Classical Viewing

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Classical Viewing

Ed Angel

Professor of Computer Science, Electrical and Computer

Engineering, and Media Arts

University of New Mexico



• Introduce the classical views

• Compare and contrast image formation by computer with how images have been formed by architects, artists, and


• 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


• 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



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)




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