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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
3-D Viewing

Assist. Prof. Dr. AhmetSayar

Computer Engineering Department

Computer Graphics Course

Kocaeli University

Fall 2013

geometric projection systems
Geometric Projection Systems

geometric projections














3d viewing
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
parallel projections
Parallel Projections
  • Projectors are parallel
  • Projectors meet at infinity

Projection plane

Center of projection

parallel projections orthographic projections
Parallel Projections-Orthographic Projections-
  • Actually more restricted parallel projection
  • Projection plane is perpendicular to one of the coordinate axis

Top view

parallel projections orthographic projections1
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
parallel projections axonometric projections 1
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
parallel projections axonometric projections 2
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
parallel projections axonometric projections 3
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
parallel projections axonometric projections 4
Parallel Projections-Axonometric Projections- 4
  • ISOMETRIC Projections
    • Lets make an example – Apply T transformations calculated before on unit matrix
parallel projections axonometric projections 5
Parallel Projections-Axonometric Projections- 5
  • Lets compute foreshortenings
  • Remember in isometric projection tx = ty = tz
    • Solving equations and find α Ɵ and t
parallel projections oblique projections 1
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
parallel projections oblique projections 2
Parallel Projections-Oblique Projections- 2
  • Depending on the values of α, we get particular types of oblique projections
parallel projections oblique projections 3
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
perspective projections
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
perspective projections1
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
perspective projections example
Perspective ProjectionsExample

Projected point after having transformation

perspective projection matrix form
Perspective ProjectionMatrix Form
  • Pz : projection on z=0
  • Pr : Perspective projection along z axis
  • We find projected point after having projections
perspective projections in shape
Perspective Projections in shape
  • When r = -1/zc this becomes same as obtained in matrix form – see earlier slide.
  • Show it?
perspective projections finding cop on z axis
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)
  • COP on x axis and y axis can be found in similar way.
  • How do you modify T (tranformation) matrix?
perspective projections types
Perspective Projections Types
  • Till now: We have done only 1-point perspective
  • Hint: How many group of lines are converging?
2 point perspective
2-point Perspective
  • Along X and y axis
  • There will be 2 center of projections and correspondingly 2 vanishing points