u i ( u i )a v a = i ( u i )a v i n = x u 1 x u 2

Transcription

1 parametrc surfaces defne a surface x(u 1, u 2 ) n R 3, over some patch of the U doman n R 2. f all goes well, are two ln nd tangent vectors n R 3 x u super/sub scrpts (..n) wll represent ndces. they form a bass for the tangent plane at x(u). tp: unon of all tangents of curves n the surface at the pont. fancy termnology: drop the x u even fancer: add a superscrpt (a..h) to tell us that t s a tangent vector any tangent vector can be wrtten as u )a v a = u )a v normal we can compute the (non unt) normal vector as n = x u 1 x u 2 change of parametrzaton cob suppose have a one to one mappng from a 2d patch (ûˆ1, ûˆ2 ) to the 2d patch (u 1, u 2 ). then the same surface can be parametrzed as x(ûˆ1, ûˆ2 ) = x(u 1 (ûˆ1, ûˆ2 ), u 2 (ûˆ1, ûˆ2 )) at every pont then there s a Jacoban (matrx) of ths mappng ( s row, j s column) we wll assume J s be non-sngular. and we also have the tangent bass J ĵ = u ûĵ ûĵ )a we can wrte the relatonshp between our two tangent bases as ûĵ )a = u )a J ĵ 1

2 just use the chan rule thus we can change the coordnates of a vector as v a = j ûĵ )aˆv j = j = =: u )a J ĵ ˆvĵ u )a j u )a v J ĵ ˆvĵ so J can be used to push forward tangent coordnates wrt û to coordnates wrt u v = j J ĵ ˆvĵ the hats n the J symbol remnd us ths. covectors covector: an operator that takes n vectors and gves out numbers n a lnear fashon s called a covector w a. apply a covector to a vector by wrtng them next to each other (n any order) usng a shared scrpt: v a w a. lets calculate n a bass v a w a = w a u )a v = w a u )a v =: w v dual bass gven our bass for vectors, we have a natural dual bass (du ) a, wth (du ) a u j )a = δ j lets take the w numbers from above and wrte a covector usng coordnates x a = w (du ) a we can compute n coordnates: x a v a = w v so x a = w a so w a v a = w v, where w and v are the coordnates n a fxed bass. so we now know how to caculate the coordnates of a covector n the dual bass. Smply apply the co-vector to the prmal bass vectors. dual change of paremeterzaton gven our earler change of parametrzaton (same J matrx), we have the the assocated dual transformaton rule (du ) a = j J ĵ (dûĵ) a 2

3 so the same J can also be used to transform covector coordnates wrt u to coordnates wrt û. w a = = = j = j w (du ) a w J ĵ (dûĵ) a j (dûĵ) a w J ĵ (dûĵ) a wĵ and so wĵ = w J ĵ note the drectons. to complete the dctonary, we would have to compute the nverse matrx to J. tensors 1 a covector w a takes one vector and gves us a number lnearly we can defne scalar multply and addton for covectors. conversely a vector v a takes one covector and gves us a number lnearly we can generalze ths. a tensor w ab takes n an ordered par of two vectors and gve out a number, blnearly. here we are namng the frst slot a and the second slot b. superscrpts on the nput vectors tell us whch slot to put them n. let w ab take n two covectors and gve out a number, blnearly. let w b a take n one vector and one covector and gve out a number takes a vector and covector and gves a number takes a vector and gves out a vector such an object can have egenvectors one (not the only) way to get these s by takng an outerproduct say m ab := w a v b tensors 2 ths basc dea can be generzed to (, j) tensors that take vectors and co-vectors and turn them n to numbers. eg T abc de once we pck a bass (whch gves us a dual bass as well), we can represent a tensor wth coordnates T jk lm. a matrx wth one dmenson for each slot. n order to change to the hat bass, we sum aganst J mˆm on the subscrpts and ts nverse Kî aganst the superscrpts. computng to apply to nput co/vectors, as Td abvd w a x b, we can compute ths as jk T j k vk w x j answer s bass nvarant we can also defne N a b M b c as the (1, 1) tensor represented by j N j M j k answer s bass nvarant we can also defne contracton of an upper aganst a lower N a a usng N. 3

4 also bass nvarant! frst fundamental form suppose want to take the dot product n R 3 between two tangent vectors, v a, and w b. the dot operator takes two tangent vectors and gves out a number. the dot operator s lnear n each of ts two slots. so we can wrte such an operator as g ab g s the preferred letter for ths operator. defned at each pont n the surface. we wrte the dottng as g ab v a w b the a,b tell us whch slot of g to put n v and w, so the order of how we lst the symbols s not relevant t also happens to be the case that g s symmetrc. g ab = g ba queston can a tensor T a b be symmetrc? g s called the metrc tensor of the surface. we defne the Frst fundamental form, whch maps a tangent vector to ts squared length metrc n coordnates I(v a ) := g ab v a v b suppose we wrte our two vectors wrtten wth coordnates n a bass v a = w b = j u )a v u j )b w j then we can compute the dot of any two vectors as g ab v a w b = g ab ( = j = g j v w j j u )a v )( j u j )b w j ) v w j g ab u )a u j )b where g j := g ab u )a u j )b remember, these 4 numbers g j depend on our choce of parametrzaton/bass f we had chosen a dfferent bass, then we would have a dfferent set of numbers ˆv, ŵ j ĝ j that represent the same tensors. sothermal coordnates not amazng fact: one can always fnd a parameterzton where g 11 = g 22 = 1, and g 12 = 0 = g 21. at a sngle pont. 4

5 ths s a useful tool sothermal parametrzaton: where g 11 = g 22 = λ 2, and g 12 = 0 = g 21. tangent bass s ortho wth same scale λ s spatally varyng amazng fact: one can always fnd an sothermal parametrzaton over a patch area of patch fx our parametrzaton u compute our 4 numbers g j at every pont put them nto a matrx G at every pont we can compute area as det(g) A = du 1 du 2 Ω nverse metrc there s a well defned dentty tensor I a b. not true of other scrpt types. we can defne an nverse metrc g ab to be the tensor wth the property g ab g bc = I a c. we use the same letter g, but now t has superscrpts. curves on surfaces we can represent a curve on a surface as wth two functons u (t) we can compute ts tangent vector c(t) = x(u 1 (t), u 2 (t)) ( dc dt )a = x du )a u dt so we see that the coordnates of the tangent vector ( dc dt )a n the bass u ) a are drectonal dervatve suppose one has a real valued functon f over every pont on the surface. here s how to defne the drectonal dervatve of f as one moves n the tangent drecton v a. we fnd some curve c(t) = x(u 1 (t), u 2 (t)), that has ts tangent v a at the pont n queston ths defnes a scalar functon f(t) whch we can dfferentate du dt df dt = = f du u dt v f u 5

6 note that the result only depends on the tangent of the curve! at a pont, and fxng f, the result s a lnear functonal of the tangent drecton. so we wrte the drectonal dervatve as v a ( a f) where a f s ths lnear functonal and so has meanng even before we contract t wth v a. knds of dervatves we get the same answer no matter how we parametrze, so we use the generc symbol snce we were just computng ds/dt. t also can be wrtten as v a (df) a, D v f, or v a [f] v a ( a f) a means partal dervatve operator. a means covarant dervatve. lttle d here means exteror dervatve. D v means drectonal dervatve. we can even just thnk of ths as smply applyng the vector feld to f. n ths nterpretaton, we can thnk of tangent vectors, not as geometry, but as a operator on functons that behaves lke a dervatve. that s why we used the bass notaton. when appled to scalar functons over a surface, they all do the same thng. gradant the gradant of f s a vector w a such that g ab w a v b = v a a f clearly the gradant s g ab b f note: gradant vector needs the metrc for ts defnton, unlke the dervatve co-vector. (only) n an orthonormal bass, w = f u steepest ascent gradant vector gves the drecton of steepest ascent max x a a f x x a g ab x b =1 work n orthonormal bass max x x x =1 x f u maxmzed when the coordnate-vector x s a scale of the coordnate-vector f u whch, n these coordnates, s the gradant drecton notce: that the drecton of steeptest ascent requres the use of the metrc. dervatve of normal let call the unt normal n. thnk of t as 3 scalar functons. lets look at all the drectonal dervatves of n on the surface. ths gves us a mappng from from the tangent plane to a vector n R 3 6

7 fact: the dervatve of normal as we move along any tangent drecton always les n the (embedded) tangent plane of the surface pont. ntuton: change of a unt length vector feld s orthogonal to the vector tself shape operator so we now have a mappng from tangent vectors to tangent vectors. the mappng s lnear n the nput. we negate ths and call ths mappng the shape operator S c a the scrpts means that t takes n one tangent vector and outputs one tangent vector. so the negated normal dervatve as one moves along a tangent vector v a s second fundamental form z c = S c av a lets defne an operater that takes the the negated normal dervatve n the drecton of one vector v a, and then dots t wth a second vector w b to obtan a number w b g cb S c av a ths defnes a bllnear form on a par of vectors, whch we can represent as fact: q ab happens to be symmetrc. n ths case we say that S c a s self adjont (wrt the metrc). q ab := g cb S c a we defne the second fundamental form operatng on a sngle tangent vector as II(v a ) := q ab v a v b amazng fact (bonnet): f two surfaces have the same I and II at all ponts, then they must be the same geometry up to a eucldean transform. normal sectons and curvature defne the normal curvature n the drecton v a. as k n (v a ) := II(v a )/I(v a ) g bc S b av a v c g ac v b v c suppose ntersect a surface wth a plane that ncludes the normal and one tangent vector v a at a pont nterpretaton, the normal/tangent slced wth the surface gves us a planar curve. ths curve has a well defned noton of curve-curvature (uses osculatng crcle). fact: ths curve s curvature agrees wth the just defned normal curvature! prncple curvatures d lke to get a handle on the normal curvature as spn the tangent vectors around. compute egenvalues k 1, k 2 and egenvectors e a 1, e a 2 of S a b parametrzaton ndependent they exst snce S a b s self adjont 7

8 and evecs wll be orthogonal wrt g ab (spectral theorem) call these prncpal curvatures and prncpal drectons no normal twstng as walk n these two drectons! mn-max thm; : the extrema of II(v a ) subject to I(v a ) = 1 are n the prncpal drectons detals: spectral thm work n orthonormal coordnates, g ab represented by the dentty. (a) q ab as well as (b) S a b are now represented by a matrx S. (a) tells us that S s a symmetrc matrx (b) tells us that the egevectors of S gve us the coordnates of the egenvectors of S a b. from matrx theory, snce S s symmetrc, t has a real egendecomposton, S = U t DU, where the columns of U t are orthogonal n E 2. these are the coordnates of the egenvectors of S a b and are clearly orthogonal wrt g ab. ths gves us the spectral theorem detals: mn max thm n these coordates, our optmzaton problem s max x x t Sx s.t. x t x = 1 max x x t Sx max x x t x (x t U t )D(Ux) (x t U t )(Ux) max u max u (u t )D(u) (u t )(u) λ u 2 u2 now λ u 2 u2 λ M u 2 u2 = λ M u2 u2 = λ M max s acheved f x s max egenvector smlar for mn gaussan curvature we call the product k = k 1 k 2 the gaussan curvature at a pont ts sgn -0+ determnes f we call the pont hyperbolc (saddle), parabolc, or ellptcal (convex or concave) so the surface s broken up nto hyperbolc, convex, and concave regons, separated by parabolc curves. amazng fact (gauss): gaussan curvature can be computed completely from knowng g ab and ts dervatves. amazng fact (gauss-bonnet): f have a closed surface, then the area ntegral of ts gaussan curvature s completely determned by ts euler characterstc. mean curvature H = 1/2(k 1 + k 2 ) s called mean curvature 8

9 a patch s called a mnmal surface f t has zero mean curvature. must be saddle everywhere. fact: can t make ts area smaller by deformng t f one has an sothermal paramterzaton, then the normal vector, wth length of the mean curvature can be computed usng the laplacan x := 2 x u u = 2λ2 Hn there s actually an operator called laplace-beltram, whch can be defned wthout fxng any specal parametrzaton, whch also gves Hn conjugacy we say two tangent vectors v a, w b are conjugate f q ab v a w b = 0 t means that the as move n drecton v, the normal turns n a drecton that s orthogonal to w. conjugacy and slhouettes a slhouette pont on a surface has ts normal orthogonal to the vewng drecton. ths means that the vewng vector s a tangent vector at a slhouette pont, so we can call t v a. the slhouette curve on the surface separates out regons that face towards and away from the vewer. so the slhouette must generally form a smooth smple closed curve on the surface. the slhouette curve has a tangent drecton, say w b. as we move n the drecton w b, the normal stays orthogonal to the vew ray, so t can t be bendng towards or away from v a so these must be conjugate drectons. q ab v a w b = 0 conjugacy also shows up when we look at sophotes of shadng as compared to the drecton to the lght. asymptotcs and cusps of slhouettes f a tangent s self conjugate, we call t an asymptotc drecton only hyperbolc regons have asymptotes as walk n an asymptotc drecton, the normal twsts q ab v a v b = 0 for a smooth slhouette to look non-smooth n projecton, t must be the case that we are lookng at the slhouette curve edge on the slhouette tangent equals the vew drecton. the vew ray must be an asymptotc drecton. because of vsblty, ths s where the slhouette ends n the mage gauss map the space of normal drectons can be thought of as ponts on a sphere usng the normal, we can map each pont to the sphere, called the gauss map n(p). 9

10 ellptcal regons map to the sphere wth preserved ortentaton and hyperbolc regons map to the sphere wth flpped orentaton so the gauss map has folds along parabolc curves. sllhouettes (orthogonal vew) are the premage of a great crcle on the gauss sphere so lps and beaks slhouette sngulartes must happen at parabolc ponts. dervatves of mappngs suppose have a mappng m from one surface Ŝ to another S. then f place a curve on the frst surface, t maps to a curve on the second surface. so can look at how the tangent vector of one curve at a pont maps to a tangent vector at m(p) fact: t does not make a dfference how chose the curve, as long as t has the desred tangent on the frst surface. so we have a dfferental mappng between the two tangent planes fact: the dfferental mappng s lnear n the nput tangent vectors so we can wrte ths mappng as dm b â the hats tell us that t maps a hatted vector to an unhatted vector. gauss map the dfferental of n(p) can be wrtten as dn b â the tangent of the gauss sphere must match the tangent at a pont. so we can dentfy both tangent planes makng dnˆbâ a lnear mappng from a tangent plane to tself. ths s another way to defne the shape operator Sˆbâ = dnˆbâ 10