The 3-D Graphics Rendering Pipeline

Transcription

1 The 3-D Graphics Rendering Pipeline Modeling Trival Rejection Illumination Viewing Clipping Projection Almost ever discussion of 3-D graphics begins here Seldom are an two versions drawn the same wa Seldom are an two versions implemented the same wa Primitives are processed in a series of steps Each step forwards its result on to the net step Rasterization & Piel Operation Displa

2 Modeling transformations Modeling Trival Rejection We start with 3-D models defined in their own model space Illumination Viewing Modeling transformations orient models within a common coordinate sstem called world space Clipping Projection Rasterization & Piel Operation Displa All objects, light sources, and the viewer live in world space Trivial rejection attempts to eliminate objects that cannot possibl be seen (an optimization) 2

3 Illumination Modeling Trival Rejection Net we illuminate potentiall visible objects Illumination Viewing Object colors are determined b their material properties, and the light sources in the scene Clipping Projection Illumination algorithm depends on the shading model and the surface model Rasterization & Piel Operation Displa 3

4 Viewing Modeling Another change of coordinate sstems Trival Rejection Illumination Viewing Clipping Maps points from world space into ee space Viewing position is transformed to the origin Projection Rasterization & Piel Operation Displa Viewing direction is oriented along some ais A viewing volume is defined 4

5 Clipping and Projection Modeling Trival Rejection Net we perform clipping of the scene's objects against a three dimensional viewing volume called a viewing frustum Illumination Viewing Clipping This step totall eliminates an objects (and pieces of objects) that are not visible in the image A clever trick is used to straighten out the viewing frustum in to a cube Projection Rasterization & Piel Operation Displa Net the objects are projected into twodimensions from ee space to screen space 5

6 Rasterization & Piel Operation Modeling Trival Rejection One last transformation from our screenspace coordinates into a viewport coordinates Illumination Viewing Clipping The rasterization step scan converts the object into piels Involve interpolating parameters as we go Projection Rasterization & Piel Operation Displa Purel 2D operation A lot going on here 6

7 . 2D 2. 3D 3. Viewing Projection 7

8 2D Translation ' ' = + t = + t 8

9 2D Rotation ' ' = = cos cos - sin sin Matri and Vector format: ' ' M cos cos sin sin 9

10 Back to Translation = + t ' = + t ' Matri format? M???????? ' ' t t M ' ' m m m m m m m m m M w w w Homogenous coordinates!

11 2D Translation Properties.There eists an inverse mapping for each function 2.There eists an identit mapping t t M ) ( I Identit t t M t t

12 2D Translation Properties These properties might seem trivial at first glance, but the are actuall ver important, because when these conditions are shown for an class of functions it can be proven that such a class is closed under composition (i.e. an series of translations can be composed to a single translation). In mathematical parlance this is the same as saing that translations form an algebraic group. ' 2 TT T n T ' 2

13 3 Back to Rotation M cos sin sin cos ' ' cos sin sin cos M R cos sin sin cos R M Identit M R

14 Order Order matters! translation ---> rotation rotation ---> translation 4

15 Other 2D s X-shear Y-shear scaling And more. 5

16 2D Rotation b Shears 6

17 3D Rotation b Shears 7

18 Affine transformation Propert: preserve parallel lines Remember affine function on vector is equal to linear plus translation m m m2 M m m m2 The coordinates of three corresponding points uniquel determine an Affine Transform!! ( 3, 3) ' ( ' 3, ' 3 ) (, ) ( 2, 2) 8 ( ' 2, ' 2 ) ( ', ' ) '

19 From 2D to 3D: Preliminar Right-handed (out of page) Z Y X vs. left-handed Y (into page) Z X Z-ais determined from X and Y b cross product: Z=X Y Z X Y X 2 Y 3 X 3 Y 2 X 3 Y X Y 3 X Y 2 X 2 Y Cross product follows right-hand rule in a right-handed coordinate sstem, and left-hand rule in left-handed sstem. 9

20 . 2D 2. 3D 3. Viewing Projection 2

21 3D Translation T = t t t 2 2

22 3D Scaling S = s s s 2 22

23 3D Rotation R = cos sin sin cos cos sin cos sin R = sin cos R z = sin cos 23