Geometric Modeling

Transcription

1 Geometrc Modelng Notes on Crve and Srface Contnty Parts of Mortenson, Farn, Angel, Hll and others

2 From Prevos Lectres

3 Contnty at Jon Ponts (from Lectre ) Dscontnos: hyscal searaton Parametrc Contnty Postonal (C 0 ): no hyscal searaton C : C 0 and matchng frst dervatves C : C and matchng second dervatves Geometrc Contnty Postonal (G 0 ) = C 0 Tangental (G ) : G 0 and tangents are roortonal, ont n same drecton, bt magntdes may dffer Crvatre (G ) : G and tangent lengths are the same and rate of length change s the same sorce: Mortenson, Angel (Ch 9), Wk

4 Contnty at Jon Ponts Hermte crves rovde C contnty at crve segment jon onts. matchng arametrc st dervatves Bezer crves rovde C 0 contnty at crve segment jon onts. Can rovde G contnty gven collnearty of some control onts (see next slde) Cbc B-slnes can rovde C contnty at crve segment jon onts. matchng arametrc nd dervatves sorce: Mortenson, Angel (Ch 9), Wk

5 Comoste Bezer Crves Jonng adjacent crve segments s an alternatve to degree elevaton. (from Lectre 3) Collnearty of cbc Bezer control onts rodces G contnty at jon ont: Evalate at =0 and = to show tangents related to frst and last control olygon lne segment. 0) 3( ) ) 3( 3 ) ( 0 ( For G contnty at jon ont n cbc case, 5 vertces mst be colanar. (ths needs frther exlanaton see later slde) sorce: Mortenson

6 Comoste Bezer Srface Bezer srface atches can rovde G contnty at atch bondary crves. For common bondary crve defned by control onts 4, 4, 34, 44, need collnearty of: { 5, 3,,4,, }, Two adjacent atches are C r across ther common bondary ff all rows of control net vertces are nterretable as olygons of C r ecewse Bezer crves. (from Lectre 5) [: 4] Cbc B-slnes can rovde C contnty at srface atch bondary crves. sorce: Mortenson, Farn

7 Slemental Materal

8 x( x'( y'( Contnty wthn a (Sngle) Crve Segment Parametrc C k Contnty: ) ) ) Refers to the arametrc crve reresentaton and arametrc dervatves Smoothness of moton along the arametrc crve A crve P(t) has kth-order arametrc contnty everywhere n the t-nterval [a,b] f all dervatves of the crve, to the kth, exst and are contnos at all onts nsde [a,b]. A crve wth contnos arametrc velocty and acceleraton has nd -order arametrc contnty. b Ke ( Ke b ( Ke b )( cos sn ) y( aly rodct rle )(cos ) ) (cos (sn b Ke )( Ke )( Ke b b sn )( be )( be st dervatves of arametrc exresson are contnos, so sral has st -order (C ) arametrc contnty. b b ) ) Note that C k contnty mles C contnty for < k. Examle sorce: Hll, Ch 0

9 Contnty wthn a (Sngle) Crve Segment (contned) Geometrc G k Contnty n nterval [a,b] (assme P s crve): Geometrc contnty reqres that varos dervatve vectors have a contnos drecton even thogh they mght have dscontnty n seed. G 0 = C 0 G : P (c-) = k P (c+) for some constant k for every c n [a,b]. Velocty vector may jm n sze, bt ts drecton s contnos. G : P (c-) = k P (c+) for some constant k and P (c-) = m P (c+) for some constants k and m for every c n [a,b]. Both st and nd dervatve drectons are contnos. Note that, for these defntons, G k contnty mles G contnty for < k. These defntons sffce for that textbook s treatment, bt there s more to the story sorce: Hll, Ch 0

10 Rearameterzaton Relatonsh Crve has G r contnty f an arc-length rearameterzaton exsts after whch t has C r contnty. sorce: Farn, Ch 0 Two crve segments are G k geometrc contnos at the jonng ont f and only f there exst two arameterzatons, one for each crve segment, sch that all th dervatves, k, comted wth these new arameterzatons agree at the jonng ont. sorce: cs.mt.ed

11 Addtonal Persectve Parametrc contnty of order n mles geometrc contnty of order n, bt not vce-versa. Sorce: htt://

12 Contnty at Jon Pont Parametrc Contnty Geometrc Contnty Defned sng arametrc dfferental roertes of crve or srface C k more restrctve than G k Defned sng ntrnsc dfferental roertes of crve or srface (e.g. nt tangent vector, crvatre), ndeendent of arameterzaton. G : common tangent lne G : same crvatre, reqrng condtons from Hll (Ch 0) & (see dfferental geometry sldes) Osclatng lanes concde or Bnormals are collnear. sorce: Mortenson Ch 3,. 00-0

13 Parametrc Cross-Plot For Farn s dscsson of C contnty at jon ont, cross-lot noton s sefl. sorce: Farn, Ch 6

14 Comoste Cbc Bezer Crves (contned) sorce: Farn, Ch 5 Doman volates (5.30) for y comonent. crves are dentcal n x,y sace Doman satsfes (5.30) for y comonent. Parametrc C contnty, wth arametrc domans consdered, reqres (for x and y comonents): ( b 3 a) b 3 b ( c 3 b) b 4 b 3 (5.30)

15 Comoste Bezer Crves sorce: Mortenson, Ch 4, ,,,, q q q m m m For G contnty at jon ont n cbc case, 5 vertces mst be colanar. (follow- from ror slde) Achevng ths mght reqre addng control onts (degree elevaton) crvatre at endonts of crve segment 3 consstent wth:

16 C Contnty at Crve Jon Pont Fll C contnty at jon ont reqres: Same rads of crvatre* Same osclatng lane* These condtons hold for crves () and r() f: r r r * see later sldes on tocs n dfferental geometry sorce: Mortenson, Ch

17 Pecewse Cbc B-Slne Crve Smoothness at Jont famlar staton looks ncorrect looks ncorrect looks ncorrect famlar staton crvatre dscontnty sorce: Mortenson, Ch 5

18 Control Pont Mltlcty Effect on Unform Cbc B-Slne Jont C and G control ont mltlctes = C and G One control ont mltlcty = C and G One control ont mltlcty = 3 ( ) ( ) 0 (3 4) ( 3 ) 3 C 0 and G 0 0 (3 ) ( 3 ) One control ont mltlcty = 4 One crve segment degenerates nto a sngle ont. Other crve segment s a straght lne. Frst dervatves at jon ont are eqal bt vansh. Second dervatves at jon ont are eqal bt vansh. ( ) ( ) 3

19 Knot Mltlcty Effect on Nonnform B-Slne If a knot has mltlcty r, then the B- slne crve of degree n has smoothness C n-r at that knot. sorce: Farn, Ch 8

20 A Few Dfferental Geometry Tocs Related to Contnty

21 Local Crve Tocs Prncal Vectors Tangent Normal Bnormal Osclatng Plane and Crcle Frenet Frame Crvatre Torson Revstng the Defnton of Geometrc Contnty sorce: Ch Mortenson

22 Intrnsc Defnton (adated from earler lectre) No relance on external frame of reference Reqres eqatons as fnctons of arc length* s: ) Crvatre: ) Torson: f ( s) g(s) Torson (n 3D) measres how mch crve devates from a lane crve. For lane crves, alternatvely: d ds *length measred along the crve Treated n more detal n Chater of Mortenson and Chater 0 of Farn. sorce: Mortenson

23 Calclatng Arc Length Aroxmaton: For arametrc nterval to, sbdvde crve segment nto n eqal eces. L n l where l l sng L d s more accrate. sorce: Mortenson,. 40

24 Tangent nt tangent vector: t sorce: Mortenson,. 388

25 Normal Plane Plane throgh erendclar to t 0 ) ( z z y y x x z z y y x x ),, ( z y x q sorce: Mortenson,

26 Prncal Normal Vector and Lne Movng slghtly along crve n neghborhood of cases tangent vector to move n drecton secfed by: Use dot rodct to fnd rojecton of onto Prncal normal vector s on ntersecton of normal lane wth (osclatng) lane shown n (a). Bnormal vector b t n les n normal lane. sorce: Mortenson,

27 Osclatng Plane Lmtng oston of lane defned by and two neghborng onts j and h on the crve as these neghborng onts ndeendently aroach. Normal vector les n osclatng lane. Tangent vector les n osclatng lane. Note:, j and h cannot be collnear. x y z x y z x y z x y z 0 sorce: Mortenson,

28 Frenet Frame Rectfyng lane at s the lane throgh and erendclar to the rncal normal n : ( q ) n 0 Note changes to Mortenson s fgre.5. sorce: Mortenson,

29 Crvatre Rads of crvatre s and crvatre at ont on a crve s: 3 Recall that vector osclatng lane. les n the Crvatre of a lanar crve n x, y lane: d y / dx ( dy / dx) 3/ Crvatre s ntrnsc and does not change wth a change of arameterzaton. sorce: Mortenson,

30 Torson Torson at s lmt of rato of angle between bnormal at and bnormal at neghborng ont h to arc-length of crve between h and, as h aroaches along the crve. Torson s ntrnsc and does not change wth a change of arameterzaton. sorce: Mortenson,

31 Rearameterzaton Relatonsh Crve has G r contnty f an arc-length rearameterzaton exsts after whch t has C r contnty. Ths s eqvalent to these condtons: C r- contnty of crvatre C r-3 contnty of torson Local roertes torson and crvatre are ntrnsc and nqely determne a crve. sorce: Farn, Ch 0,.89 & Ch,. 00

32 Local Srface Tocs Fndamental Forms Tangent Plane Prncal Crvatre Osclatng Parabolod sorce: Ch Mortenson

33 Local Proertes of a Srface Fndamental Forms Gven arametrc srface (,w) Form I: d d Ed Fddw Gdw E F w G w w Form II: L n d (, w) dn(, w) Ld Mddw Ndw M w n Usefl for calclatng arc length of a crve on a srface, srface area, crvatre, etc. N ww n n w w Local roertes frst and second fndamental forms are ntrnsc and nqely determne a srface. sorce: Mortenson,

34 Local Proertes of a Srface Tangent Plane 0 w q 0 w w w z z z z y y y y x x x x sorce: Mortenson,. 406 w )/, ( w w w )/, ( q (,w ) comonents of arametrc tangent vectors (,w ) and w (,w )

35 Local Proertes of a Srface Prncal Crvatre Derve crvatre of all arametrc crves C on arametrc srface S assng throgh ont wth same tangent lne l at. contans l normal crvatre vector k n = rojecton of crvatre vector k onto n at k ( k n) n n normal crvatre: n k n n tangent lane wth arametrc drecton dw/d L( d / dt) M ( d / dt)( dw / dt) N( dw / dt) n E( d / dt) F( d / dt)( dw / dt) G( dw / dt) sorce: Mortenson,

36 Local Proertes of a Srface Prncal Crvatre (contned) Rotatng a lane arond the normal changes the crvatre n. crvatre extrema: rncal normal crvatres tyograhcal error? sorce: Mortenson,

37 Local Proertes of a Srface Osclatng Parabolod Second fndamental form hels to measre dstance of srface from tangent lane. d (q ) n As q aroaches : d f Ld Mddw Ndw Osclatng Parabolod sorce: Mortenson,. 4

38 Local Proertes of a Srface Local Srface Characterzaton sorce: Mortenson, a) LN M 0 b) LN M 0 Elltc Pont: locally convex Hyerbolc Pont: saddle ont c) LN M L M N 0 0 L M N Planar Pont (not shown) 0 tyograhcal error? Parabolc Pont: sngle lne n tangent lane along whch d =0