Computer Graphics. Transformation

Transcription

1 (SBE 36) Dr. Aman Eldeib Spring 2 SBE 36

2 i a fundamental corner tone of computer graphic and i a central to OpenGL a well a mot other graphic tem.(2d and 3D ) Given an object, tranformation i to change the object Poition (tranlation) Sie (caling) Orientation (rotation) Spring 2 Shape (hear) SBE 36

3 Tranlation We can tranlate or move point to a new poition b adding offet to their coordinate Tranlate individual vertice Spring 2 SBE 36

4 SBE 36 Tranlation Spring 2 We can tranlate or move point to a new poition b adding offet to their coordinate Tranlate individual vertice t t t t t t 2D

5 Spring 2 Homogeneou Coordinate Homogeneou coordinate: repreenting coordinate in 2 dimenion with a 3-vector and coordinate in 3 dimenion with a 4-vector (,, ) w (Note that tpicall w in object coordinate) SBE 36

6 Spring 2 Homogeneou Coordinate Homogeneou coordinate eem unintuitive, but the make graphic operation much eaier Our tranformation matrice are now 44 for 3D T T T P T P T Tranlation matri SBE 36

7 SBE 36 Tranlation Spring 2 We can tranlate or move point to a new poition b adding offet to their coordinate Tranlate individual vertice t t t 3D t t t t t t P P T

8 Tranlation We can tranlate or move point to a new poition b adding offet to their coordinate To tranform an object multipl each verte b the ame matri A quare tranlation can be done a follow: [P P2 P3 P4] T [P P2 P3 P4] Tranlate individual vertice Spring 2 SBE 36

9 Scaling Scaling a coordinate mean multipling each of it component b a calar Uniform caling mean thi calar i the ame for all component 2 Spring 2 SBE 36

10 Scaling Scaling a coordinate mean multipling each of it component b a calar 2 Spring 2 SBE 36

11 SBE 36 Scaling Scaling a coordinate mean multipling each of it component b a calar Spring 2 2D 3D P P S Scaling matri

12 Scaling Scaling a coordinate mean multipling each of it component b a calar Non-uniform caling: different calar per component X 2, Y.5 Spring 2 SBE 36

13 Fied Point Scaling Spring 2 SBE 36

14 Fied Point Scaling Spring 2 SBE 36

15 SBE 36 Fied Point Scaling Spring 2

16 SBE 36 Fied Point Scaling Spring 2 Step : Tranlate to the origin

17 SBE 36 Fied Point Scaling Spring 2 Step 2: Scale the object

18 SBE 36 Fied Point Scaling Spring 2 Step 3: Tranlate the object back

19 . Tranlate to origin 2. Scale 3. Tranlate back Fied Point Scaling p ( T S T) p Spring 2 SBE 36

20 2D (, ) (, ) Rotation co() - in() in() + co() co in ( ) in( ) ( ) co( ) Spring 2 SBE 36

21 SBE 36 Rotation Spring 2 3D 3-D i more complicated Need to pecif an ai of rotation There are four wa to pecif a 3D rotation Simple cae: rotation about X, Y, Z ae 3-D rotation matri look like for a rotation about the X-ai ) co( ) in( ) in( ) co( for a rotation about the Y-ai for a rotation about the Z-ai ) co( ) in( ) in( ) co( ) co( ) in( ) in( ) co(

22 SBE 36 Rotation Spring 2 3D for a rotation about the X-ai for a rotation about the Y-ai for a rotation about the Z-ai co in in co ) ( R co in in co ) ( R co in in co ) ( R 3-D i more complicated Need to pecif an ai of rotation There are four wa to pecif a 3D rotation Simple cae: rotation about X, Y, Z ae 3-D rotation matri look like

23 3D 3-D i more complicated Spring 2 Rotation Need to pecif an ai of rotation There are four wa to pecif a 3D rotation 3D rotation ai angle repreentation 3-D rotation matri look like 2 2 R( a, ) co ( co ) in Rotate a point about an arbitrar ai a (,,) going through the origin Note: the ai a hould be of unit length a SBE 36

24 . Tranlate to origin 2. Rotate 3. Tranlate back Fied Point Rotating p ( T R T) p Spring 2 SBE 36

25 Matri Compoition Matri multiplication doe not commute product ma not be commutative AB BA Spring 2 SBE 36

26 SBE 36 Shearing Y coordinate are unaffected, but coordinate are tranlated linearl with, i.e. + * h Spring 2 h

27 SBE 36 Shearing X coordinate are unaffected, but coordinate are tranlated linearl with, i.e. + * g Spring 2 g

28 Invere Tranlate Scale Shear Rotate Spring 2 SBE 36

29 OpenGL: Modeling OpenGL provide everal command for performing modeling tranform: gltranlate{fd}(,, ) Create a matri T that tranform an object b tranlating (moving) it b the pecified,, and value glrotate{fd}(angle,,, ) Create a matri R that tranform an object b rotating it counterclockwie angle degree about the vector {,, } glscale{fd}(,, ) Create a matri S that cale an object b the pecified factor in the,, and direction Spring 2 SBE 36

30 OpenGL: Matri Manipulation Each of thee pot multiplie the current matri E.g., if current matri i C, then CCS The current matri i either the modelview matri or the projection matri (alo a teture matri, won t dicu for now) Set thee with glmatrimode(), e.g.: glmatrimode(gl_modelview); glmatrimode(gl_projection); WARNING: common mitake ahead! Be ure that ou are in GL_MODELVIEW mode before making modeling or viewing call! Ugl mitake becaue it can appear to work, at leat for a while Spring 2 SBE 36

31 OpenGL: Matri Manipulation More matri manipulation call To replace the current matri with an identit matri: glloadidentit() Potmultipl the current matri with an arbitrar matri: glmultmatri{fd}(float/double m[6]) Cop the current matri and puh it onto a tack: glpuhmatri() Dicard the current matri and replace it with whatever on top of the tack: glpopmatri() Note that there are matri tack for both modelview and projection mode Spring 2 SBE 36

32 OpenGL: Matri Manipulation glmatrimode (GL_MODELVIEW) glloadidentit ( ); glmultmatrtif (m2); M M M 2 glmultmatrif (m); OpenGL ue pot multiplication when multipling matrice thu, tranformation are applied in the invere order. The lat one pecified i the firt one applied. Spring 2 SBE 36

33 OpenGL: Matri Manipulation glmatrimode (GL_MODELVIEW) glloadidentit ( ); glrotationf (theta,,,); drawcube ( ); gltranlatef (a,b,c); drawcube ( ); Spring 2 SBE 36

34 OpenGL: Matri Manipulation gltranlatef(a,b,c); T glpuhmatri ( ); glrotatef(theta,a2,b2,c2); glscale(a3,b3,c3); T R 2 T R 2S3 glpopmatri ( ); gltranlatef(a4,b4,c4); T T 2 Spring 2 SBE 36

35 OpenGL: Hierarchie glpuhmatri(); // tranlate to houlder poition // rotate b houlder joint // draw houlder (circle and rectangle) glpuhmatri(); // tranlate to elbow poition // rotate b elbow joing // draw elbow (circle and rectangle) glpopmatri(); glpopmatri(); Spring 2 SBE 36

36 OpenGL: Specifing Color Can pecif other propertie uch a color To produce a ingle aqua-colored triangle: glcolor3f(.,.5,.); Spring 2 glverte3fv(v); glverte3fv(v); glverte3fv(v2); To produce a Gouraud-haded triangle: glcolor3f(,, ); glverte3fv(v); glcolor3f(,, ); glverte3fv(v); glcolor3f(,, ); glverte3fv(v2); In OpenGL, color can alo have a fourth component α (opacit) Generall want α. (opaque); SBE 36

37 Quetion? Spring 2 SBE 36