Introduction to 3D Graphics Lecture 3: General Camera Model

1. Introduction to 3D GraphicsLecture 3: General Camera Model Anthony Steed University College London

2. Overview • More Maths • Rotations and translations • Homogenous co-ordinates • General Camera • Specification • Mapping to world coordinates

3. Vectors and Matrices • Matrix is an array of numbers with dimensions M (rows) by N (columns) • 3 by 6 matrix • element 2,3 is (3) • Vector can be considered a 1 x M matrix

4. Identity matrices - I Diagonal Symmetric Diagonal matrices are (of course) symmetric Identity matrices are (of course) diagonal Types of Matrix

5. Operation on Matrices • Addition • Done elementwise • Transpose • “Flip” (M by N becomes N by M)

6. Operations on Matrices • Multiplication • Only possible to multiply of dimensions • x1 by y1 and x2 by y2 iff y1 = x2 • resulting matrix is x1 byy2 • e.g. Matrix A is 2 by 3 and Matrix by 3 by 4 • resulting matrix is 2 by 4 • Just because A x B is possible doesn’t mean B x A is possible!

7. A is n by k , B is k by m C = A x B defined by BxA not necessarily equal to AxB Matrix Multiplication Order

8. Example Multiplications

9. Inverse • If A x B = I and B x A = I then A = B-1 and B = A-1

10. 3D Transforms • In 3-space vectors are transformed by 3 by 3 matrices

11. Scale • Scale uses a diagonal matrix • Scale by 2 along x and -2 along z

12. Rotation • Rotation about z axis • Note z values remain the same whilst x and y change Y X

13. About X About Y Scale (should look familiar) Rotation X, Y and Scale

14. Homogenous Points • Add 1D, but constrain that to be equal to 1 (x,y,z,1) • Homogeneity means that any point in 3-space can be represented by an infinite variety of homogenous 4D points • (2 3 4 1) = (4 6 8 2) = (3 4.5 6 1.5) • Why? • 4D allows as to include 3D translation in matrix form

15. Homogenous Vectors • Vectors != Points • Remember points can not be added • If A and B are points A-B is a vector • Vectors have form (x y z 0) • Addition makes sense

16. Translation in Homogenous Form • Note that the homogenous component is preserved (* * * 1), and aside from the translation the matrix is I

17. Putting it Together • R is rotation and scale components • T is translation component

18. Order Matters • Composition order of transforms matters • Remember that basic vectors change so “direction” of translations changed

19. Overview • More Maths • Rotations and translations • Homogenous co-ordinates • General Camera • Specification • Mapping to world coordinates

20. Simple Camera (Cross Section) Y d ymax Z -Z COP ymin

21. General Camera • View Reference Point (VRP) • where the camera is • View Plane Normal (VPN) • where the camera points • View Up Vector (VUV) • which way is up to the camera • X (or U-axis) forms LH system

22. UVN Co-ordindates • View Reference Point (VRP) • origin of VC system • View Plane Normal (VPN) • Z (or N-axis) of VC system • View Up Vector (VUV) • determines Y (or V-axis) of VCS • X (or U-axis) forms LH system

23. World Coords and Viewing Coords Y V U V U V V R P N X We want to find a general transform (EQ1) of the above form that will map WC to VC Z

24. View from the Camera N and VPN into the page V xmax, ymax VUV Z Y X U xmin, ymin

25. Finding the basis vectors • Step 1 - find n • Step 2 - find u • Step 3 - find v

26. Finding the Mapping (1) • u,v,n must rotate under R to i,j,k of viewing space • Both basis are normalised so this is a pure rotation matrix • recall in this case RT = R-1

27. Finding the Mapping (2) • In uvn system VRP (q) is (0 0 0 1) • And we know from EQ1 so

28. Complete Mapping • Complete matrix

29. For you to check • If • Then

30. Using this for Ray-Casting • Use a similar camera configuration (COP is usually, but not always on -n) • To trace object must either • transform spheres into VC • transform rays into WC

31. Ray-casting • Transforming rays into WC • Transform end-point once • Find direction vectors through COP as before • Transform vector by • Intersect spheres in WC

32. Ray-casting • Transforming spheres into VC • Centre of sphere is a point so can be transformed as usual (WC to VC) • Radius of sphere is unchanged by rotation and translation (and spheres are spheroids if there is a non-symmetric scale)

33. Tradeoff • If more rays than spheres do the former • transform spheres into VC • For more complex scenes e.g. with polygons • transform rays into WC

34. Alternative Forms of the Camera • Simple “Look At” • Give a VRP and a target (TP) • VPN = TP-VRP • VUV = (0 1 0) (i.e. “up” in WC) • Field of View • Give horizontal and vertical FOV or one or the other and an aspect ratio • Calculate viewport and proceed as before

35. Animated Cameras • Animate VRP (observer-cam) • Animate VPN (look around) • Animate TP (track-cam) • Animate COP • along VPN - zoom • orthogonal to VPN - distort

36. Summary • We set up the mathematics of transformations between co-ordinate spaces • We created a more general camera which we can use to create views of our scenes from arbitrary positions • Formulation of mapping from WC to VC (and back)