The Graphics Pipeline. Principle of 3D Pipeline. The Graphics Pipeline Steps. What happens to graphics data? Modeling? Rendering?

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1 The Graphics Pipeline What happens to graphics data? Principle of 3D Pipeline Modeling? Rendering? Displaying User? Internal 3D? Internal 2D? Images representation representation The Graphics Pipeline Steps input devices modeler Modeling transformations projection clipping visibility Rendering shading rasterization Output device drivers output devices 1

2 Display System Overview AGP CPU Memory Devices BUS Graphics processor Graphics memory used as frame buffer, display list storage, texture memory,... Graphics memory Video controller screen Modeling Construction of graphics objects with modeler Graphics input devices (interactive) Graphics program (passive) Result: 3D data (file or data structure) Rendering Generation of image data from model data Transformations: geometric object manipulations Projection: object space? image space Clipping: cutting off outside windows Visibility: elimination of invisible parts Shading: calculation of shade and color Rasterization: image space? pixels Result: image data (character sequence/file) 2

3 Output Display of image data with device drivers Device independent data? device dependent data (often only Window-Viewport transform) Output to file Result: visible images (or image file) 2D and 3D Graphics Models How to define a model? Data Flow in a Graphics System Graphics Pipeline User User Interface Modeling Program Description Interface Rendering Program Image Interface Devices object data image data 3

4 2D and 3D models objects 2D objects 3D objects elements 1D elements Line model Wire frame model 2D elements Area model Boundary model 3D elements Volume model 2 1 / 2 D = pseudo 3D : 2D with simple 3 rd dimension Line Model 2D objects are made up of 1D lines examples: circuit diagrams, flow charts,... Typical basic elements: points, lines, circles, arcs, letters, splines,... Typical operations: position, connect, together, translate, rotate,... Area Model 2D objects are made up of 2D surfaces example: paint programs Typical basic elements: points, lines, disc, closed polygonal area, pixel matrixes, textures,... Typical operations: position, logical combinations (AND, OR, XOR,...), translation, rotation,... 4

5 example: Wire Frame Model 3D objects are made up of 1D lines Typical basic elements: points, lines, circles, arcs, letters, splines,... Properties: hidden surface removal impossible, surfaces undefined, not good for CAD? Boundary Model 3D objects are made up of 2D surfaces example: Typical basic elements: polygons, discs, math. defined surfaces, free form surfaces... Properties: hidden surface removal possible, large memory consumption, consistency tests necessary, o.k. for some CAD applications 5

6 Volume Model (solid representation) example: 3D objects are made up of 3D solids Typical basic elements: box, sphere, cylinder, octahedron, pyramid, cone, torus, sweeps,... Properties: hidden surface removal simple, always consistent, small memory consumption, great for mechanical design 2 Kinds of Volume Model 1. Construction: complex objects are assembled with set operations Example:? = 2. Enumeration: space is subdivided into voxels, content definition for every voxel Example: = Shading of Volume Data 6

7 Basic Modeling How are graphics primitives defined? Polygons Polygon: defined by a sequence of vertices Every polygon can be decomposed into triangles Often used as basic primitive Vertices usually in counter-clockwise (ccw) order Oriented Polygons Definition of front-facing polygon: Vertices in counterclockwise (CCW) order. Normal-vector given by cross-product of first two edges (different in LH vs. RH coord sys) v3 N v2 v0 v1 7

8 Quadrics (implicit form) Sphere: x 2 + y 2 + z 2 = 1 Cylinder: x 2 + y 2 = 1 0? z? height x 2 + y 2? 1 z=0, z=height (top/bottom) Cone: x 2 + y 2 = z 2 0? z? height x 2 + y 2? 1 z=height (top) Torus: (x 2 + y 2 + z r 2 ) 2 4(x 2 +y 2 ) = 0 (Quartic) r Note: all solids in normalized form Quadrics (explicit form) Quadrics can also be described in parametric form e.g. 2D circle? x = r cos??????????????????? y = r sin? e.g. 3D sphere?? x = r?cos??cos? y = r?cos??sin? z = r?sin????????? azimuth -?/2?????? elevation Superquadrics The trigonometric functions of the quadrics are changed to non-integer exponents e.g. circle? super-circle x = cos?? y = sin?? (implicit form x n + y n = 1 n=2/? ) outside in: n=50,6,2,1,0.5 Example: sphere becomes x n + y n + z n = 1 Or with 2 degrees of freedom:? 1 and? 2 x = r?cos????cos??? y = r?cos????sin? 0??? 2??? z = r?sin??? -?/2?????? 8

9 Sweeps Sweeps are objects produced by moving a generating simple object (shape) through space The resulting object is the trace of this procedure Translational sweep (= Extrusion): 2D shape 3D object Rotational sweep: Sweeps (2) Definition of an object by a contour (e.g. curve, polygon) and a rotational axis 2D shape 3D object Conical sweep: Sweeps (3) Definition of an object by a contour (e.g. curve, polygon) and a vertex which is connected to the contour by straight lines 2D shape 3D object 9

10 Height Fields z y x z = f (x, y) For every point in the x-y-plane (discrete) a height value z is defined. These height values for the points (x,y) are given in a matrix. Soft Objects key points soft object influencing radius R C(r) threshold Boundary of soft object is the iso-surface, at which the sum of all weighting functions C i (r) is above the threshold. R??C i (r) - threshold = 0 r etc. Fractals Example Fractal mountain Simple rule: Every edge recursively replaced by two random edges. Random number should depend on: length of the edge the fractal dimension, i.e. how rough the mountain will be 10

11 Freeform primitives Idea: influence curve/surface shape by moving control points Control points specify the general shape of a curve red: interpolating, green: approximating curve 11