Snap, Crackle, and Pop Speed is the magnitude of velocity s ẋ All are dependent on parameterization and not intrinsic Note that, e.g., velocity is a vector field on t 9 9 Arclength Let s A(t) A(t) is the arclength of the curve 0 t x( ) d The arclength reparameterization of the curve is ˆx(s) x(a (s)) The arclength parameterization is unique up to sign change and translation dˆx(s) dx(t) dx(t) and ds dt dt dˆx(s) ds Closed form arclength parameterization may be hard to find. 0 0

11 Tangent Vector Tangent vector is a geometric property of the curve Does not depend on parameterization Tangent may exist where velocity is zero or may be undefined T dˆx(s) ds Curvature and Normal Note: T T (T T) () T T 0 Therefore: T T We can write: T N Curvature of the curve at this point Normal of the curve at this point Taylor expansion implies that if curvature is zero curve must be locally a straight line.

12 Frenet Frame Define binormal by B T N Gives us orthonormal coordinate frame: Frenet Frame Moves along curve Give local frame of reference T N B Not defined at inflection points where there is no curvature Frenet Frame Image from Wikipedia Osculating Plane Defined by N and T Locally contains the curve Normal Plane Defined by N and B Locally perpendicular to the curve 4 4

13 Torsion B B B B 0 B T 0 B T + B T 0 B T B T B. N 0 B B and B T Change in binormal is then B N Torsion If torsion is zero, we have a planar curve. The minus sign is to make positive torsion CCW w.r.t tangent. 5 5 Evolution of Frenet Frame N N N T + B N T Recall it s an orthonormal basis. N B Differentiate Yields Therefore We know N T 0 and N B 0 N T N N N B N ( )N N T + B T N and B B 6 6

14 Evolution of Frenet Frame T N N T + B B N T N B T N B ODE for evolution of Frenet Frame Given starting point, if you know curvature and torsion, then you can build curve. (Need speed also if not arclength parameterized.) Discrete analogy: stacking up macaroni 7 7 Radius of Curvature ˆx(s) [r cos T [ sin s r T [ r cos s r s r,rsin s r ], cos, Note that ˆx s r ] r sin s r ] r T r Curvature is inverse of radius of curvature. 8 8

15 Some Formulae For arclength parameterized curve ˆx(s) ˆx (ˆx ˆx ) ˆx For arbitrarily parameterized curve x (t) x (t) x (t) 3 x (t) x (t) x (t) x (t) x (t) 9 9 Field Evaluated Along a Curve Curve defined in some space x(t) Function on embedding space of curve f(x) Composition function f(x(t)) df dt f dx dt 30 30

16 Parametric Surfaces Surface is a geometric entity Set of points in space In neighborhood of any point it is isomorphic to a plane Generator function: x(u) A vector valued function (careful with vector ) A scalar function for each dimension of embedding space Dimension of parameter is two A particular parameterization is arbitrary and not unique Parameterization is not intrinsic 3 3 Derivatives Given function for curve we can take derivatives w.r.t. the parameter: x(u) x(u) u v All are dependent on parameterization and not intrinsic Note that each one is a vector field on u Examples of degeneracies [v 3,u,v ] [v 3,u, v 5 ] [v(u + ),u( v), 0] 3 3

17 Tangent Space The tangent space at a point on a surface is the vector space spanned by x(u) u x(u) v Definition assumes that these directional derivatives are linearly independent. Tangent space of surface may exist even if the parameterization is bad For surface the space is a plane Generalized to higher dimension manifolds Non Orthogonal Tangents cos( ) cos( pi/) sin( ) cos( /) sin( /) [0..] [..] cos( ) cos ) cos(6 ) + ( ) cos(6 ) + sin( cos ( sin ) cos(6 ) + ( )

18 Normals The normal at a point is the unit vector perpendicular to the tangent space N u x vx u x vx The normal direction is determined Up to a sign change Relative to surface First Fundamental Pick a direction in parametric space: du [du, dv] Corresponding direction in the tangent plane: dx x u du + x v dv dx du x(u) For unit speed in parametric space, the sped in the embedding space is s dx dx du T ( x) ( x) T du dx dx du T I du I ux ux vx ux ux vx vx vx I ij ( i x k )( j x k ) 36 36

19 First Fundamental I ux ux vx ux ux vx vx vx I ij ( i x k )( j x k ) Encodes distance metric on the surface If tangents are orthonormal it reduces to identity Used as metric by Green s Strain Invariant w.r.t. translations and rotations of surface: ( i x k )( j x k ) ( i R kp x p )( j R kq x q ) e.g. R kp R kq ( i x p )( j x q ) x i R ij x j pq ( i x p )( j x q ) ( i x p )( j x p ) First Fundamental I ux ux vx ux ux vx vx vx I ij ( i x k )( j x k ) Transforms like a tensor in parameter space: u i R ij u j u i R ji u j Assume orthonormal transform... x k u i x k u j x k u p u p u i x k u q u q u j R ip x k u p x k u q R jq I ij R ip I pq R jq 38 38

20 Arclength Over Surface c(t) x(u(t)) l b a b a b a b a dc(t) dt dt dx dt dx dx dt du T I du dt x, y, z u, v Principle Tangents x, y, z u, v Bottom row is eigenvectors of I Not intrinsic features of the surface! 40 40

21 Principle Tangents 4 4 Orthonormal Parameterization Eigen decomposition of Fist Fundamental I RS R T AA T Define coordinate transform by du SR T du A T du du R(/S)du A T du In transformed parameterization is the identity. du T I du du T (/S) R T R S R T R (/S) du du T (/S) R T R S R T R (/S) du I Similar to definition of arclength reparameterization. 4 4

22 Second Fundamental Let dx be some tangent direction dx du x(u) The directional derivative of the normal is un N u du + N v dv The normal is unit length so it is perpendicular to its derivative. N dn dx N dn dx As shown in top-down view, the three vectors may not be co-planar. Surface may tilt to side as point moves Second Fundamental Let dx be some tangent direction dx du x(u) The directional derivative of the normal is un N u du + N v dv The change in normal restricted to the plane containing the tangent and normal is given by T N T dx u N (du x) (du N) du T u x un ux vn vx un vx vn du N dn dx 44 44

23 Second Fundamental T N T du T u x un ux vn vx un vx vn du du T II du Matches definition of curvature for curve defined by cutting surface with the normal-tangent plane, but scaled by the surface metric. N N dx dx Second Fundamental II ux un ux vn vx un vx vn uux N vux N uv x N vv x N Symmetry Easy to show second version by expanding normal Box product with repeat is zero Any change in normal length will be perpendicular to surface Permutation of box product does not change results 46 46

24 Osculating Paraboloid Tangent plane is linear approximation to surface at a point Osculating paraboloid is quadratic approximation to surface at a point Matches surface s First and Second Fundamentals at the point P(u) c 0 + c u + c v + c 3 u + c 4 uv + c 5 v Nature of Surface P(u) c 0 + c u + c v + c 3 u + c 4 uv + c 5 v Elliptic Hyperbolic Parabolic c 3 c 4 (c 5 /) > 0 c 3 c 4 (c 5 /) < 0 c 3 c 4 (c 5 /) 0 Includes planar case 48 48

25 Normal Curvature Curvature adjusted for surface metric and for velocity in parameter space: dut II du du T I du Normal Curvature dut II du du T I du du T I du du T II du Recall I RS R T AA T du R(/S)du A T du du T A I A T du du T A II A T du du T du du T A II A T du du T A II A T du du 50 50

26 Principal Curvatures du T II du du II A II A T Dot product projects away twisting curvature Eigenvectors are where there is nothing to project away Notice that it s a real and symmetric matrix II v v 5 5 Principal Curvatures II v v Elliptic Hyperbolic Parabolic > 0 < 0 0 Includes planar case 5

27 Weingarten Operator W I II A T A II A T A A II A T A T II A T If and u are an eigenvalue/vector pair of Then u A T u is an eigenvector of W with the eigenvalue The eigenvectors are expressed in the original parameterization II Gaussian curvature Measure of intrinsic flatness of the surface Imagine flat-landers computing π on the surface K detw det II det I 54 54

28 Mean curvature Average curvature of the surface Will be zero for minimal surfaces H + Tr (I II ) deti Parabolic Lines Curves on surface where Gaussian curvature is zero 56 56

29 Contours Surface normal perpendicular to view direction Generator curve: f(u, v) ( u S(u, v) vs(u, v)).v Contours Surface normal perpendicular to view direction Generator curve: f(u, v) ( u S(u, v) vs(u, v)).v

30 Geodesic Curves Given a curve, C, on a surface, S C(t) S(u(t), v(t)) The geodesic curvature is g + n n (N s N c ) Separates d C S 0 dt ui curvature into What s necessary to stay on surface What s wiggling in tangent plane Geodesics are curves with g i ODE for curve 0 Generalize straight lines Locally shortest path between points On a sphere they are great arcs u q (I )qp S k Sk u i u j up ui uj Geodesic Curves

31 Geodesic Curves 6 6 Geodesic Curves Note integration errors when passing near poles. 6 6

32 Geodesic Curves Flat w/ bump Hyperbolic w/ bump Lines of Curvature A line of curvature on a surface is tangent everywhere to one of the principal curvatures Except at umbilic points where the two principal curvatures are equal 64 64

33 Implicit Surfaces {x f(x) 0} See 005 paper by Ron Goldman N(x) f f K G f ( T f) f f 4 K M f ( T f) f f Tr( T f) f 3 K M ± K M K G 65 65