6. Modelview Transformations

Transcription

1 6. Modelview Transformations Transformation Basics Transformations map coordinates from one frame of reference to another through matri multiplications Basic transformation operations include: - translation - rotation - scaling E.R. Bachmann & P.L. McDowell MV 422 Page of 3

2 2D Translation Points in the - plane can be translated to new positions b adding translation amounts to the coordinates of each point. Y P (, ) d P (, ) d X = + d (d, change along the -ais) (6.) = + d (d, change along the -ais) (6.) - This can be epressed notationall as P = P + T (6.3) where P = [ ], P = [ ], and T = [d d] E.R. Bachmann & P.L. McDowell MV 422 Page 2 of 3

3 - Objects described in terms of points and lines can be translated b appling = + d (6.4) = + d (6.5) to each verte describing the object Eample 6. Appl [ ] = [ ] + [ 2 3 ] to ever point of the object In other words, add [ 2 3 ] to ever end point of the object. Y (3,4) d = 2 d = 3 (, ) X E.R. Bachmann & P.L. McDowell MV 422 Page 3 of 3

4 2D Scaling Points can be scaled b S along the ais and S along the ais b the following formula: = * S (6.6) = * S (6.7) - In matri form 6.6 and 6.6 become S = S (6.8) - Scaling is denoted notationall b: P = S * P (6.9) Eample 6.2 Scaling for a point. Y (2, 3) S = 2 S = 3 (, ) X E.R. Bachmann & P.L. McDowell MV 422 Page 4 of 3

5 Eample 6.3 Scaling for a line. S=2 S=3 (2,) (8,) (,) (4,) Eample 6.4 Non-uniform scaling of a square. Y S = 2 S = (, ) (2, ) (4, ) X E.R. Bachmann & P.L. McDowell MV 422 Page 5 of 3

6 2D Rotations Vertices and the objects the describe can be rotated through an angle about an origin via the following formulas: = cos - sin (6.) = sin + cos (6.) - In matri form, 6. and 6. become cos = sin sin cos (6.2) - Notationall, this can be written: P = R * P (6.3) where R represents the rotation matri - In 2D space, positive angles are measured "counterclockwise" from the ais towards the ais E.R. Bachmann & P.L. McDowell MV 422 Page 6 of 3

7 Eample degree rotation of a point about the origin. Y P (4.7, 3.9) 3 deg. P (6, ) X E.R. Bachmann & P.L. McDowell MV 422 Page 7 of 3

8 Eample degree rotation about the origin of a square with its corner at the origin. (.,.44) (, ) (, ) (-.77,.77) (.77,.77) (, ) (, ) (, ) Eample degree rotation about the origin of a square centered at the origin. (., -.77) (-.5,.5) (.5,.5) (-.77,.) (.77,.) (-.5, -.5) (.5, -.5) (., -.77) E.R. Bachmann & P.L. McDowell MV 422 Page 8 of 3

9 Eample degree rotation about the origin of a square with its corner at (3, 5). (3, 6) (4, 6) (3, 5) (4, 5) (, ) (-.44, 7.7) (-2.2, 6.36) (-.77, 6.36) (-.44, 5.657) (, ) E.R. Bachmann & P.L. McDowell MV 422 Page 9 of 3

10 Homogeneous Coordinates Scaling and rotation can be accomplished through matri multiplication - In order to allow translation to be accomplished through matri multiplication all coordinates are represented in Homogeneous form - Point (, ) is represented as point (w*, w*, w), for an scale factor w - Thus (6.4) becomes w w w (6.5) in homogeneous form. - w is tpicall one and 6.5 becomes (6.6) E.R. Bachmann & P.L. McDowell MV 422 Page of 3

11 E.R. Bachmann & P.L. McDowell MV 422 Page of 3 Matri Representation of 2D Transformations in Homogeneous Coordinates In homogeneous coordinates the matri operations are defined as follows: - Translation: = d d (6.7) - Scaling: = S S (6.8) - Rotation: = cos sin sin cos (6.9)

12 Composite Transformations Homogeneous coordinates allow transformations to be performed successivel or combined into a single matri Eample 6.9 Translate an object P from the origin to a point p and rotate P about the arbitrar point p b an angle The following transformation must be applied to each verte v describing P: v = p cos p sin sin cos v (, ) (p,p ) (p,p ) E.R. Bachmann & P.L. McDowell MV 422 Page 2 of 3

13 Eample 6. Translate an object P from the origin to a point p and rotate P about the origin b an angle The following transformation must be applied to each verte v describing P: cos sin p v = sin cos p (6.2) (, ) ( p cos - p sin, p sin + p cos ) P P E.R. Bachmann & P.L. McDowell MV 422 Page 3 of 3

14 E.R. Bachmann & P.L. McDowell MV 422 Page 4 of 3 Matri Representation of 3D Transformations 3D transformations are a simple etension of 2D transformations - 3D transformations using homogeneous coordinates are performed with a 4 4 matri - Verte coordinates are represented with a 4 column vector z (6.22) - Translation: = z dz d d z (6.23) - Scaling: = Sz S S z (6.24)

15 E.R. Bachmann & P.L. McDowell MV 422 Page 5 of 3 - Rotations: 3D Rotations ma be performed about an arbitrar ais Rotations of an angle about each of the coordinate aes are as follows: ais: = cos sin sin cos z z (6.25) ais: = cos sin sin cos z z (6.26) z ais: = cos sin sin cos z z (6.27) Note the similarities between 6.27 and 6.9.

16 Sign of Rotation The sign of the angle of rotation is found using the right hand rule:. Point the thumb of the right hand along the ais of rotation in the positive direction 2. The direction in which the fingers curl is the direction of positive rotation z E.R. Bachmann & P.L. McDowell MV 422 Page 6 of 3

17 Modeling Transformations in OpenGL Modeling transformations are used to position and orient models or objects - Models ma be rotated, translated or scaled - Modeling transformations ma also be used to position and orient the parts of a model relative to each other - Modeling transformations are closel related to viewing transformations - All modeling transformations are specified after the viewing transformations are complete Transformations can be thought of as transforming both and - an object - the local coordinate sstem associated with that object Subsequent transformations will be relative to the local coordinate sstem of that object - Each subsequent transformation establishes a new local coordinate sstem - The first modeling transformation takes place relative to the origin associated with the viewpoint E.R. Bachmann & P.L. McDowell MV 422 Page 7 of 3

18 Transformations ma also be thought of as occurring relative to a single, grand, fied coordinate sstem - The transformations when written in the source code will appear to occur in reverse order Translate Translations are performed using void gltranslate{fd}( TYPE, TYPE, TYPE z ); where,, and z represent the translations in the,, and z directions relative to the current local coordinate sstem Eample 6.4 Translate unit in the direction, 2 units in the direction and -3 units in the z direction. Draw a cube one unit high. gltranslatef(.f, 2.f, -3.f ); ausolidcube(. ); // Draw a cube three units above the previous one. gltranslatef(.f, 3.f,.f ); ausolidcube(. ); E.R. Bachmann & P.L. McDowell MV 422 Page 8 of 3

19 Rotate Rotations are performed using void glrotate{fd}( TYPE angle, TYPE, TYPE, TYPE z ); where - angle specifies the angle of rotation in degrees -,, and z specif the ais of rotation relative to the origin of the local coordinate sstem Eample 6.2 Draw a cube rotated -75 degrees about the ais glrotatef( -75.f,.f,.f,.f ); ausolidcube(. ); Eample 6.3 Rotate 45 degrees about the ais at the point (, 2, -3) relative to the current local coordinate sstem. Draw a cube one unit high. gltranslatef(.f, 2.f, -3.f ); glrotated( 45.,.,.,. ); ausolidcube(. ); E.R. Bachmann & P.L. McDowell MV 422 Page 9 of 3

20 Scaling Scaling is performed using where void glscale{fd}( TYPE, TYPE, TYPE z ); -,, and z are the scale values in the,, and z directions relative to the current local coordinate sstem - Scale values greater than. stretch the object - Scale values less than. shrink the object - Negative scale values reflect the object across an ais Keep in mind scale operations affect the coordinate sstems associated with all subsequent transformations. Eample 6.4 Draw a bo centered at the origin of the current local coordinate sstem. glscalef( 4.f, 2.f, 3.f ); ausolidcube(. ); The cube will be a rectangular bo due to the different scales of component aes. E.R. Bachmann & P.L. McDowell MV 422 Page 2 of 3

21 Viewing Transformations in OpenGL Viewing transformations change the position and orientation of the viewpoint - Default viewpoint is at the origin, looking down the negative z ais, with the positive ais defining the up orientation - Viewing transformations must be completed before all modeling transformations Viewing transformations can be thought of in one of two was: - moving the objects relative to a fied viewpoint - setting the viewpoint relative to a fied origin from which all modeling transformations take place E.R. Bachmann & P.L. McDowell MV 422 Page 2 of 3

22 Using gltranslate*( ) and glrotate*( ) to Set the Viewpoint To achieve a certain viewpoint either move the camera in one direction or move the objects in the opposite direction - A modeling transformation that rotates an object counterclockwise is equivalent to a viewing transformation that rotates the viewpoint clockwise - A modeling transformation that translates an object forward units is equivalent to a viewing transformation that moves the viewpoint back units gltranslate and glrotate when used as modeling transformations should be thought of as moving the objects and their local coordinate sstem awa from the viewer Eample 6.5 View a teapot looking down the z ais from 5 units awa. gltranslate(.f,.f, -5.f ); ausolidteapot( 3. ); E.R. Bachmann & P.L. McDowell MV 422 Page 22 of 3

23 Eample 6.6 View a Octahedron from the right side. gltranslatef(.,., -5. ); glrotatef( 9.f,.f,.f,.f ); ausolidoctahedron(. ); Using glulookat( ) to Set the Viewpoint glulookat( ) allows programmers to construct a scene around a fied origin and set an arbitrar viewing position, direction and up orientation void glulookat( GLdouble ee, GLdouble ee, GLdouble eez, GLdouble cent, GLdouble cent, GLdouble centz, GLdouble up, GLdouble up, GLdouble upz ); - ee, ee and eez define the viewing position - cent, cent and centz define the specif an point along the line of sight - up, up and upz specif the up vector glulookat encapsulates a gltranslate* command and perhaps several glrotate* commands E.R. Bachmann & P.L. McDowell MV 422 Page 23 of 3

24 Eample 6.7 View a torus looking down the z ais from 5 units awa. glulookat(.,., 5.,.,., -.,.,.,. ); ausolidtorus(., 2. ); Eample 6.8 View a tetrahedron from it s left side. glulookat( 5.,.,., -.,.,.,.,.,. ); ausolidtetrahedron(. ); E.R. Bachmann & P.L. McDowell MV 422 Page 24 of 3

25 The ModelView Matri Stack gltranslate*( ), glrotate*( ), glscale*( ) and glulookat( ) specif a matri and multipl it times the current matri For modelview transformations: - The current matri is on top of the modelview matri stack - The current matri represents the result of all previous modelview transformations - The result of all previous transformations represents the local coordinate sstem from which the new transformation will act There are three matri stacks in OpenGL: - modelview, projection and teture - Transformations ma be specified for an stack E.R. Bachmann & P.L. McDowell MV 422 Page 25 of 3

26 The stack on which transformations will act is set using: where void glmatrimode( GLenum mode ); - mode specifies the stack on which subsequent transformations will act - mode is GL_MODELVIEW to specif the modelview matri glmatrimode( GL_MODELVIEW ); Before begining modelview transformations the stack is normall cleared using void glloadidentit( ); glloadidentit ( ) clears the current matri and sets it to the Identit matri I = I (6.28) The result of multipling the identit matri times an matri M is M = I( M ) M ( I ) (6.29) M = E.R. Bachmann & P.L. McDowell MV 422 Page 26 of 3

27 The current matri on the current stack ma be saved and restored using the following void glpushmatri( ); void glpopmatri( ); This allows the current local coordinate sstem to be saved b a push operation and restored later b a pop operation - glpushmatri( ) does the following: Pushes all matrices in the current stack down one level Copies the top matri (current matri) Places the duplicate cop on top of the stack The original cop in effect saves the state of the matri stack - glpopmatri( ) does the following: Pops the top matri off the stack and discards it Makes the second-from-the-top-matri, the top matri (current matri) Popping the top matri restores the state of the stack to its condition prior to the most recent push The modelview matri stack is capable of holding at least thirt-two 4 4 matrices E.R. Bachmann & P.L. McDowell MV 422 Page 27 of 3

28 Eample 6.9 Three triangles drawn relative to the same origin: // COpenGLView drawing void COpenGLView::OnDraw( CDC* pdc ) { COpenGLDoc* pdoc = GetDocument(); ASSERT_VALID( pdoc ); glclear( GL_COLOR_BUFFER_BIT ); glmatrimode( GL_MODELVIEW ); glloadidentit(); glcolor3f(.f,.f,.f ); draw_triangle(); glpushmatri(); gltranslatef( -2.f,.f,.f ); glrotatef( 45.f,.f,.f,.f ); draw_triangle(); glpopmatri(); glpushmatri(); glscalef(.5f,.5f,.f ); draw_triangle(); glpopmatri(); glrotatef( 9.f,.f,.f,.f ); draw_triangle(); glflush(); SwapBuffers( m_pdc->getsafehdc() ); } // end OnDraw E.R. Bachmann & P.L. McDowell MV 422 Page 28 of 3

29 The current matri ma be set to an eplicitl specified matri using void glloadmatri{fd}( const TYPE *m ); where m specifies a 6 element matri in column major order m m m m 2 3 m m m m m m m m 8 9 m m m m (6.3) Eample 6.2 The following code fragments are equivalent. GLdouble m[ ] = {.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,. }; glloadmatrid( m ); and glloadidentit( ); E.R. Bachmann & P.L. McDowell MV 422 Page 29 of 3

30 A matri ma also be eplicitl specified and multiplied b the current matri using void glmultmatri{fd}( const TYPE *m ); where m specifies a 6 element matri in column major order Eample 6.2 The following code fragments are equivalent. GLdouble m[ ] = {.,.,.,.,.,.,.,.,.,.,.,., 5., 2., 3.,. }; glloadidentit( ); glmultmatrid ( m ); and glloadidentit( ); gltranslated( 5., 2., 3. ); E.R. Bachmann & P.L. McDowell MV 422 Page 3 of 3

31 Modelview Transformation Review General algorithm for positioning objects and setting the viewpoint:. Clear the frame buffer and whatever other buffers are in use. glclear( GL_COLOR_BUFFER_BIT ); 2. Set the matri mode to modelview. glmatrimode( GL_MODELVIEW ); 3. Clear the matri stack to the Identit matri. glloadidentit(); 4. Perform viewing transformations using gltranslate and glrotate or glulookat. glulookat(.,., 5.,.,., -.,.,.,. ); 5. Perform modeling transformations using gltranslate, glrotate, glscale, glpushmatri, glpopmatri and glloadidentit. 6. Draw objects in the transformed coordinate sstems using vertices specified between glbegin and glend pairs. 7. Force completion of all drawing commands using glflush and SwapBuffers. glflush(); SwapBuffers( m_pdc->getsafehdc( ) ); E.R. Bachmann & P.L. McDowell MV 422 Page 3 of 3