GR2 Advanced Computer Graphics AGR

GR2Advanced Computer GraphicsAGR Lecture 2 Basic Modelling

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

Polygonal Representation • Objects with curved surfaces can be approximated by polygons - improved approximation by more polygons

Scene Organisation • Scene = list of objects • Object = list of surfaces • Surface = list of polygons • Polygon = list of vertices scene vertices object surfaces polygons

Polygon Table Vertex Table V1 X1, Y1, Z1 P1 V1, V2, V3, V4 V2 X2, Y2, Z2 P2 V1, V5, V6, V2 . ... . ... Polygon Data Structure V7 V6 Object Obj1 V5 V8 V3 V2 P2 V4 V1 P1 Object Table Obj1 P1, P2, P3, P4, P5, P6

Typical Primitives • Graphics systems such as OpenGL typically support: • triangles, triangle strips and fans • quads, quad strips • polygons • Which way is front? • convention is that normal points towards you if vertices are specified counter-clockwise

Modelling Regular Objects • Sweeping • Spinning sweep axis 2D Profile R1 R2 spinning axis

Sweeping a Circle to Generate a Cylinder as Polygons V12 V13 V11 V18 V3 V10 V4 V14 V2 V17 V15 V5 V1 vertices at z=depth V16 V6 V8 V7 V9 vertices at z=0 V1[x] = R; V1[y] = 0; V1[z] = 0 V2[x] = R cos ; V2[y] = R sin ; V2[z] = 0 (q=p/4) Vk[x] = R cos k; Vk[y] = R sin k; Vk[z] = 0 where k = 2 p (k - 1 )/8, k=1,2,..8

Exercise and Further Reading • Spinning: • Work out formulae to spin an outline (in the xy plane) about the y-axis • READING: • Hearn and Baker, Chapter 10

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

Automatic Generation of Polygonal Objects • 3D scanners - or laser rangers - are able to generate computer representations of objects • object sits on rotating table • contour outline generated for a given height • scanner moves up a level and next contour created • successive contours stitched together to give polygonal representation

A Puzzle

Modelling Objects and Creating Worlds • We have seen how boundary representations of simple objects can be created • Typically each object is created in its own co-ordinate system • 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

Transformations • 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 • As in 2D graphics, we use homogeneous co-ordinates in order to express all transformations as matrices and allow them to be combined easily

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

Translation • Suppose we want to translate P (x,y,z)T by a distance (Tx, Ty, Tz)T • We express P as (x, y, z, 1)T and form a translation matrix T as below • The translated point is P’ x y z 1 = x’ y’ z’ 1 1 0 0 Tx 0 1 0 Ty 0 0 1 Tz 0 0 0 1 = x + Tx y + Ty z + Tz 1 = T P P’

= x’ y’ z’ 1 Sx 0 0 0 0 Sy 0 0 0 0 Sz 0 0 0 0 1 x y z 1 P’ S P = Scaling • Scaling by Sx, Sy, Sz relative to the origin: = Sx . x Sy . y Sz . z 1

x’ y’ z’ 1 = 1 0 0 0 0 cos -sin 0 0 sin cos 0 0 0 0 1 x y z 1 P’ = Rz () P Rotation • Rotation is specified with respect to an axis - easiest to start with co-ordinate axes • 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

Composite Transformations • The attraction of homogeneous co-ordinates is that a sequence of transformations may be encapsulated as a single matrix • 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) • Thus: P’ = T(a,b,c) S T(-a,-b,-c) P = M P

Rotation about a Specified Axis • It is useful to be able to rotate about any axis in 3D space • This is achieved by composing 7 elementary transformations

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

Inverse Transformations • As in this example, it is often useful to calculate the inverse of a transformation • ie the transformation that returns to original state • Translation: T-1 (a, b, c) = T (-a, -b, -c) • Scaling: S-1 ( Sx, Sy, Sz ) = S ............ • Rotation: R-1z () = Rz (-)

Rotation about Specified Axis • Thus the sequence is: T-1 R-1x() R-1 y() Rz() Ry() Rx() T • EXERCISE: How are  and  calculated? • READING: • Hearn and Baker, chapter 11.

GR2 Advanced Computer Graphics AGR