Math Review. Week 1, Wed Jan 10

Transcription

1 Uniersity of British Colbia CPSC 4 Coter Grahics Jan-Ar 007 Taara Mnzner Math Reiew Week, Wed Jan 0 htt://

2 News sign sheet with nae, eail, rogra

3 Reiew: Coter Grahics Defined CG ses oies, gaes, art/design, ads, VR, isalization CG state of the art hotorealis achieable (in soe cases) htt://

4 Correction: Exectations hard corse! heay rograing and heay ath fn corse! grahics rograing addictie, create great deos rograing rereq CPSC (Basic Algoriths and Data Strctres) or CPSC 6 (Progra Design and Data Strctres) corse langage is C/C ath rereq MATH 00 (Calcls III) MATH / (Matrix Algebra/Linear Algebra) 4

5 Reiew: Rendering Caabilities 5

6 Readings Mon (last tie) FCG Cha Wed (this tie) FCG Cha excet.5.,.5.,.7.,.7.,.8,.9,.. FCG Cha excet 5.., 5..4 Fri (next tie) RB Cha Introdction to OenGL RB Cha State Manageent and Drawing Geoetric Objects RB A Basics of GLUT (Ax in.) 6

7 Today s Readings FCG Chater : Miscellaneos Math ski. (sets and as),. (qadratic eqns) iortant:. (trig),.4 (ectors),.5-6 (lines).0 (linear interolation) ski.5.,.5.,.7.,.7.,.8,.9 ski. now (coered later) FCG Chater : Linear Algebra ski 5. (deterinants) iortant: , 5..5 (atrices) ski 5..-4, (atrix nerical analysis) 7

8 Notation: Scalars, Vectors, Matrices scalar a (lower case, italic) ector (lower case, bold) a [ a a... ] a n atrix (er case, bold) A a a a a a a a a a 8

9 Vectors arrow: length and direction oriented segent in nd sace offset / dislaceent location if gien origin 9

10 Coln s. Row Vectors row ectors a [ a a... ] row a n a coln ectors a a col... an switch back and forth with transose T col a a row 0

11 Vector-Vector Addition add: ector ector ector arallelogra rle tail to head, colete the triangle geoetric exales: algebraic (,) (6,4) (9,6) (,5,) (,, ) (5,6,0)

12 Vector-Vector Sbtraction sbtract: ector - ector ector ( ) (,) (6,4) (, ) (,5,) (,, ) (,4,)

13 Vector-Vector Sbtraction sbtract: ector - ector ector ( ) (,) (6,4) (, ) (,5,) (,, ) (,4,) argent reersal

14 Scalar-Vector Mltilication ltily: scalar * ector ector ector is scaled a* a * ( a*, a *, a* ) *(,).5*(,5,) (6,4) (,.5,.5) 4

15 5 Vector-Vector Mltilication ltily: ector * ector scalar dot rodct, aka inner rodct ( ) ( ) ( )

16 6 Vector-Vector Mltilication ltily: ector * ector scalar dot rodct, aka inner rodct ( ) ( ) ( ) ( ) ( ) ( )

17 7 Vector-Vector Mltilication ltily: ector * ector scalar dot rodct, aka inner rodct ( ) ( ) ( ) cosθ θ geoetric interretation lengths, angles can find angle between two ectors

18 Dot Prodct Geoetry can find length of rojection of onto cosθ θ cosθ as lines becoe erendiclar, cosθ 0 8

19 9 Dot Prodct Exale (*) (*7) (6*) 7 6 ( ) ( ) ( )

20 0 Vector-Vector Mltilication, The Seqel ltily: ector * ector ector cross rodct algebraic

21 Vector-Vector Mltilication, The Seqel ltily: ector * ector ector cross rodct algebraic

22 Vector-Vector Mltilication, The Seqel ltily: ector * ector ector cross rodct algebraic blah blah

23 Vector-Vector Mltilication, The Seqel ltily: ector * ector ector cross rodct algebraic geoetric arallelogra area erendiclar to arallelogra a b a b sinθ b a θ

24 RHS s. LHS Coordinate Systes right-handed coordinate syste conention y z left-handed coordinate syste x right hand rle: index finger x, second finger y; right thb oints z x y y x z left hand rle: index finger x, second finger y; left thb oints down z x y 4

25 Basis Vectors take any two ectors that are linearly indeendent (nonzero and nonarallel) can se linear cobination of these to define any other ector: c w a w b 5

26 Orthonoral Basis Vectors if basis ectors are orthonoral (orthogonal (tally erendiclar) and nit length) we hae Cartesian coordinate syste failiar Pythagorean definition of distance orthonoral algebraic roerties x y, x y 0 6

27 Basis Vectors and Origins coordinate syste: jst basis ectors can only secify offset: ectors coordinate frae: basis ectors and origin can secify location as well as offset: oints j o i o x i yj 7

28 Working with Fraes j o F i o x i yj F 8

29 Working with Fraes j o F i o x i yj F (,-) 9

30 Working with Fraes j o F i o x i yj F i j o F (,-) F 0

31 Working with Fraes j o F i o x i yj F i j o F (,-) F (-.5,)

32 Working with Fraes j o F i o x i yj F i j o j F i o F (,-) F (-.5,) F

33 Working with Fraes j o F i o x i yj F i j o j F i o F (,-) F (-.5,) F (,)

34 Naed Coordinate Fraes origin and basis ectors o ick canonical frae of reference then don t hae to store origin, basis ectors jst ( a, b, c) conention: Cartesian orthonoral one on reios slide handy to secify others as needed airlane nose, looking oer yor sholder,... really coon ones gien naes in CG object, world, caera, screen,... a x by cz 4

35 sloe-intercet for y x b ilicit for y x b 0 Ax By C 0 f(x,y) 0 Lines 5

36 Ilicit Fnctions find where fnction is 0 lg in (x,y), check if 0: on line < 0: inside > 0: otside analogy: terrain sea leel: f0 altitde: fnction ale too a: eqal-ale contors (leel sets) 6

37 7 Ilicit Circles circle is oints (x,y) where f(x,y) 0 oints on circle hae roerty that ector fro c to dotted with itself has ale r oints oints on the circle hae roerty that sqared distance fro c to is r oints on circle are those a distance r fro center oint c ) ( ) ( ), ( r y y x x y x f c c 0 ) ( ) ( ) :, ( ),, ( r y x c y x c c c c 0 r c 0 r c

38 Paraetric Cres araeter: index that changes continosly (x,y): oint on cre t: araeter ector for f (t) x y g( t h( t ) ) 8

39 D Paraetric Lines x x 0 t(x x 0 ) y y 0 t(y y 0 ) ( t) t( ) 0 0 ( t ) o t( d) start at oint 0, go towards, according to araeter t (0) 0, () 9

40 Linear Interolation araetric line is exale of general concet ( t) t( ) 0 0 interolation goes throgh a at t 0 goes throgh b at t linear weights t, (-t) are linear olynoials in t 40

41 4 Matrix-Matrix Addition add: atrix atrix atrix exale n n n n n n n n ) ( 7 5 4

42 Scalar-Matrix Mltilication ltily: scalar * atrix atrix a exale a* a* a * a* 4 5 * * *4 *

43 4 Matrix-Matrix Mltilication can only ltily (n,k) by (k,): nber of left cols nber of right rows legal ndefined l k j i h g f e c b a k j i h q g f o e c b a

44 Matrix-Matrix Mltilication row by coln n n n n n n 44

45 Matrix-Matrix Mltilication row by coln n n n n n n n n 45

46 Matrix-Matrix Mltilication row by coln n n n n n n n n n n 46

47 Matrix-Matrix Mltilication row by coln n n n n n n n n n n n n 47

48 Matrix-Matrix Mltilication row by coln n n n n n n n n n n n n noncotatie: AB! BA 48

49 Matrix-Vector Mltilication oints as coln ectors: ostltily x' y' z' h' 4 4 x y z h oints as row ectors: reltily [ x' y' z' h' ] [ x y z h] ' M T T ' T M T 49

50 50 Matrices transose identity inerse not all atrices are inertible T AA I

51 5 Matrices and Linear Systes linear syste of n eqations, n nknowns atrix for Axb z y x z y x z y x z y x