Course Outline. Course Outline. Computer Graphics (Fall 2008) Motivation. Outline of Unit. Bezier Curve (with HW2 demo)

Transcription

1 Compter Graphics (Fall 2008) COMS 4160, Lectre 6: Crves 1 3D Graphics Pipelie Modelig (Creatig 3D Geometry) Corse Otlie Rederig (Creatig, shadig images from geometry, lightig, materials) Corse Otlie Motivatio 3D Graphics Pipelie Modelig (Creatig 3D Geometry) Uit 1: Trasformatios Wees 1,2. Ass 1 de Sep 25 Uit 2: Splie Crves Modelig geometric objects Wees 3,4 hw2.exe Ass 2 de Oct 7 (demo) Rederig (Creatig, shadig images from geometry, lightig, materials) How do we model complex shapes? I this corse, oly 2D crves, bt ca be sed to create iterestig 3D shapes by srface of revoltio, loftig etc Techiqes ow as splie crves This it is abot mathematics reqired to draw these splie crves, as i HW 2 History: From sig compter modelig to defie car bodies i ato-mafactrig. Pioeers are Pierre Bezier (Realt), de Castelja (Citroe) Otlie of Uit Bezier crves decastelja algorithm, explicit form, matrix form Polar form labelig (ext time) B-splie crves (ext time) Not well covered i textboos (especially as taght here). Mai referece will be lectre otes. If yo do wat a prited ref, hadots from CAGD, Seidel Bezier Crve (with HW2 demo) Motivatio: Draw a smooth ititive crve (or srface) give a few ey ser-specified cotrol poits Cotrol poits (all that ser specifies, edits) hw2.exe Cotrol polygo Smooth Bezier crve (draw atomatically)

2 Bezier Crve: (Desirable) properties Iterpolates, is taget to ed poits Crve withi covex hll of cotrol polygo hw2.exe Cotrol polygo Cotrol poits (all that ser specifies, edits) Smooth Bezier crve (draw atomatically) Isses for Bezier Crves Mai qestio: Give cotrol poits ad costraits (iterpolatio, taget), how to costrct crve? Algorithmic: decastelja algorithm Explicit: Berstei-Bezier polyomial basis 4x4 matrix for cbics Properties: Advatages ad Disadvatages decastelja: : Liear Bezier Crve decastelja: : Qadratic Bezier Crve Jst a simple liear combiatio or iterpolatio (easy to code p, very merically stable) F(1) Liear (Degree 1, Order 2) F(0) =, F(1) = F() F() =? F(0) 1- F() = (1-) Qadratic Degree 2, Order 3 F(0) =, F(1) = F() =? F() = (1-) 2 + 2(1-) + 2 Geometric iterpretatio: Qadratic Geometric Iterpretatio: Cbic 1-1-

3 decastelja: : Cbic Bezier Crve Smmary: decastelja Algorithm P3 Cbic Degree 3, Order 4 F(0) =, F(1) = P3 P F() = (1-) 3 +3(1-) (1-) + 3 P3 Liear Degree 1, Order 2 F(0) =, F(1) = 1- Qadratic Degree 2, Order 3 F(0) =, F(1) = 1-1- F() = (1-) + 1- F() = (1-) 2 + 2(1-) + 2 P3 Cbic Degree 3, Order 4 F(0) =, F(1) = P3 P F() = (1-) 3 +3(1-) (1-) + 3 P3 DeCastelja Implemetatio Ca be optimized to do withot axiliary storage Smmary of HW2 Implemetatio Bezier (Bezier2 ad Bsplie discssed ext time) Arbitrary degree crve (mber of cotrol poits) Brea crve ito detail segmets. Lie segmets for these Evalate crve at locatios 0, 1/detail, 2/detail,, 1 Evalatio doe sig decastelja Key implemetatio: decastelja for arbitrary degree Is ayoe cofsed? Abot hadlig arbitrary degree? Ca also se alterative formla if yo wat Explicit Berstei-Bezier polyomial form (ext) Qestios? Isses for Bezier Crves Mai qestio: Give cotrol poits ad costraits (iterpolatio, taget), how to costrct crve? Algorithmic: decastelja algorithm Explicit: Berstei-Bezier polyomial basis 4x4 matrix for cbics Properties: Advatages ad Disadvatages Recap formlae Liear combiatio of basis fctios Liear: F ( ) = P(1 ) + P Qadratic: F ( ) = P(1 ) + P[2 (1 )] + P Cbic: F ( ) = P(1 ) + P[3 (1 ) ] + P[3 (1 )] + P 2 Degree : F ( ) = PB ( ) B( ) are Berstei-Bezier polyomials Explicit form for basis fctios? Gess it?

4 Recap formlae Smmary of Explicit Form Liear combiatio of basis fctios Liear: F ( ) = P(1 ) + P Qadratic: F ( ) = P(1 ) + P[2 (1 )] + P 2 Cbic: F ( ) = P(1 ) + P[3 (1 ) ] + P[3 (1 )] + P Liear: F ( ) = P(1 ) + P Qadratic: F ( ) = P(1 ) + P[2 (1 )] + P 2 Cbic: F ( ) = P(1 ) + P[3 (1 ) ] + P[3 (1 )] + P Degree : F ( ) = PB ( ) B( ) are Berstei-Bezier polyomials Explicit form for basis fctios? Gess it? Biomial coefficiets i [(1-)+] Degree : F ( ) = PB ( ) B( ) are Berstei-Bezier polyomials! B ( ) = (1 )!( )! Isses for Bezier Crves Mai qestio: Give cotrol poits ad costraits (iterpolatio, taget), how to costrct crve? Algorithmic: decastelja algorithm Explicit: Berstei-Bezier polyomial basis 4x4 matrix for cbics Properties: Advatages ad Disadvatages Cbic 4x4 Matrix (derive) F ( ) = P(1 ) + P[3 (1 ) ] + P[3 (1 )] + P = M =? P 2 P ( 1) Cbic 4x4 Matrix (derive) F ( ) = P(1 ) + P[3 (1 ) ] + P[3 (1 )] + P = ( 1) P P 3 Isses for Bezier Crves Mai qestio: Give cotrol poits ad costraits (iterpolatio, taget), how to costrct crve? Algorithmic: decastelja algorithm Explicit: Berstei-Bezier polyomial basis 4x4 matrix for cbics Properties: Advatages ad Disadvatages

5 Properties (brief discssio) Demo: hw2.exe Iterpolatio: Ed-poits, bt approximates others Sigle piece, movig oe poit affects whole crve (o local cotrol as i B-splies later) Ivariat to traslatios, rotatios, scales etc. That is, traslatig all cotrol poits traslates etire crve Easily sbdivided ito parts for drawig (ext lectre): Hece, Bezier crves easiest for drawig