Computer Graphics Conceptual Model

1. Computer GraphicsConceptual Model API Output Devices Application Model Graphics System Application Program Input Devices Function Calls or Protocol Data

2. Modeling in Three DimensionsTwo Approaches • Constructive Solid Geometry (CSG) • Primitives are a small set of volumetric objects such as cube, pyramid, sphere, etc. • Objects built from normalized set operations on these objects • Boundary (surface) representation • Objects are described by their surfaces and can be thought of as being hollow • Surfaces composed (approximated) by simple, convex, planar polygons • Objects can be specified through a set of vertices in three dimensions • Implies underlying coordinate system • Duality: Object <--> Object representation

3. 0 A 3 B 2 C 1 D E F Representation of a CubeSurface Representation Data Structure VERTEX LISTS FACES VERTICES 5 6 X0, Y0, Z0 X1, Y1, Z1 1 X2, Y2 , Z2 2 X3, Y3 , Z3 4 7 X4, Y4 , Z4 3 X5, Y5 , Z5 7 0 3 X6, Y6 , Z6 6 X7, Y7 , Z7 2 Representation: data structure . . . Object: set of points

4. Inward and Outward Pointing Faces:Right Hand Rule Outside 3 2 0 1

5. Transformations • Maps an object into another object • In general, a transformation maps every point on an object to another point in the underlying coordinate space. • Why do we want/need transformations? • Change size, location, orientation of objects without changing underlying model (or primitive drawing commands) • Animation • Instancing Transformation Object New Object Representation New Representation Operation

6. P1’ P2’ What Do We Require of Transforms?Line Preserving P1 Line Not Preserved P2

7. Abstract Data Types:Affine Space • Scalar a, b, d, e • Magnitude (e.g. real numbers) • Vector u, v, w, x • Direction • Magnitude • No Position • Point P, Q, R, X • Position • No Direction • No Magnitude

8. Defined Operations:Scalars • Addition: a+b • Additive identity: zero • Additive inverse: - b • Subtraction defined in terms of additive inverse • Multiplication: d * j • Multiplicative identity: 1 • Multiplicative inverse: 1/ d • Division defined in terms of multiplicative inverse

9. u q v Defined Operations:Vectors Dot Product Cross Product Addition v u u + v q v u Produces vector w orthogonal to u and v. u · v = | u | | v | cos q Magnitude (produces scalar) | u | = b w = u x v | w | = | u | | v | sin q (produces scalar)

10. Defined Operations:Scalars and Vectors Multiplication of vector by scalar: Produces vector u v = a u |a u | = a | u |

11. Defined Operations:Points and Vectors Affine Addition: Point + Vector = Point Q v P Q = P + v Related to point subtraction.

12. Defined Operations:Points Subtraction (Addition of points has no meaning.) P u = P - R R Produces vector

13. Lines in Affine Space R’ R Q v Q = P + v P R = P + a v - ¥ < a < ¥ Line segment from P to Q defined when: 0 £ a £ 1

14. Planes in Affine Space Planes are defined uniquely by three points: P, Q, R W(l) Q u = Q - P v = R - P u T( b) R P v S(a) S(a) = P + a v T(b) = P + b u W(a, b, l) = S +l (T - S) = P + a v + l (b u - a v) W (a, b, l) = P + a ( 1 - l ) v + l b u W = P + d v + c u

15. Representations of Points and Vectors Coordinate system: y P (x, y, z) v x z where:

16. Homogeneous Coordinates Frame is specified by: where are linearly independent basis vectors and P0 arbitrary point Frame Homogeneous coordinate representation:

17. Representation of VectorsIn Homogeneous Coordinates

18. Affine AdditionIn Homogeneous Coordinate Representation Point Point Scalar times vector

19. y x Transform as aChange of Frame y’ Want to translate by:Dx = 2, Dy=4 (2, 3) in x’y’ (4,7) in x y y x’ (2, 3) Dx = 2, Dy=4 x 1. Transform frame 2. Find representation in original frame

20. Transform as aChange of Frame Representation: Frame:

21. Transformation Matrix Original Frame: Transformed Frame: Express elements of transformed frame in original frame: where:

22. is representation in transformed frame is representation in original frame Transformation of Representation where is of the form:

23. Affine Transformations T(point) -> point T(vector) -> vector

24. P1’ P2’ What Do We Require of Transforms?Line Preserving P1 Line Not Preserved P2

25. Affine Transformation:Line Preservation Matrix multiplication is linear. Therefore: T(p)+aT(v) p+av v T(v) p T(p) Meets our requirements!

26. Translation Assumptions: Translate the new frame:

27. Scaling Scale the new frame:

28. Rotation Rotation around the z axis: q

29. Rotation About x and y axes Similar derivations give:

30. Concatenation of Transformations Apply T1, Then apply T2:

31. OpenGL Matrix Operations glLoadIdentity(); Loads an identity matrix onto the top of the current stack glLoadMatrixf(pointer_to_matrix); Loads arbitrary matrix onto top of the current stack glMultMatrixf(pointer_to_matrix); Postmultiplies current matrix by arbitrary matrix Matricies are one-dimensional arrays of type GLfloat in column major order.

32. Predefined Postmultiplier Operatorsin OpenGL glTranslatef(dx, dy, dz); Multiplies current matrix with translation matrix. dx, dy, and dz are translations along x,y, and z axes. glRotatef(angle, x, y, z); Multiplies current matrix with rotation about the line from the origin through the point (x, y, z) by angle. Right hand rule applies. Angle in degrees. glScalef(sx, sy, sz); Multiplies current matrix with scaling matrix. sx, sy, sz are the scale factors along the x, y, and z axes.

33. Load Matrix Current Current Stack Stack Transformation Matrices in OpenGL Matrix Mode 3D Model Vertices 2D 3D Vertices Modelview Projection