Computer Graphics. Computer Graphics. Rendering 4/1/2009 CS551

Transcription

1 4//009 Computer Grphics CS55 Computer Grphics Wht do it refer to? Rederig odelig Aimtio Imge Processig Also Virtul relity Visuliztio Artistic rederig Ad my my more Rederig Wht it is rederig Oe type of simultio Wht re issuse tht we eed to del with i the equtio

2 4//009 Rederig How to compute the color of pixel? How to recoer iformtio from scee? Represet the oject? How to compute occlusio? How to do this fst? th Bsics

3 4//009 y x 3

4 4//009 y (x, y) x ρ (x, y) θ (x, y) y ρ siθ ρ θ x ρ cosθ How do we compute (ρ, θ) from (x, y)? 4

5 4//009 (x, y) - (-x, -y) k (kx, ky) (x, y) (x, y) 5

6 4//009 (x, y) w (x, y ) (x, y) w (x, y ) V+w (x+x, y+y ) (x, y) w (x, y ) V+w (x+x, y+y ) x ρ cosθ Why ot he +w (ρ+ρ, θ+θ )? 6

7 4//009 Lier Iterpoltio of k l w w (x, y ) (x, y) θ w ( xx' + yy') w cosθ w (x, y ) (x, y) θ Why do we eed dot product? 7

8 4//009 i N-dimesiol Spces (x, y, z) How do these opertios chge lgericlly d geometriclly? + w k w w (x, y ) N (x, y) θ y z yz' zy' z w zx' xz' x xy' yx' x y y' z' z' w siθn x' x' y' w (x, y ) N (x, y) θ How do we exted the ide of cross product to higher dimesios? (geometriclly d lgericlly) 8

9 4//009 w (x, y ) (x, y) θ Why do we eed cross product? Chges of Coordite Systems (x, y) (x, y ) Chges of Coordite Systems (x, y) (x, y ) We eed mtrices. 9

10 4//009 trices Lier opertors o ectors ( + w) + w ( k) k Wht opertios re lier opertors Wht opertios re NOT lier opertors How do we represet mtrix? (0, ) trices (c, d) c d (x, y) (x, y ) (, 0) (, ) trices c x x + cy d y x + dy 0

11 4//009 trices How do we determie the compoets of mtrix? How my ectors do we eed to test it o? Lier Idepedece of I D I ND Lier Idepedece of Bsis Not ecessrily the sme s the coordite system

12 4//009 trices Determie the compoets of mtrix (z, w) (z, w ) u (x, y ) u(x, y) trices Determie the compoets of mtrix (z, w) x x' y y' (z, w ) z z' w w' u(x, y) u (x, y ) trices Determie the compoets of mtrix (z, w) (z, w ) x y z x' z' w y' w' u(x, y) u (x, y )

13 4//009 3 trices ' ' z x z x Determie the compoets of mtrix (z, w) y' w' w y u(x, y) u (x, y ) (z, w ) trix d c Determit: d-c Trce: +d Trce: +d Idetity mtrix Ierse: 0 0 I Determit ) ( () () ) ( S σ σ σ σ σ L Wht is σ?

14 4//009 Permuttio Permuttio Permuttio 4

15 4//009 Permuttio Permuttio Permuttio 5

16 4//009 6 Permuttio Permuttio Determit ) ( () () ) ( S σ σ σ σ σ L Wht is the formul of determit for 3?

17 4//009 7 Trce L How do determit d trce chge uder chges of coordites? Ierse Wht re these ectors, d how re they relted to ech ( ) w w w L j i j i w j i 0 other? trices uder Coordite Chges w w

18 4//009 trices uder Coordite y cx+dy y Chges w x w x x+y trices uder Coordite Chges Trce d determit do ot chge uder chge of sis Why? Powers of mtrix trices Whe the determit is zero, there exists umer N such tht A to the -th power is zero. Exmple: 8

19 4//009 trices For o-zero mtrix, wht do we do? Chrcteristic polyomil Do ot chge uder chge of sis Trce d determit re coefficiets of this polyomil Solutios re eigelues There re N solutios i the complex domi trices Eigelues d eigeectors trices Wht mtrices re gurteed to he oly rel eigelues? Symmetric oes Digolizle i the rel domi uder orthoorml sis Symmetric d ti-symmetric decompositio How? Wht s i the ti-symmetric compoet? 9

20 4//009 Symmetric trices Isotropy-deitor decompositio How to determie the eigeectors? Symmetric trices A geometric iterprettio of eigelues d eigeectors Eistei Coetio Corit d cotrrit tesors Higher-order tesors Upper d lower idices 0