Evening’s Goals

2. Evening’s Goals • Discuss viewing and modeling transformations • Describe matrix stacks and their uses • Show basic geometric rasterization and clipping algorithms COEN 290 - Computer Graphics I

3. Modeling Objects • Recall objects are composed of geometric primitives • each primitive is composed of vertices • Model around the origin • model just means “determine an object’s vertices” COEN 290 - Computer Graphics I

4. Modeling Transformations • Position objects in world coordinates • Move coordinate systems, not objects • Affects all objects rendered after transformation • Types • translation • scale • rotation • shear COEN 290 - Computer Graphics I

5. Translation • Move the origin to a new location COEN 290 - Computer Graphics I

6. glTranslate[fd]() glTranslatef( tx, ty, tz ); COEN 290 - Computer Graphics I

7. Scale • Stretch, mirror or decimate a coordinate direction Note, there’s a translation applied here to make thingseasier to see COEN 290 - Computer Graphics I

8. glScale[fd]() glScalef( sx, sy, sz ); COEN 290 - Computer Graphics I

9. Rotation • Rotate coordinate system about an axis in space Note, there’s a translation applied here to make thingseasier to see COEN 290 - Computer Graphics I

10. glRotate[fd]() glRotatef( angle, x, y, z ); M COEN 290 - Computer Graphics I

11. Some Additional Rotation Examples COEN 290 - Computer Graphics I

12. Shear • Scaling in one dimension depends on another • For example • No OpenGL command • load custom matrix COEN 290 - Computer Graphics I

13. Shear • Making a shear matrix COEN 290 - Computer Graphics I

14. Other OpenGL Matrix Commands • glMultMatrix( m ) • multiples the current matrix by m • glLoadMatrix( m ) • replaces the current matrix by m • glLoadIdentity() • replace the current matrix with an identity matrix COEN 290 - Computer Graphics I

15. OpenGL Matrix Format GLfloat m[16]; COEN 290 - Computer Graphics I

16. Modeling Transforms and our Pipeline World Coordinates Eye Coordinates ClipCoordinates NDC’s Model Coordinates Perspective Divide ModelingTransformations ProjectionTransformation COEN 290 - Computer Graphics I

17. Using Multiple Modeling Transforms • Multiple modeling transforms form a composite modeling transform • Individual transform matrices are multiplied together • for example COEN 290 - Computer Graphics I

18. Multiplying Matrices • Matrix multiplication is not commutative • in general, • order of operations is important • OpenGL multiples matrices on the right COEN 290 - Computer Graphics I

19. Independent vs. Dependent Transforms • Every modeling transform affects the model coordinate system • Transformations accumulate • we record every transformation every made • to undo a transform, we’d need to do the inverse transform • this is very inconvenient COEN 290 - Computer Graphics I

20. A Stack Based Solution • We create a stack containing matrices • Any transform multiples the top of stack • Push makes a copy and pushes it onto the stack • Pop discards the top of stack COEN 290 - Computer Graphics I

21. ModelingTransformations A Stack Based Solution ( cont. ) ProjectionTransformation COEN 290 - Computer Graphics I

22. OpenGL Matrix Stack Commands • Top of stack matrix is called the current matrix • glPushMatrix() • copy the current matrix and pushes it • glPopMatrix() • pop the current matrix COEN 290 - Computer Graphics I

23. OpenGL Matrix Stacks • Why multiple matrix stacks? • certain techniques are done in different spaces glMatrixMode( mode ) • choose which stack to manipulate • GL_MODELVIEW • GL_PROJECTION COEN 290 - Computer Graphics I

24. Model Coordinates Model Coordinates Model Coordinates Current Transformation Pipeline World Coordinates Eye Coordinates ClipCoordinates NDC’s Perspective Divide ProjectionTransformation ModelingTransformations COEN 290 - Computer Graphics I

25. Eye coordinates are nice, but ... • Eye coordinates are restrictive • eye’s positioned at the origin • looking down the -z axis • What if we want to look at the scene from somewhere else? • use a viewing transformation COEN 290 - Computer Graphics I

26. Viewing Transformations • Reorient world coordinates to match eye coordinates • Basically just a modeling transform • affects the entire scene • usually a translation and a rotation • Usually set up after the projection transform, but before any modeling transforms COEN 290 - Computer Graphics I

27. The Simplest Viewing Transform • “Push” the origin into the viewing frustum near far COEN 290 - Computer Graphics I

28. polarview() • Useful if you want to view an entire object polarview(dist,elev,azim,twist){ glTranslatef( 0, 0, -dist ); glRotatef( -twist, 0, 0, 1 ); glRotatef( -elev, 1, 0, 0 ); glRotatef( azim, 0, 0, 1 );} • update elev and azim for interactive viewing COEN 290 - Computer Graphics I

29. Transforming World to Eye Coordinates • Viewing transform gluLookAt( eyex, eyey, eyez, lookx, looky, lookz, upx, upy, upz ); • Creates an orthonormal basis • a set of linearly independent vectors of unit length COEN 290 - Computer Graphics I

30. Creating an Orthonormal Basis COEN 290 - Computer Graphics I

31. gluLookAt( eyex, eyey, eyez, lookx, looky, lookz, upx, upy, upz ){ /* Create matrix m */ glMultMatrixf( m ); glTranslatef( -eyex, -eyey, -eyez );} COEN 290 - Computer Graphics I

32. Model Coordinates Model Coordinates Model Coordinates Our Final Transformation Pipeline ViewingTransformation World Coordinates Eye Coordinates ClipCoordinates NDC’s Perspective Divide ProjectionTransformation ModelingTransformations COEN 290 - Computer Graphics I

33. Retrieving Matrix information GLfloat m[16]; glGetFloatv( which, m ); • which is which matrix stack to access • GL_MODELVIEW_MATRIX • GL_PROJECTION_MATRIX COEN 290 - Computer Graphics I

34. Putting it all Together • General flow of transformations • projection transformation • viewing transformation • push matrix • modeling transformation • render • pop matrix • goto  as necessary COEN 290 - Computer Graphics I