3D Viewing Pipeline Week 7, Lecture 13

Transcription

1 CS 43/536 Compute Gaphics I 3D Viewing Pipeline Week 7, Lectue 3 David Been, William Regli and Maim Pesakhov Geometic and Intelligent Computing Laboato Depatment of Compute Science Deel Univesit Oveview Pojection Mathematics Canonical View Volume Paallel Pojection Pipeline Pespective Pojection Pipeline Lectue Cedits: Most pictues ae fom Fole/VanDam; Additional and etensive thanks also goes to those cedited on individual slides Pics/Math coutes of Dave UMD-CP Pojection Mathematics What is the set of tansfomations needed to map 3D lines/planes onto a D sceen positioned in 3D? Basic pocedue 4D homogeneous coodinates to 3D homogeneous coodinates fo eve pimitive in the 3D view volume Pojection Mathematics We pesent 44 matices to be used fo implementing pojections Pespective Pojections How much to scale objects as a function of distance? Paallel Pojections simplified case) Just chop out the coodinate 3 4 The Pespective Pojection Detemining scale Conside Point P Pojected onto pojection plane as point P p Idea: compute atios via simila tiangles The Pespective Pojection In the diection atio is d p p /d 5 6

2 The Pespective Pojection In the diection atio is d p p /d The Pespective Pojection Homogenous pespective pojection mati M pe d Assumes is ais. 7 8 The Pespective Pojection Homogenous pespective pojection The Pespective Pojection Homogenous pespective pojection to 3D /d d /d d /d /d d 9 The Othogaphic Pojection Implementing Pojections Fole et al.) M ot Assumes DOP is ais. Just makes ais a sequence of mati opeations and a pocedue

3 Implementing Pojections Fole et al.). Etend 3D coodinates to homogenous coods. Appl nomaliing tansfomation, N pa o N pe 3. Divide b W to map back down to 3D 4. Clip in 3D against canonical view volume paallel o pespective view volume 5. Etend 3D coodinates back to homogenous 6. Pefom paallel pojection using M ot o Pefom pespective pojection M pe 7. Divide b W to map fom homogenous to D coodinates division effects pespective pojection) 8. Tanslate and scale in D) to device coodinates 3 Canonical View Volume: Paallel Pojection Defined b 6 planes: Eas to clip against 4 Canonical View Volume: Pespective Pojection Defined b 6 planes: - - min - Eas to clip against 5 Paallel Pojection Pipeline Tansfoming an abita view volume into the canonical one. Tanslate VRP to the oigin. Rotate so becomes, VUP becomes and u becomes 3. Shea to make diection of the pojection become paallel to 4. Tanslate and scale into a canonical view volume 6. Tanslate VRP to the oigin. Rotate Simple tanslation T-VRP) vp vp T vp otated to VUP R VUP otated to VUP R R R R [, ],..., 3 R VRP, VUP in Wold Coodinates,,) Top Pojection Off-Ais Pojection Top Pojection Off-Ais Pojection 7 8 3

4 . Rotate 3. Shea Computing diection of pojection DOP pp u PRP pp v,cw pp n PRP in VRC Coodinates u,v,n) u mau min v mav min, DOP u mau min pp u v mav min pp v pp n sh sh, SH pa 9 sh sh SH pa 3. Shea Cont.) sh sh Fo Sh Find Sh such that: 3. Shea Cont), sh, sh DOP [,,DOP,] T Sh DOP DOP DOP sh, DOP DOP sh sh DOP DOP, sh DOP DOP Old Fole et al. is wong Fo Sh, sh, sh 3. Shea Finall) 3. Shea Given DOP u mau min pp u v mav min pp v pp n sh sh DOP sh, SH sh DOP pa DOP DOP, We can compute the SH pa mati /u u ) pp ma min u pp n /v v ) pp ma min v pp n 4 5 4

5 4. Tanslate and Scale Tanslate the cente of the volume to the oigin Scaling to Needed fo 3D Fole et al. is wong u ma u min ) / v ma v min ) / T pa F /u ma u min ) /v S pa ma v min ) /F B) Paallel Pojection View Volume Tansfomation N pa S pa T pa SH pa T-VRP))))) Appl to all model vetices P N pa P Top Pojection Off-Ais Pojection Top Pojection Off-Ais Pojection 6 7 Paallel Pojection Summa Pespective Pojection Pipeline Tansfoming an abita view volume into the canonical one. Tanslate VRP to the oigin. Rotate so becomes, VUP becomes and u becomes 3. Tanslate COP to oigin 4. Shea so volume centeline becomes ais 5. Scale into a canonical view volume fo 8 9. Tanslate VRP to the oigin. Rotate Simple tanslation T-VRP) VRP, VUP in Wold Coodinates,,) vp vp T vp otated to VUP R VUP otated to VUP R R R R [, ],..., 3 R Top Pojection Off-Ais Pojection Top Pojection Off-Ais Pojection 3 Top Pojection Off-Ais Pojection Top Pojection Off-Ais Pojection 3 5

6 3. Tanslate 4. Shea Simple tanslation T-PRP) PRP in VRC Coodinates u,v,n) pp u pp v T pp n Goal is to tansfom the /u ma u min ) pp u cente line to the ais pp n Same as paallel SH pa /v v ) pp ma min v pojection pp n Shea mati is the same Top Pojection Off-Ais Pojection Top Pojection Off-Ais Pojection 3 Top Pojection Off-Ais Pojection Top Pojection Off-Ais Pojection Scaling We can define VRP afte tansfomation as VRP SH pa T-PRP) [,,,] T vp -pp n since shea does not affect coodinates Fist, we scale diffeentiall in and to set plane slopes to and - vp, vp u u v ma min ma v min, [ ] Second, we scale unifoml b vp B pp n B Top Pojection Off-Ais Pojection Top Pojection Off-Ais Pojection Scaling Cont) We can combine the tansfomations S pe VRP ma a) Befoe scaling min ma min Back vp u ma u min )vp B) View plane at poj, ) vp v ma v min )vp B) vp B b) Afte scaling Font min X,Y scaling vp, vp [ u u v ma min ma v min, ] Scaling Cont) We can combine the tansfomations S pe pp n ) u ma u min )pp n B), ) pp n v ma v min )pp n B) pp n B Impotant Pojection Info Back plane is - Font plane is min vp F vp B pp F n B pp n Pojection plane is d vp poj vp B pp n B pp n ma Back Font min View plane at poj VRP a) Befoe scaling min X,Y scaling ma min v ma v min, pp n pp u ma u min, n b) Afte scaling

7 Pespective Pojection Pipeline Pespective Pojection Summa N pe S pe SH pa T-PRP) T-VRP))))) Appl to all model vetices P N pe P 39 4 Summa of 3D Tansfoms We know how to take an pojection and convet it into a canonical View Volume 3D edges can be clipped against it and pojected onto sceen Implementing Pojections Without 3D Clipping. Etend 3D coodinates to homogeneous coodinates. Appl nomaliing tansfomation, N pa o N pe 3. Pefom tivial eject test with view volume 4. Pefom paallel pojection o pefom pespective pojection 5. Clip against D wold view plane) window 6. Tanslate and scale in D) to device coodinates i.e. into viewpot) 4 4 Tansfomed Window Paallel Pojection -,-),) Pespective Pojection - poj, - poj ) poj, poj ) Use these fo wold window paametes in viewpot mapping and D Pogamming assignment 4 ead SMF file Implement paallel pojection Implement pespective pojection Daw pojected and clipped polgon edges