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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

parallel

perspective

orthographic

axonometric

oblique

trimetric

cavalier

cabinet

dimetric

isometric

single-point

two-point

three-point

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)
question
Question
  • 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