Reading. 14. Subdivision curves. Recommended:

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Derek Dixon
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1 eadng ecommended: Stollntz, Deose, and Salesn. Wavelets for Computer Graphcs: heory and Applcatons, 996, secton 6.-6., A Subdvson curves Note: there s an error n Stollntz, et al., secton A.5. Equaton A. should read: MV VΛ

2 Subdvson curves Idea: repeatedly refne the control polygon P P P curve s the lmt of an nfnte process Q lm P Chakn s algorthm Chakn ntroduced the followng corner-cuttng scheme n 974: Start wth a pecewse lnear curve Insert new vertces at the mdponts (the splttng step) Average each vertex wth the next (clockwse) neghbor (the averagng step) Go to the splttng step 4

3 Averagng masks Can we generate other B-splnes? he lmt curve s a quadratc B-splne! Instead of averagng wth the nearest neghbor, we can generalze by applyng an averagng mask durng the averagng step: r (, r, r, r, ) In the case of Chakn s algorthm: r 0 Answer: Yes ane-esenfeld algorthm (980) Use averagng masks from Pascal s trangle: Gves B-splnes of degree n+. n0: n n n r,,, n 0 n n: n: 5 6

4 Subdvde ad nauseum? After each splt-average step, we are closer to the lmt curve. How many steps untl we reach the fnal (lmt) poston? Can we push a vertex to ts lmt poston wthout nfnte subdvson? Yes! ocal subdvson matrx Consder the cubc B-splne subdvson mask: ( ) 4 Now consder what happens durng splttng and averagng: 7 We can wrte equatons that relate ponts at one subdvson level to ponts at the prevous: ( ) * 0 0 Q Q + QC * 0 0 Q ( QC + Q) Q Q Q Q Q Q Q Q * 0 * QC ( Q + QC+ Q) ( Q + 6QC+ Q) 4 8 Q Q Q Q Q Q Q Q ( C ) ( C) ( 4 4 C) 0 * 0* ( C ) ( C ) ( 4 C 4 ) 0 *

5 ocal subdvson matrx We can wrte ths as a recurrence relaton n matrx form: Q Q 6 Q C QC Q Q Q SQ Where the Q s are (for convenence) row vectors and S s the local subdvson matrx. We can thnk about the behavor of each coordnate ndependently. For example, the x- coordnate: x x 6 x C xc x x - X SX ocal subdvson matrx, cont d rackng ust the x components through subdvson: 0 X SX SSX SSSX SX he lmt poston of the x s s then: X lm SX OK, so how do we apply a matrx an nfnte number of tmes??

6 Egenvectors and egenvalues o solve ths problem, we need to look at the egenvectors and egenvalues of S. Frst, a revew et v be a vector such that: Sv λv We say that v s an egenvector wth egenvalue λ. An nxn matrx can have n egenvalues and egenvectors: Sv λ v Sv n λ v If the egenvectors are lnearly ndependent (whch means that S s non-defectve), then they form a bass, and we can re-wrte X n terms of the n egenvectors: X av n n o nfnty, but not beyond Now let s apply the matrx to the vector X: n n n 0 X SX S av asv aλv Applyng t tmes: n n n X S X S av asv aλ v et s assume the egenvalues are non-negatve and sorted so that: Now let go to nfnty: If λ >, then: If λ <, then: If λ, then: λ > λ λ λ n > 0 n 0 lm lm λ X SX a v 6

7 Evaluaton masks What are the egenvalues and egenvectors of our cubc B-splne subdvson matrx? λ v We re OK! λ v 0 But where dd the x-coordnates end up? What about the y-coordnates? λ 4 v Evaluaton masks, cont d o fnsh up, we need to compute a. Frst, we can reorganze the expanson of X nto the egenbass: a a V a 0 X av + av + + anvn v v vn A We can then solve for the coeffcents n ths new bass: 0 A V X a u a X a u u 0 n n Now we can compute the lmt poston of the x- coordnate: 0 xc a ux n We call u the evaluaton mask. 4 7

8 Evaluaton masks, cont d eft egenvectors Note that we need not start wth the 0 th level control ponts and push them to the lmt. If we subdvde and average the control polygon tmes, we can push the vertces of the refned polygon to the lmt as well: x S X u X he same result obtans for the y-coordnate: y S Y u Y What are these u-vectors? Consder the egenvector relaton: Sv λ v We can re-wrte ths as a matrx: SV VΛ where Λ s a dagonal matrx flled wth the egenvalues of S. Now lets multply both sdes by V - from the left and rght and then smplfy: V ( SVV ) V ( VΛ) V V SΛV USΛU 5 hus, we fnd that the u-vectors obey the relaton: us λ u hese are the left egenvectors of S. (Alternatvely, they are the egenvectors of S.) 6 8

9 ecpe for subdvson curves he evaluaton mask for the cubc B-splne s: ( 4 ) 6 Now we can cook up a smple procedure for creatng subdvson curves: Subdvde (splt+average) the control polygon a few tmes. Use the averagng mask. Push the resultng ponts to the lmt postons. Use the evaluaton mask. angent analyss What s the tangent to the cubc B-splne curve? Frst, let s consder how we represent the x and y coordnate neghborhoods: 0 X av + av + av 0 Y bv + bv + bv We can vew the pont neghborhoods then as: After subdvsons, we would get: [ ] [ ] [ ] Q X Y v a b + v a b + v a b { [ ] [ ] [ ] } [ ] [ ] [ ] Q S v a b + v a b + v a b λ v a b + λ v a b + λ v a b We can wrte ths more explctly as: Q v, v, v, QC λ v C a b + λ v C a b + λ v C a b Q v, v, v, [ ] [ ] [ ],,, 7 8 9

10 angent analyss (cont d) he tangent to the curve s along the drecton: What s wrong wth ths defnton? Instead, we ll fnd the normalzed tangent drecton : Q t lm Q Now, let s look at the rght and center ponts n solaton: [ ] [ ] [ ] [ ] [ ] [ ] Q λv, a b + λv, a b + λv, a b Q λ v a b + λ v a b + λ v a b C, C, C, C he dfference between these s: ( )[ ] λ ( v, v, C)[ a b ] + λ ( v, v, C)[ a b ] ( v v C)[ a b ] λ ( v v C)[ a b ] Q Q λ v v a b + C,, C ( Q QC) t lm Q Q λ +,,,, C C 9 he tangent mask And now computng the tangent: lm ( v v C)[ a b ] + λ ( v v C)[ a b ] ( )[ ] λ ( )[ ] Q λ QC,,,, lm lm Q Q λ v v a b + v v a b C,, C,, C ( v, v, C)[ a b ] + ( v, v, C)[ a b ] ( v, v, C)[ a b ] + ( v, v, C)[ a b ] ( v, v, C)[ a b ] ( v, v, C) [ a b] [ a b] [ a b] 0 0 ux uy 0 0 ux uy 0 uq 0 uq λ λ λ λ hus, we can compute the tangent usng the second left egenvector! hs analyss holds for general subdvson curves and gves us the tangent mask. 0 0

11 Approxmaton vs. Interpolaton of Control Ponts Prevous subdvson scheme approxmated control ponts. Can we nterpolate them? Yes: DG nterpolatng scheme (987) Slght modfcaton to subdvson algorthm: splttng step ntroduces mdponts averagng step only changes mdponts For DG (Dyn-evn-Gregory), use: r (,5,0,5, ) 6 Snce we are only changng the mdponts, the ponts after the averagng step do not move.

Subdivision curves. University of Texas at Austin CS384G - Computer Graphics

Subdivision curves. University of Texas at Austin CS384G - Computer Graphics Subdivision curves University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Reading Recommended: Stollnitz, DeRose, and Salesin. Wavelets for Computer Graphics: Theory and Applications,