Transforms II. Overview. Homogeneous Coordinates 3-D Transforms Viewing Projections. Homogeneous Coordinates. x y z w

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1 Transforms II Overvie Homogeneous Coordinates 3- Transforms Vieing Projections 2 Homogeneous Coordinates Allos translations to be included into matri transform. Allos us to distinguish beteen a vector and a point. In perspective transformations the etra coordinate can be thought to contain the perspective information or scaling. X Y Z

2 2 3 Homogeneous Coordinates As gets smaller eal space X gets Larger When reaches X is no at infinit Homogeneous coordinates allos us to deal mathematicall ith infinit Homogeneous Coordinates Man different homogeneous space vectors for the same real space point Vectors can not be translated but can be scaled and rotated Point Vector

3 Homogeneous Coordinates Attribute Vector Point epresents Magnitude and Location irection Origin Unique Arbitrar Transformation Liner Scale and otate Affine Move Vector Point 5 3- Transforms otation cosθ sinθ sinθ cosθ otation about X cosφ sinφ sinφ cosφ otation about Y cosλ sinλ sinλ cosλ otation about Z 6 3

4 3- Transforms otation cosφ sinφ cosλ sin λ cosθ sinθ sin λ cosλ sinθ cosθ sinφ cosφ 7 3- Transforms Scale scale scale scale 8

5 Transforms Translation 3- Transforms Composite Transform Matri [ ] [ ][ ][ ] M S T [ ] [ ][ ][ ] oll Pitch Ya

6 Vieing Projections Ho do e see in 3- Parallel Projections Perspective Projections Vieing Projections Parallel Projections Elevations: Projection plane is perpendicular to a principle ais. Front, Top (Plan), Side. 2 6

7 Vieing Projections Parallel Projections Aonometric: Projection plane is not orthogonal to a principle ais. Isometric: irection of projection makes equal angles ith each principal ais.. 3 Vieing Projections Parallel Projections Oblique: irection of projection is not orthogonal to the projection plane. 7

8 Vieing Projections Perspective Projection One-point: One principle ais cut b projection plane. One ais vanishing point. 5 Vieing Projections Perspective Projection To-point: To principle aes cut b projection plane. To vanishing points. 6 8

9 Vieing Projections Perspective Projection Three-point 7 Vieing Projections efinitions VC - Vieing eference Coordinate VP - Vie eference Point VPN - Vie Plane Normal VUP - Vie Up irection OP - irection of Projection PP - Projection eference Point Center of Projection VP - Vieing Plane BCP - Back Clipping Plane FCP - Front Clipping Plane 8 9

10 Parallel Projections. Translate VP to the origin 2. otate VC such that n ais (VPN) becomes, u ais becomes, and v ais becomes. 3. Shear such that the direction of projection becomes parallel to the ais.. Translate and scale into the parallel-projection canonical vie volume. 9 Parallel Projections Step: Simpl the negative of the VP vector vrp vrp vrp 2

11 Parallel Projections Step2: VPN VPN VUP VUP Parallel Projections Step3: u v CW ma + umin 2 ma + vmin 2 prpu prp v PP prpn OP CW PP shear shear shear shear 22

12 Parallel Projections Step: Translation uma + 2 vma + 2 F umin vmin Scale u ma 2 umin 2 vma v min F B 23 Perspective Projections. Translate VP to the origin 2. otate VC such that n ais (VPN) becomes, u ais becomes, and v ais becomes. 3. Translate such that the center of Projection (COP), given b the PP, is at the origin.. Shear such that the center line fo the vie volume becomes the ais. 5. Scale such that the vie volume becomes the canonical perspective vie volume, the truncated right pramid defined b the si planes 2 2

13 Perspective Projections Step: Step2: Simpl the negative of the VP vector VUP VPN VPN VUP vrp vrp vrp 25 Perspective Projections Step3: prpu prp v prpn 26 3

14 Perspective Projections Step: shear shear shear shear 27 Step5: Perspective Projections VP Sstep Tstep 3 2vrp ( )( ) uma umin vrp + B 2vrp ( )( ) vma vmin vrp + B vrp B 28

15 5 29 Perspective Projections Matri Projection Othrographic 2 cot : fov Where ( ) [ ] Z Y X