(x, y) (ρ, θ) ρ θ. Polar Coordinates. Cartesian Coordinates

Transcription

1 Coordinate Sstems Point Representation in two dimensions Cartesian Coordinates: (; ) Polar Coordinates: (; ) (, ) ρ θ (ρ, θ) Cartesian Coordinates Polar Coordinates p = CPS1, 9: Computer Graphics D Geometric Transforms Page 1

2 Transformations (a) d (b) CPS1, 9: Computer Graphics D Geometric Transforms Page

3 Translation in D Moving the object along a line from one location to another. p = (; );t=(t ;t ); p 0 =( 0 ; 0 ) 0 =+t ; 0 =+t, p 0 =p+t. 0 0 = t t { } + T (t) (,) p t (, ) p t CPS1, 9: Computer Graphics D Geometric Transforms Page

4 Translation in D Translating a rigid bod: Translate ever point of b t. 0 = fp + t j p g: Enough to translate the vertices. CPS1, 9: Computer Graphics D Geometric Transforms Page

5 Rotation in D Rotation b with respect to origin. (, ) θ φ (,) = r cos ; = r sin : 0 = r cos( + ) = r cos cos, r sin sin = cos, sin : 0 = r sin( + ) = sin + cos : CPS1, 9: Computer Graphics D Geometric Transforms Page

6 Rotation in D 0 0 = cos, sin sin cos { } R() p 0 = R p Suppose R() = [R 1 () R ()]. R() 1 0 = cos sin = R 1 () R() 0 1 =, sin cos = R () CPS1, 9: Computer Graphics D Geometric Transforms Page

7 Rotation in D Rotation w.r.t. (; ): (, ) r (α,β) θ r φ (,) = + r cos ) r cos =, = + r sin ) r sin =, 0 = + r cos( + ) = + r cos cos, r sin sin = + (, ) cos, (, ) sin 0 = + (, ) sin + (, ) cos : CPS1, 9: Computer Graphics D Geometric Transforms Page

8 Scaling in D Scale b factor (s ;s ) 0 =s 0 =s 0 0 = s 0 0 s { } S(;) Uniform scaling s = s CPS1, 9: Computer Graphics D Geometric Transforms Page 8

9 Matri Representation of Transforms P 0 = AP + B Translation b (t ;t ): A = B = t t Rotation b : A = cos, sin sin cos B = 0 0 Scaling b (s ;s ): A = s 0 0 s B = 0 0 CPS1, 9: Computer Graphics D Geometric Transforms Page 9

10 Composite Transformation T 1 = (A 1 ;B 1 ) T = (A ;B ) T T 1 (P) = A (A 1 P +B 1 )+B T T 1 CPS1, 9: Computer Graphics D Geometric Transforms Page 10

11 Noncommutativit of Transforms T : Translate (; 0); R: Rotate b = R T T R T R T CPS1, 9: Computer Graphics D Geometric Transforms Page 11

12 Homogeneous Coordinates Homogeneous coordinates unif translation, rotation, and scaling in one transformation matri. Transformation matri is matri. Translation p = (; ; w)! t = (t ;t ) w ; w p 0 = p+t= w +t ; w +t p 0 = (+t w; + t w; w) 0 0 w 0 = 1 0 t 0 1 t { } w T(t ;t ) T (t ;t ),1 =T(,t ;,t ) CPS1, 9: Computer Graphics D Geometric Transforms Page 1

13 Homogeneous Coordinates Rotation 0 0 w 0 = cos, sin 0 sin cos { } w R() R(),1 = R(,) Scaling 0 0 w 0 = s s { } w S(s ;s ) S(s ;s ),1 =S(1=s ; 1=s ) CPS1, 9: Computer Graphics D Geometric Transforms Page 1

14 D Rotation revisited Rotation w.r.t a point q (; ) b angle Regard as a composite transform. (α,β) (α,β) T (,;,) (α,β) (α,β) R()T (,;,) T (;)R()T (,;,) CPS1, 9: Computer Graphics D Geometric Transforms Page 1

15 D Rotation Translate b T (,;,) so that q becomes the origin. Rotate b with respect to the origin; R(). Translate b T (; ) to bring q back to its original position. R(; ; ) = T(; ) R() T (,;,): CPS1, 9: Computer Graphics D Geometric Transforms Page 1

16 D Rotation: Few Remarks Rotation involves either trigonometric functions or p Use Power series to approimate rotation for small values of Consider the upper-left matri of R() and regard the rows as vectors in the plane v 1 = (cos ;, sin ); v = (sin ; cos ): p (sinθ,cosθ) p R(θ) p (cos θ,-sin θ) CPS1, 9: Computer Graphics D Geometric Transforms Page 1

17 Euclidean Transform Euclidean transformation Rigid Bod transformations Translation and Rotation Angles and distances are unchanged T (; t ;t )= cos, sin t sin cos t CPS1, 9: Computer Graphics D Geometric Transforms Page 1

18 Reection w.r.t -ais S(1;,1) w.r.t -ais S(,1; 1) What about w.r.t the line =,? CPS1, 9: Computer Graphics D Geometric Transforms Page 18

19 Shear Transform SH (a) = 1 a = + a; 0 = : SH(a) Horiontal segment at height is shifted in the -direction b a. Not a Euclidean transform CPS1, 9: Computer Graphics D Geometric Transforms Page 19

20 D Transforms Homogeneous coordinates (; ; ; w) =) w ; w ; : w Right Hand Sstem: Positive rotation dened as: Look toward origin from a positive ais, = counter-clockwise rotation transforms one ais to another. :!, :!, :! Some graphics sstems use the left hand sstem. CPS1, 9: Computer Graphics D Geometric Transforms Page 0

21 Translation Translation b (t ;t ;t ) p = (; ; ; w); t = (t ;t ;t ) p 0 = ( 0 ; 0 ; 0 ;w 0 ) = (+wt ;+wt ;+wt ;w): w 0 = t t t { } w T(t ;t ;t ) T (t ;t ;t ),1 =T(,t ;,t ;,t ) CPS1, 9: Computer Graphics D Geometric Transforms Page 1

22 Scaling Scaling b (s ;s ;s ) p = (; ; ; w) s = (s ;s ;s ) p 0 = ( 0 ; 0 ; 0 ;w 0 )=(s ;s ;s ;w): w 0 = s s s { } w S(s ;s ;s ) S(s ;s ;s ),1 =S(1=s ; 1=s ; 1=s ) CPS1, 9: Computer Graphics D Geometric Transforms Page

23 Rotation in D Rotation w.r.t. a line: Ais of rotation Ais of rotation: -ais 0 = cos, sin 0 = sin + cos 0 = w 0 = cos, sin 0 0 sin cos { } w R () a (a) (b) CPS1, 9: Computer Graphics D Geometric Transforms Page

24 Rotation w.r.t. -ais Substitute! ;! ;!. 0 = cos, sin 0 = sin + cos 0 0 = w 0 = cos, sin 0 0 sin cos { } w R () g (a) (b) CPS1, 9: Computer Graphics D Geometric Transforms Page

25 Rotation w.r.t. -ais Substitute! ;! ;! in the previous equations. 0 = cos, sin 0 = sin + cos 0 = w 0 = cos 0 sin , sin 0 cos { } w R () b (a) (b) CPS1, 9: Computer Graphics D Geometric Transforms Page

26 Rotation wrt a Point p f u p f (a) (b) p f p f R(; p) = T(p) R () T(,p) CPS1, 9: Computer Graphics D Geometric Transforms Page

27 Transformations in OpenGL Vertices CTM Vertices glmatrimode (mode) Species the matri that will be modied. GL MODELVIEW, GL PROJECTION, GL TEXTURE. glloadidentit () Sets the current matri to the identit matri. glloadmatriffdg (tpe *M) Sets the current matri to M. glmultmatriffdg (tpe *M) Multiplies the current matri b M; C: current matri. C = C M: CPS1, 9: Computer Graphics D Geometric Transforms Page

28 Transformations in OpenGL gltranslateffdg (,, ) Translates an object b (; ; ). C = C T (; ; ) glscaleffdg (,, ) Scales an object b (; ; ). C = C S(; ; ) glrotateffdg (,,, ) Rotates an object b angle w.r.t. the ra emanating from the origin to the point (; ; ). C = C R(; ; ; ) CPS1, 9: Computer Graphics D Geometric Transforms Page 8

29 Order of multiplication Last transformation is applied rst! glloadidentit(); glmultmatrif (L); glmultmatrif (M); glmultmatrif (N); glbegin(gl POINT) glvertef(v); glend(); C = I C = L C = L M C = L M N v 0 = C v = L M N v CPS1, 9: Computer Graphics D Geometric Transforms Page 9

30 Order of multiplication T R gltranslatef (T ); glrotatef (R); Assume the object comes with its own coordinate sstem. Keep the object coordinates the same as world coordinates and transform the object. Transform the object coordinate sstem. CPS1, 9: Computer Graphics D Geometric Transforms Page 0

31 Multiple Coordinate sstems C i : i-th coordinate sstem. P (i) : Coordinates of P in C i M ij : Transformation C j! C i. C P (i) = M ij P (j) : C C1 M ik = M ij M jk M ij = M,1 ji : Q (j) : A geometric transform in C j. Q (i) = M ij Q (j) M ji = M ij Q (j) M,1 ij : CPS1, 9: Computer Graphics D Geometric Transforms Page 1

32 Multiple Coordinate Sstems wo World-coordinate sstem tr Triccle-coordinate sstem wo wh tr tr wo wh wh Wheel-coordinate sstem Coordinate sstems: World, triccle, wheels As the front wheel rotates around its -ais: Both wheels rotate about the -aes of wheels coordinate sstem. Triccle moves as a whole. Both triccle and wheels move w.r.t world coordinates. Wheel and triccle coordinates are related b translation in and directions and rotation in -direction. CPS1, 9: Computer Graphics D Geometric Transforms Page