Computer Graphics (CS 4731) Lecture 7: Linear Algebra for Graphics (Points, Scalars, Vectors)

Transcription

1 Computer Grphics (CS 4731) Lecture 7: Liner Alger for Grphics (Points, Sclrs, Vectors) Prof Emmnuel Agu Computer Science Dept. Worcester Poltechnic Institute (WPI)

2 Annoncements Project 1 due net Tuesd, 11.59PM. Midterm in clss net Frid SAs will give the em. I will e w Smple midterm (A 14) lred on clss wesite All code from ook (working progrms) on ook wesite. Quite useful. Tke look

3 Points, Sclrs nd Vectors Points, vectors defined reltive to coordinte sstem Point: Loction in coordinte sstem Emple: Point (5,4) Cnnot dd or scle points 5 (5,4) 4 (0,0)

4 Vectors Mgnitude Direction NO position Cn e dded, scled, rotted CG vectors:, 3 or 4 dimensions Length Angle

5 Vector-Point Reltionship Sutrct points = vector v = Q P P point + vector = point P + v = Q v Q

6 Vector Opertions Define vectors ( 1,, 3), ) ( 1, 3 Then vector ddition: ( 1 1,, 3 3 ) +

7 Vector Opertions Define sclr, s Scling vector sclr s ( s, s, 3 1 s ) Note vector sutrction: ( 1 ( 1 ), ( ), 3 ( 3 )) -.5

8 Vector Opertions: Emples Scling vector sclr Vector ddition: s ( s, s, 3 1 s ) ( 1 1,, 3 3 ) For emple, if =(,5,6) nd =(-,7,1) nd s=6, then ( 3 1 1,, 3 ) (0,1,7) s ( 3 1s, s, s) (1,30,36)

9 Affine Comintion Given vector (,, 3,..., 1 n 1... n 1 Affine comintion: Sum of ll components = 1 ) Conve ffine = ffine + no negtive component i.e 1,,... n non negtive

10 Mgnitude of Vector Mgnitude of 1... n Normliing vector (unit vector) â vector mgnitude Note mgnitude of normlied vector = 1. i.e 1... n 1

11 Mgnitude of Vector Emple: if = (, 5, 6) Mgnitude of Normliing â 65, 5 65, 6 65

12 Conve Hull Smllest conve oject contining P 1,P,..P n Formed shrink wrpping points 1

13 Dot Product (Sclr product) Dot product, d For emple, if =(,3,1) nd =(0,4,-1) then ( 0) (3 4) (1 1)

14 Properties of Dot Products Smmetr (or commuttive): Linerit: Homogeneit: And ) ( ) ( s s c c ) (

15 Angle Between Two Vectors c cos, sin c c cos, c sin c c c c c cos Sign of.c: c c c.c > 0.c = 0.c < 0

16 Angle Between Two Vectors Prolem: Find ngle /w vectors = (3,4) nd c = (5,) Step 1: Find mgnitudes of vectors nd c Step : Normlie vectors nd c 5 4, 5 3 ˆ 9, 9 5 ĉ c

17 Angle Between Two Vectors Step 3: Find ngle s dot product ˆ cˆ ˆ cˆ 3 5, , 9 ˆ cˆ Step 4: Find ngle s inverse cosine cos( 0.854) 31.36

18 Stndrd Unit Vectors Define i j 1,0,0 0,1,0 k i k 0,0,1 0 j So tht n vector, v, c i j ck,

19 Cross Product (Vector product),,,, If Then k j i ) ( ) ( ) ( Rememer using determinnt k j i Note: is perpendiculr to nd

20 Cross Product Note: is perpendiculr to oth nd 0

21 Cross Product (Vector product) Clculte if = (3,0,) nd = (4,1,8) 3,0, 4,1,8 Using determinnt i j k Then ( 0 ) i (4 8) j (3 0) k i 16j 3k

22 Norml for Tringle using Cross Product Method plne n (p - p 0 ) = 0 n p n = (p - p 0 ) (p 1 - p 0 ) normlie n n/ n p p 0 p 1 Note tht right-hnd rule determines outwrd fce

23 Newell Method for Norml Vectors Prolems with cross product method: clcultion difficult hnd, tedious If vectors lmost prllel, cross product is smll Numericl inccurc m result p 1 Proposed Mrtin Newell t Uth (tepot gu) Uses formule, suitle for computer Compute during mesh genertion Roust! p 0 p

24 Newell Method Emple Emple: Find norml of polgon with vertices P0 = (6,1,4), P1=(7,0,9) nd P = (1,1,) Using simple cross product: ((7,0,9)-(6,1,4)) X ((1,1,)-(6,1,4)) = (,-3,-5) P1 - P0 P - P0 P (1,1,) PO (6,1,4) P1 (7,0,9)

25 Newell Method for Norml Vectors Formule: Norml N = (m, m, m) m m m N 1 i0 N 1 i0 N 1 i0 i net( i) i net( i) i net( i) i net( i) i net( i) i net( i)

26 Newell Method for Norml Vectors Clculte component of norml m N 1 i0 i net( i) i net( i) P m m m (1)(13) ( 1)(11) (0)(6) P1 P P

27 Newell Method for Norml Vectors Clculte component of norml m N 1 i0 i net( i) i net( i) P m m m ( 5)(13) (7)(8) ( )(7) P1 P P

28 Newell Method for Norml Vectors Clculte component of norml m m m m N 1 i0 i net( i) i net( i) ( 1)(1) (6)(1) ( 5)() P0 P1 P P Note: Using Newell method ields sme result s Cross product method (,-3,-5)

29 Finding Vector Reflected From Surfce = originl vector n = norml vector r = reflected vector m = projection of long n e = projection of orthogonl to n n Note: Θ 1 = Θ e m r r e m m Θ 1 Θ -m r m e e

30 Forms of Eqution of Line Two-dimensionl forms of line Eplicit: = m +h Implicit: + +c =0 Prmetric: () = 0 + (1-) 1 () = 0 + (1-) 1 Prmetric form of line 1 - α More roust nd generl thn other forms Etends to curves nd surfces P o α P α P 1

31 Conveit An oject is conve iff for n two points in the oject ll points on the line segment etween these points re lso in the oject P P Q Q conve not conve

32 References Angel nd Shreiner, Interctive Computer Grphics, 6 th edition, Chpter 3 Hill nd Kelle, Computer Grphics using OpenGL, 3 rd edition, Sections