**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)