1. 2.1 si31-2001 SI31 Advanced Computer Graphics AGR Lecture 2 Basic Modelling

2. 2.2 si31-2001 Polygonal Representation n Any 3D object can be represented as a set of plane, polygonal surfaces V1 V2 V3 V4 V5 V8 V7 V6 Note: each vertex part of several polygons

3. 2.3 si31-2001 Polygonal Representation n Objects with curved surfaces can be approximated by polygons - improved approximation by more polygons

4. 2.4 si31-2001 Scene Organisation n Scene = list of objects n Object = list of surfaces n Surface = list of polygons n Polygon = list of vertices scene object surfacespolygons vertices

5. 2.5 si31-2001 Polygon Data Structure V1 V2 V3 V4 V5 V8 V7 V6 P1 P2 Object Table Obj1 P1, P2, P3, P4, P5, P6 Object Obj1 Vertex Table V1X1, Y1, Z1 V2X2, Y2, Z2.... Polygon Table P1V1, V2, V3, V4 P2V1, V5, V6, V2.....

6. 2.6 si31-2001 Typical Primitives Order of Vertices n Graphics systems such as OpenGL typically support: – triangles, triangle strips and fans – quads, quad strips – polygons n How are vertices ordered? – convention is that vertices are ordered counter-clockwise when looking from outside an object – allows us to distinguish outer and inner faces of a polygon

7. 2.7 si31-2001 Modelling Regular Objects n Sweeping n Spinning 2D Profile sweep axis spinning axis R1 R2

8. 2.8 si31-2001 Sweeping a Circle to Generate a Cylinder as Polygons vertices at z=0 vertices at z=depth V1 V2 V3 V4 V5 V6V8 V7 V10 V9 V11 V12 V13 V14 V15 V16 V17 V18 V1[x] = R; V1[y] = 0; V1[z] = 0 V2[x] = R cos  ; V2[y] = R sin  ; V2[z] = 0 (  =  /4) Vk[x] = R cos  k ; Vk[y] = R sin  k ; Vk[z] = 0 where  k = 2  (k - 1 )/8, k=1,2,..8

9. 2.9 si31-2001 Exercise and Further Reading n Spinning: – Work out formulae to spin an outline (in the xy plane) about the y-axis n READING: – Hearn and Baker, Chapter 10 – Watt, Chapter 2 (2.1)

10. 2.10 si31-2001 Complex Primitives n Some systems such as VRML have cylinders, cones, etc as primitives – polygonal representation calculated automatically n OpenGL has a utility library (GLU) which contains various high-level primitives – again converted to polygons n For conventional graphics hardware: – POLYGONS RULE!

11. 2.11 si31-2001 Automatic Generation of Polygonal Objects n 3D laser scanners are able to generate computer representations of objects – for successive heights, 2d outline generated as object rotates – contours stitched together into 3D polygonal representation n Cyberware Cyberscanner in Med Physics at LGI able to scan human faces

12. 2.12 si31-2001 A Puzzle

13. 2.13 si31-2001 Modelling Objects and Creating Worlds boundary representations n We have seen how boundary representations of simple objects can be created co-ordinate system n Typically each object is created in its own co-ordinate system translationscaling rotation n To create a world, we need to understand how to transform objects so as to place them in the right place - translation, at the right size - scaling, in the right orientation- rotation

14. 2.14 si31-2001 Transformations n The basic linear transformations are: – translation: P = P + T, where T is translation vector – scaling: P’ = S P, where S is a scaling matrix – rotation: P’ = R P, where R is a rotation matrix homogeneous n As in 2D graphics, we use homogeneous co-ordinates in order to express all transformations as matrices and allow them to be combined easily

15. 2.15 si31-2001 Homogeneous Co-ordinates n In homogeneous coordinates, a 3D point P = (x,y,z) T is represented as: P = (x,y,z,1) T n That is, a point in 4D space, with its ‘extra’ co-ordinate equal to 1 n Note n Note: in homogeneous co-ordinates, multiplication by a constant leaves point unchanged – ie (x, y, z, 1) T = (wx, wy, wz, w) T

16. 2.16 si31-2001 Translation n Suppose we want to translate P (x,y,z) T by a distance (Tx, Ty, Tz) T n We express P as (x, y, z, 1) T and form a translation matrix T as below n The translated point is P’ TP x’ y’ z’ 1 P’ = 1 0 0 Tx 0 1 0 Ty 0 0 1 Tz 0 0 0 1 xyz1xyz1 = x + Tx y + Ty z + Tz 1 =

17. 2.17 si31-2001 Scaling relative to the origin n Scaling by Sx, Sy, Sz relative to the origin: x’ y’ z’ 1 Sx 0 0 0 0 Sy 0 0 0 0 Sz 0 0 0 0 1 xyz1xyz1 P’ = SP = = Sx. x Sy. y Sz. z 1

18. 2.18 si31-2001 Rotation with respect to an axis n Rotation is specified with respect to an axis - easiest to start with co-ordinate axes n To rotate about the x-axis:  a positive angle corresponds to counterclockwise direction looking at origin from positive position on axis EXERCISE: write down the matrices for rotation about y and z axes x’ y’ z’ 1 =1 0 0 0 0 cos  -sin  0 0 sin  cos  0 0 0 0 1 xyz1xyz1 P’= R x (  ) P

19. 2.19 si31-2001 Composite Transformations n The attraction of homogeneous co-ordinates is that a sequence of transformations may be encapsulated as a single matrix with respect to a fixed position (a,b,c) n For example, scaling with respect to a fixed position (a,b,c) can be achieved by: – translate fixed point to origin- say, T(-a,-b,-c) – scale- S – translate fixed point back to its starting position- T(a,b,c) n Thus: P’ = T(a,b,c) S T(-a,-b,-c) P = M P

20. 2.20 si31-2001 Rotation about a Specified Axis n It is useful to be able to rotate about any axis in 3D space n This is achieved by composing 7 elementary transformations

21. 2.21 si31-2001 Rotation through  about Specified Axis x y z x y z rotate through requ’d angle,  x y z x y z P2 P1 x y z P2 P1 x y z initial position translate P1 to origin rotate so that P2 lies on z-axis (2 rotations) rotate axis to orig orientation translate back

22. 2.22 si31-2001 Inverse Transformations n As in this example, it is often useful to calculate the inverse of a transformation – ie the transformation that returns to original state n Translation: T -1 (a, b, c) = T (-a, -b, -c) n Scaling: S -1 ( Sx, Sy, Sz ) = S............ Rotation: R -1 z (  ) = R z (…….) Exercise: Check T -1 T = I (identity matrix)

23. 2.23 si31-2001 Rotation about Specified Axis n Thus the sequence is: T -1 R -1 x (  ) R -1 y (  ) R z (  ) R y (  ) R x (  ) T EXERCISE: How are  and  calculated? n READING: – Hearn and Baker, chapter 11. – Watt, Chapter 1 (1.1)