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3-D Viewing

3-D Viewing. Assist. Prof. Dr. Ahmet Sayar Computer Engineering Department Computer Graphics Course Kocaeli University Fall 2013. Geometric Projection Systems. geometric projections. parallel. perspective. orthographic. axonometric. oblique. trimetric. cavalier. cabinet.

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3-D Viewing

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  1. 3-D Viewing Assist. Prof. Dr. AhmetSayar Computer Engineering Department Computer Graphics Course Kocaeli University Fall 2013

  2. Geometric Projection Systems geometric projections parallel perspective orthographic axonometric oblique trimetric cavalier cabinet dimetric isometric single-point two-point three-point

  3. 3D Viewing • Projections • Projection plane – view plane • Center of projection • Projectors are the straight lines from eyes to object • Type of projection here is perspective projection • Projectors are not in parallel

  4. Parallel Projections • Projectors are parallel • Projectors meet at infinity Projection plane Center of projection

  5. Parallel Projections-Orthographic Projections- • Actually more restricted parallel projection • Projection plane is perpendicular to one of the coordinate axis Top view

  6. Parallel Projections-Orthographic Projections- • Multiviews • x=0, y=0, z=0 planes • One view is not adequate • True size and shapes for lines • On z=0 plane

  7. Parallel Projections-Axonometric Projections- 1 • Orthogonal projection that displays more than one face of an object • Example below: Additional translation, rotation (or both) and then projection on z=0 plane • Distortions are tx, ty and yz • Distortions=foreshortening = f = bozulma

  8. Parallel Projections-Axonometric Projections- 2 • Three types • Trimetric: No foreshortening is the same • Dimetric: Two foreshortening is the same • Isometric: All foreshortening is the same

  9. Parallel Projections-Axonometric Projections- 3 • ISOMETRIC Projections (Example) • Let there be two rotations • About y-axis α • About x-axis Ɵ AND PROJECT ON Z=0 PLANE

  10. Parallel Projections-Axonometric Projections- 4 • ISOMETRIC Projections • Lets make an example – Apply T transformations calculated before on unit matrix

  11. Parallel Projections-Axonometric Projections- 5 • Lets compute foreshortenings • Remember in isometric projection tx = ty = tz • Solving equations and find α Ɵ and t

  12. Parallel Projections-Oblique Projections- 1 • In axonometric projections • Projectors are parallel and vertical to the projection plane • Lets relax this condition a little (Oblique Projection) • Projectors are parallel but not perpendicular to the projection plane • The  front  or  principal  surface  of an  object  (the surface toward the plane of projection) is parallel to the plane of projection. • It carries 3D aspects of objects

  13. Oblique Projections

  14. Parallel Projections-Oblique Projections- 2 • Depending on the values of α, we get particular types of oblique projections

  15. Parallel Projections-Oblique Projections- 3 • When α = 45 (Cavalier) • Lines perpendicular to the projection planes are not foreshortened • Cot α = ? • When cot α = 1/2 (Cabinet) • Lines perpendicular to the projection planes are foreshortened by half • Ɵ is typically 30 or 45

  16. Perspective Projections • Parallel lines converge • Non-uniform foreshortening • Helps in depth perception, important for 3D viewing • Shape is not preserved. There is depth concept. • Parallel lines seem to converge

  17. Perspective Projections • Center of projection is at infinity • Direction of projection (DOP) same for all points • What happens to parallel lines they are not parallel to the projection plane? • Each set of parallel lines intersect at a vanishing point on the PP

  18. Perspective ProjectionsExample Projected point after having transformation

  19. Perspective ProjectionMatrix Form • Pz : projection on z=0 • Pr : Perspective projection along z axis • We find projected point after having projections

  20. Perspective Projections in shape • When r = -1/zc this becomes same as obtained in matrix form – see earlier slide. • Show it?

  21. Perspective ProjectionsFinding COP on z-axis • Point at infinity on +Z • Recall r = -1/zc : the vanishing point is at zc • Point [0 0 1 0] homogeneous (point at infinity)

  22. Question • COP on x axis and y axis can be found in similar way. • How do you modify T (tranformation) matrix?

  23. Perspective Projections Types • Till now: We have done only 1-point perspective • Hint: How many group of lines are converging?

  24. 2-point Perspective • Along X and y axis • There will be 2 center of projections and correspondingly 2 vanishing points

  25. 3-point Perspective

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