Lecture 3: Transformations a nd Viewing

1. Lecture 3: Transformations and Viewing

2. General Idea • Object in model coordinates • Transform into world coordinates • Represent points on object as vectors • Multiply by matrices

3. Whatnext? • 2D transformations: scale, shear, rotation • Composing transformations • 3D rotations

4. (Nonuniform) Scale

5. Shear

6. Rotations 2D rotationsaresimple • Can we deriveit? x’,y’ x,y  

7. Rotations, contd. Basic properties… • Linear • Commutative • R(X+Y)=R(X)+R(Y)

8. Outline • 2D transformations: rotation, scale, shear • Composing transforms • 3D rotations • Translation: Homogeneous Coordinates • Transforming Normals

9. Composing Transforms • Usually we build a complextransformation by composingsimpleones (for which we havematrixpatterns) • E.g. first scale then rotate • We seenowadvantageof matrix formulation: • Transformationisperformed for manypoints • Complextransformationis in advancedputinto a single matrix • In generalnotcommutative!! Order matters

10. E.g. Composing rotations, scales • In terms of functions: • In terms of matrices: RS isalso a matrix • RS ≠ SR, not commutative

11. Composingexample

12. Inverting Composite Transforms • Say,wewant to invert a combination of 3 transforms • Option 1: Find composite matrix, invert • Option 2: Invert each transform and swap order • Obvious from properties of matrices

13. Outline • 2D transformations: rotation, scale, shear • Composing transforms • 3D rotations • Translation: Homogeneous Coordinates • Transforming Normals

14. Orthogonalmatrix Matrix Aisorthogonalif and onlyif E.g. Rotationmatrixisorthogonal. Checkit.

15. Orthogonalmatrix Important and usefulproperty: It is much faster to transpose a matrix, than to invertit

16. Rotations in 3D • Rotations about coordinate axes is a simpleextension of 2D rotations • Againorthogonalmatrices

17. Interpretation 3D RotationsMatrix • Rows of matrix are thevectors of new coordinatesystem(the rotated one) • Oldcoordinateframe: x, y, z versors • Aftertransformation: • we havenewuvwcoordinateframe

18. Geometric Interpretation 3D Rotations • Rows of matrix are 3 unit vectors of new coord frame • Can construct rotation matrix from 3 orthonormal vectors • Effectively, projections of point into new coord frame • New coord frame uvw taken to cartesian components xyz • Inverse or transpose takes xyz cartesian to uvw

19. Non-Commutativity • Not Commutative (unlike in 2D)!! • Rotate by x, then y is not same as y then x • Order of applying rotations does matter • Follows from matrix multiplication not commutative • R1 * R2 is not the same as R2 * R1

20. Translation • Translation isprobably the simplesttransformation (isit?), but there’ssomethingtrickyaboutit • E.g. move x by +7units, leave y, z unchanged: • How to express it in a matrix form? I.e.

21. ?

22. Homogeneous Coordinates • We need to adda fourth coordinate (w=1). It’squitecommonway in linear algebra, to getrid off freeterms. • 4x4 matrices in 3D space -very common in computergraphics • w maylooklike a dummyvariable – but mayplayit’s role

23. Representation of Points (4-Vectors) Homogeneous coordinates • Divide by 4thcoord (w) to get (inhomogeneous) point • Multiplication by w > 0, no effect • Assume w ≥ 0. For w > 0, normal finite point. For w = 0, point at infinity (used for vectors to stop translation)

24. Whatare the Advantages of Homogeneous Coords? • Unified framework for translationand othertransformations • Only4x4 matrix – easy to concatenate • Less problems with perspectiveviewing • No special cases – homogenousformulas

25. General Translation Matrix

26. 3 basicRotationMatrices

27. Combining Translations, Rotations • Order matters!! TR is not the same as RT (demo) • General form for rigid body transforms • We show rotation first, then translation (commonly used to position objects) on next slide. Slide after that works it out the other way

28. Combining Translations, Rotations First Rotation, thenTranslation First Translation, thenRotation

29. Geometry Transformation • Transformationsarecomposed of fourelements • Viewingtransformation – positioning the camera • Modelingtransformation - movingobjects • (Viewing and Modelingarecomplementary) • Projection – on the plane • Viewport – mapping on the window Podstawy OpenGL

30. Cameraanalogy Podstawy OpenGL

31. Cameraanalogycontd. Podstawy OpenGL

32. Twoways of makingtransformation in OpenGL • First approach: we operatedirectly on matrices • Second approach: we hidematrixtransformationsinsidefunctions • Which one isbetter?

33. Twoways of makingtransformation in OpenGL • Functionstypical to ”old” OpenGL. • Cancelled in OpenGL 3.0 • Transformations and Projectionsmoved to shaderprogramming. • Not everyonelikeslow-levelshaderprogramming. • Search for newfunctions…

34. GLM – OpenGLMathematics • Can be easilyfound onglm.g-truc.net/  , along with otherinformation • Can be treated as enhancement of GLSL. • IsincludedintounofficialOpenGL SDK

35. Viewing

36. Motivation • We have seen transforms (between coord systems) • But all that is in 3D • We still need to make a 2D picture • Project 3D to 2D. How do we do this? • This lecture is about viewing transformations

37. Demo (Projection Tutorial) • Nate Robbins OpenGL tutors • http://user.xmission.com/~nate/ • Projection tutorial • OldOpenGL Style • Demo

38. What we’ve seen so far • Transforms (translation, rotation, scale) as 4x4 homogeneous matrices • Last row always 0 0 0 1. Last w component always 1 • For viewing (perspective), we will use that last row and w component no longer 1 (must divide by it)

39. Outline • Orthographic projection (simpler) • Perspective projection, basic idea • Derivation of gluPerspective (handout: glFrustum) • Brief discussion of nonlinear mapping in z

40. Projections • To lower dimensional space (here 3D -> 2D) • Preserve straight lines • Trivial example: Drop one coordinate (Orthographic)

41. Orthographic Projection • Characteristic: Parallel lines remain parallel • Useful for technical drawings etc. Perspective Orthographic

42. Example • Simply project onto xy plane, drop z coordinate

43. In general • We have a cuboid that we want to map to the normalized or square cube from [-1, +1] in all axes • We have parameters of cuboid (l,r ; t,b; n,f) t f y n b r l y y Translate Scale x x x z z z

44. Orthographic Matrix • First center cuboid by translating • Then scale into unit cube t f y n b r l y y Translate Scale x x x z z z

45. Transformation Matrix Scale Translation

46. Caveats • Looking down –z, f and n are negative (n > f) • OpenGL convention: positive n, f, negate internally t f y n b r l y y Translate Scale x x x z z z

47. Final Result

48. Outline • Orthographic projection (simpler) • Perspective projection, basic idea • Derivation of gluPerspective (handout: glFrustum) • Brief discussion of nonlinear mapping in z

49. Perspective Projection • Most common computer graphics, art, visual system • Further objects are smaller (size, inverse distance) • Parallel lines not parallel; converge to single point A Plane of Projection A’ B B’ Center of projection (camera/eye location) Slides inspired by Greg Humphreys

50. Overhead View of Our Screen (x,y,z) (x’,y’,z’) (0,0,0) d Looks like we’ve got some nice similar triangles here?