Classical Viewing
Ed Angel
Professor of Computer Science, Electrical and Computer
Engineering, and Media Arts
University of New Mexico
Objectives
• Introduce the classical views
• Compare and contrast image formation by computer with how images have been formed by architects, artists, and
engineers
• Learn the benefits and drawbacks of
each type of view
Classical Viewing
• Viewing requires three basic elements - One or more objects
- A viewer with a projection surface
- Projectors that go from the object(s) to the projection surface
• Classical views are based on the relationship among these elements
- The viewer picks up the object and orients it how she would like to see it
• Each object is assumed to constructed from flat principal faces
Planar Geometric Projections
• Standard projections project onto a plane
• Projectors are lines that either
- converge at a center of projection - are parallel
• Such projections preserve lines
- but not necessarily angles
• Nonplanar projections are needed for
applications such as map construction
Classical Projections
Perspective vs Parallel
• Computer graphics treats all projections the same and implements them with a single pipeline
• Classical viewing developed different techniques for drawing each type of projection
• Fundamental distinction is between parallel and perspective viewing even
though mathematically parallel viewing is
the limit of perspective viewing
Taxonomy of Planar Geometric Projections
parallel perspective
axonometric multiview
orthographic oblique
isometric dimetric trimetric
2 point
1 point 3 point
planar geometric projections
Perspective Projection
Parallel Projection
Orthographic Projection
Projectors are orthogonal to projection surface
Multiview Orthographic Projection
• Projection plane parallel to principal face
• Usually form front, top, side views
isometric (not multiview
orthographic view) front
top side in CAD and architecture, we often display three
multiviews plus isometric
Advantages and Disadvantages
• Preserves both distances and angles
- Shapes preserved
- Can be used for measurements
• Building plans
• Manuals
• Cannot see what object really looks like because many surfaces hidden from view
- Often we add the isometric
Axonometric Projections
Allow projection plane to move relative to object
classify by how many angles of a corner of a projected cube are the same
none: trimetric two: dimetric three: isometric
θ 1 θ 3 θ 2
Types of Axonometric
Projections
Advantages and Disadvantages
• Lines are scaled (foreshortened) but can find scaling factors
• Lines preserved but angles are not
- Projection of a circle in a plane not parallel to the projection plane is an ellipse
• Can see three principal faces of a box-like object
• Some optical illusions possible
- Parallel lines appear to diverge
• Does not look real because far objects are scaled the same as near objects
• Used in CAD applications
Oblique Projection
Arbitrary relationship between projectors and projection plane
Advantages and Disadvantages
• Can pick the angles to emphasize a particular face
- Architecture: plan oblique, elevation oblique
• Angles in faces parallel to projection plane are preserved while we can still see “around” side
• In physical world, cannot create with simple
camera; possible with bellows camera or special lens (architectural)
Perspective Projection
Projectors coverge at center of projection
Vanishing Points
• Parallel lines (not parallel to the projection plan) on the object converge at a single point in the projection (the vanishing point)
• Drawing simple perspectives by hand uses these vanishing point(s)
vanishing point
Three-Point Perspective
• No principal face parallel to projection plane
• Three vanishing points for cube
Two-Point Perspective
• On principal direction parallel to projection plane
• Two vanishing points for cube
One-Point Perspective
• One principal face parallel to projection plane
• One vanishing point for cube
Advantages and Disadvantages
• Objects further from viewer are projected
smaller than the same sized objects closer to the viewer (diminution)
- Looks realistic
• Equal distances along a line are not projected into equal distances (nonuniform foreshortening)
• Angles preserved only in planes parallel to the projection plane
• More difficult to construct by hand than parallel projections (but not more difficult by computer)
