SI23 Introduction to Computer Graphics

1. SI23Introduction to Computer Graphics Lecture 10 – Introduction to 3D Graphics

2. Graphics Programming 3D Graphics URL VRML viewer 2D vector graphics surfaces Image Display URL SVG Viewer lines, areas URL GIMP viewing, shading graphics algorithms colour interaction Course Outline • Graphics programming • Using OpenGL with C, C++ OpenGL API animation

3. SVG is a vector graphics description language <?xml version="1.0" ?> <!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 20010904//EN" "http://www.w3.org/TR/2001/REC-SVG-20010904/DTD/svg10.dtd"> <svg width = "300" height = "300"> <rect x = "100" y = "100" width = "100" height = "100" style = "fill:red" /> </svg> SVG Viewer 2D Graphics – Picture Description

4. For programming graphics, there are a number of APIs, or Application Programming Interfaces OpenGL is industry standard Both 2D and 3D 2D Graphics – Programming Approach User Program calling OpenGL functions OpenGL Library

5. Objectives for This Part • To understand how 3D scenes can be modelled - in terms of geometry and appearance - and rendered on a display • To be able to program interactive 3D graphics applications using industry standard software (OpenGL)

6. Lecture Outline - The Basics • MODELLING and VIEWING • representing objects in 3D • transforming objects and composing scenes • specifying the camera viewpoint • PROJECTION • projecting 3D scenes onto a 2D display surface • RENDERING • illumination • shading • adding realism via textures, shadows

7. Basic Modelling and Viewing y objects represented as set of faces - ie polygons- and faces as a set of points x z scenes composed by scaling, rotating, translating objects to create a 3D world Camera position specified, together with its direction of view camera

8. camera Projection • Projection • 3D scene is projected onto a 2D plane view plane

9. A Puzzle

10. Rendering illumination: how is light reflected from surfaces? ?? shading: how do we use our knowledge of illumination to shade surfaces in our world?

11. Rendering • texture • shadows

12. OpenGL is an API that allows us to program 3D graphics As well as 2D VRML is a language that allows us to describe 3D graphics scenes Cf SVG for 2D Creating 3D Graphics

13. Applications - Computer Games

14. This is Hubble Space Telescope modeled using the BRL-CAD system Uses CSG modeling and ray tracing for rendering http://ftp.arl.mil/brlcad Applications - Computer-Aided Design

15. Virtual oceanarium built for EXPO in Lisbon Example taken from Fraunhofer Institute site http://www.igd.fhg.de Applications - Virtual Reality

16. Before we begin...mathematics! • 3D Co-ordinate Systems y y z x x z LEFT RIGHT z points away z points toward Align thumb with x, first finger with y, then second finger of appropriate hand gives z direction. Common now to use a RIGHT HANDED system.

17. Points and Vectors • We shall write points as column vectors y P P = x y z x z Difference of two points gives a direction vector: D = P2 - P1 y P2 Note: If P1 and P2 are on a plane, then D lies in the plane x z P1

18. Magnitude of a Vector • The magnitude of a vector V = (v1,v2,v3)T is given by: |V| = sqrt(v1*v1 + v2*v2 + v3*v3) eg (1,2,3)T has magnitude sqrt(14) • A unit vector has magnitude 1 • A unit vector in the direction of V is V / |V|

19. Scalar or Dot Product • The scalar product, or dot product, of two vectors U and V is defined as: U.V = u1*v1 + u2*v2 + u3*v3 • It is important in computer graphics because we can show that also: U.V = |U|*|V|*cosq where q is the angle between U and V • This lets us calculate angle q as cos q = (u1*v1 + u2*v2 + u3*v3) / (|U|*|V|)

20. Diffuse Lighting • Diffuse reflection depends on angle between light direction and surface normal: reflected intensity = light intensity * cosine of angle between light direction and surface normal normal light scalar product lets us calculate cosq q

21. Vector or Cross Product • The vector or cross product is defined as: UxV = (u2v3 - u3v2, u3v1 - u1v3, u1v2 - u2v1) • We can also show that: UxV = N |U||V| sin  where N is unit vector orthogonal to U and V (forming a right handed system) and q is angle between U and V • This allows us to find the normal to a plane • cross-product of two directions lying in plane , eg (P3-P2), (P2-P1), where P1, P2, P3 are three points in the plane

22. Convince yourself that the x-axis is represented by the vector (1,0,0) What is the unit normal in the direction (2,3,4)? What is the angle between the vectors (1,1,0) and (1,0,0)? Which vector is orthogonal to the vectors (1,0,0) and (0,1,0)? What is the normal to the plane through the points (1,2,3), (3,4,5) and (0,0,0)? Exercises

23. 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

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

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

26. 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 . ...

27. Typical PrimitivesOrder of Vertices • Graphics systems such as OpenGL typically support: • triangles, triangle strips and fans • quads, quad strips • polygons • 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

28. Complex Primitives • OpenGL has utility libraries (GLU and GLUT) which contain various high-level primitives • Sphere, cone, torus • Polygonal representation constructed automatically • Similarly for VRML • For conventional graphics hardware: • POLYGONS RULE!

29. 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 Cyberware Cyberscanner in Med Physics at LGI able to scan human faces Automatic Generation of Polygonal Objects

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

31. 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

32. 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