3D Coordinates & Transformations
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1 3D Coordinates & Transformations Prof. Aaron Lanterman (Based on slides b Prof. Hsien-Hsin Sean Lee) School of Electrical and Computer Engineering Georgia Institute of Technolog
2 3D graphics rendering pipeline () Geometr Processing Rasteriation Processing Geometr Pipeline Processing Vertices Mainl floating-point operations Rasteriation Pipeline Processing Piels Mainl dealing with Integer operations 2
3 3D graphics rendering pipeline (2) Geometr Processing MMX ISA Rasteriation Processing Geometr Pipeline Processing Vertices Mainl floating-point operations Rasteriation Pipeline Processing Piels Mainl dealing with Integer operations MMX was originall designed to accelerate this tpe of functionalit 3
4 3D graphics rendering pipeline (3) SSE/SSE2 ISA Geometr Processing MMX ISA Rasteriation Processing Geometr Pipeline Processing Vertices Mainl floating-point operations SSE/SSE2 were designed for this part Rasteriation Pipeline Processing Piels Mainl dealing with Integer operations MMX was originall designed to accelerate this tpe of functionalit 4
5 Fied-function 3D graphics pipeline Geometr Processing Performed b GPU Geometr Pipeline Processing Vertices Mainl floating-point operations SSE/SSE2 were designed for this part Rasteriation Pipeline Processing Piels Mainl dealing with Integer operations MMX was originall designed to accelerate this tpe of functionalit Rasteriation Processing 5
6 3D Coord: Math tetbooks use -up Z-up, Right-Handed Sstem 6
7 3D Coord: Real games tend to use -up Left-Handed Sstem Direct3D Unit3D Right-Handed Sstem OpenGL XNA 7
8 X-Y natural for screen coordinates (awa from the viewer) (toward the viewer) Left-Handed Sstem Direct3D Unit3D Right-Handed Sstem OpenGL XNA 8
9 Some use Z-up for world coordinates Left-Handed Sstem Right-Handed Sstem Z-up, LHS: Unreal Z-up, RHS: Quake/Radiant, Source/Hammer, C4 Engine Nearl everthing still uses Y-up for screen coordinates! 9
10 Another view Left-Handed Sstem Unreal Right-Handed Sstem Quake/Radiant Source/Hammer C4 Engine
11 3D object modeling software Right-Handed Sstem 3D Studio Ma, Blender + Right-Handed Sstem Maa, Milkshape +
12 Geometr format verte coordinates +Y (X2, Y2, Z2) +Z (X3, Y3, Z3) (X, Y, Z) Y Z X +X 2
13 Geometr format verte normals +Y (NX2, NY2, NZ2) +Z (NX3, NY3, NZ3) (NX, NY, NZ) +X 3
14 Geometr format verte colors +Y (R2, G2, B2, A2) +Z (R3, B3, B3, G3) (R, G, B, A) +X 4
15 Triangle-based geometr representation V5 V V2 V4 V5 V4 V5 V4 V7 V6 V3 V9 V2 V3 V3 V V8 Careful! V V2 Triangle List (note the verte order) Triangle Strip Triangle Fan 5
16 Specifing a 3D object () V V7 V5 V6 Triangle list {v, v3, v2}, {v, v5, v3}, {v5, v6, v3}, {v4, v3, v6}, {v, v7, v6}, {v, v6, v5} V4 Triangle strip {v5, v3, v, v2}, {v5, v6, v3, v4}, {v7, v6, v, v5} V2 V3 Verte ordering is critical when culling mode enabled We will discuss normal computation later 6
17 Specifing a 3D object (2) V V7 V6 Triangle list {v, v2, v7}, {v2, v8, v7}, {v2, v3, v4}, {v2, v4, v8}, {v4, v7, v8}, {v4, v6, v7} V8 V4 Triangle strip {v, v2, v7, v8}, {v3, v4, v2, v8}, {v6, v7, v4, v8} V2 V3 Verte ordering is critical when culling mode enabled We will discuss normal computation later 7
18 3D rendering pipeline World Transform View Transform Lighting Projection Transform Backface Culling Clipping Perspective Divide Viewport Transform Rasteriation 8
19 Transformation pipeline World Transformation Model coordinates à World coordinates View Transformation World coordinates à Camera space Projection Transformation Camera space à View plane These are a series of matri multiplications 9
20 World transformation World Coordinates + Local model coordinates + Translation Rotation Scaling World origin Local model coordinates + 2
21 View transformation World Coordinates Camera position Look vector + World origin + 2
22 Projection transformation Set up camera internals Set up Field of View (FOV) View frustum View planes Will discuss in the net lecture 22
23 Homogeneous coordinates Enable all transformations to be done b multiplication Primaril for translation (see net few slides) Add one coordinate (w) to a 3D vector Each verte has [,,, w] w will be useful for perspective projection w should be in a Cartesian coordinate sstem 23
24 Transformation : translation (Offset) ( t, t, t ) + (,, )
25 Translation matri " $ [t,t,t,] = [,,,] $ $ $ # T T T % ' ' ' ' & Eample of a row-coordinate convention Direct3D, XNA, HLSL/Cg use row coordinates OpenGL & non-graphics world uses column coordinates 25
26 Transformation 2: scaling
27 Scaling matri " $ [s,s,s,] = [,,,] $ $ $ # S S S % ' ' ' ' & 27
28 Transformation 3: rotation
29 2D rotation + + (, ) + θ + + (, ) θ ϕ (, ) cosθ sin θ [ ', ',] = [,,] sin θ cosθ + Rotate along which ais? 29
30 3D rotation matri (LHS) Rotation along Z ais Rotation along Y ais Rotation along X ais cosθ sin θ [', ',',] = [,,,] cosθ [', ',',] = [,,,] sinθ [', ',',] = [,,,] cosθ sin θ sin θ cosθ sinθ cosθ sin θ cosθ 3
31 Non-commutative propert () Counter-clockwise 9 o along 2. Clockwise 9 o along. Clockwise 9 o along 2. Counter-clockwise 9 o along 3
32 32 Non-commutative propert (2) = ) 2 cos( ) 2 sin( ) 2 sin( ) 2 cos( ) 2 cos( ) 2 sin( ) 2 sin( ) 2 cos( π π π π π π π π = ) 2 cos( ) 2 sin( ) 2 sin( ) 2 cos( ) 2 cos( ) 2 sin( ) 2 sin( ) 2 cos( π π π π π π π π (,, ) = (-, -, ) (,, ) = (-, -, )
33 Non-commutative propert (3) Translation b (,, ) 2. Scale b 2 times. Scale b 2 times 2. Translation b (,, ) 33
34 34 Non-commutative propert (4) = ST ST ST S S S S S S T T T (,, ) = (*S+S*T, *S+S*T, *S+S*T) = T T T S S S T T T S S S (,, ) = (*S+T, *S+T, *S+T) Offsets were scaled as well
35 Non-commutative propert (5) Ordering matters! Be careful when performing matri multiplication 35
36 View transformation revisited World Coordinates Camera position Look vector + World origin + 36
37 Specifing the view transformation Most commonl parameteried b: Position of camera Position of point to look at Vector indicating up direction of camera In Direct3D: D3DXMatriLookAtLH! D3D uses a LHS, but also have D3DXMatriLookAtRH In XNA: Matri.CreateLookAt (RHS) In OpenGL: glulookat (RHS) Can also build a rotation+translation matri as if the camera was an object in scene, then take the inverse of that matri msdn.microsoft.com/en-us/librar/bb25342(vs.85).asp msdn.microsoft.com/en-us/librar/bb25343(vs.85).asp 37
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