3D PROJECTION

OUTLINE

• Classification of different types of projections

• Parallel projection

• Perspective projection

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PROJECTION

• Geometric projection:

• Projectors are straight

lines.

• Planar projection:

• Projection surface is a

plane.

Projection

Geometric

Planar

Non-planar

Non-

geometric

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PLANAR GEOMETRIC PROJECTION

• Parallel:

• Different projectors

parallel to each other.

• Preserve parallelism.

• Perspective:

• Different projectors

converge to the center of

projection.

• Does not preserve

parallelism.

Planar Geometric

Projection

Parallel

Orthographic

Oblique

Perspective

One-point

Two-point

Three-point

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PARALLEL PROJECTION

• Orthographic:

• Projectors are

perpendicular to the

projection plane.

• Oblique:

• Projectors are NOT

perpendicular to the

projection plane.

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Projectionsurface

Projectors

ORTHOGRAPHIC PROJECTION

• Elevation:

• Projectors are parallel to

a principle axis.

• Axonometric:

• Projectors are NOT

parallel to any principle

axis.

Orthographic

Elevation

Top

Front

Side

Axonometric

Isometric

Dimetric

Trimetric

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ELEVATIONS

• Used for:

• Engineering drawings.

• Architecture drawings.

• Advantage:

• Preserve distance and angle.

• All views are at same scale.

• Disadvantage:

• Hard to understand the 3D shape even all three elevations are given.

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X

Y

O

Z

Top

Front

MATRIX REPRESENTATIONS

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AXONOMETRIC PROJECTION

• Isometric:

• Angles between the

projections of all three

principal axes are equal

(120º).

• Dimetric:

• Angles between the

projections of two of the

principal axes are equal.

• Trimetric:

• All three angles are different.

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Isometric

Dimetric

ISOMETRIC PROJECTION

• Used for:

• Patent office records;

• Furniture design;

• Video games (SimCity).

• Advantage:

• Illustrates 3D nature of

object.

• Preserve distance along

principal axes.

• Disadvantage:

• Do not preserve angle.

• Lack of foreshortening.

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OBLIQUE PROJECTION

Oblique

Cavalier

Cabinet

Other

shear

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Projection Plane

Projectors

α

φ

Projectors

CAVALIER & CABINET PROJECTIONS

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α=45°

Cavalier

Cabinet

α=tan-12≈63°

1

½

1

1

1

½

MATRIX REPRESENTATION

• Cavalier:

• u = x + z*cos(φ)

• v = y + z*sin(φ)

• Cabinet:

• u = x + z*cos(φ)/2

• v = y + z*sin(φ)/2

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PERSPECTIVE PROJECTION

• Foreshortening effect:

• Objects further from the

center of projection

appear smaller.

• Vanishing point:

• Projections of parallel

lines that are not parallel

to the projection plane

converge to a point.

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Projectionsurface

Projectors

Center ofProjection

ONE-POINT & TWO-POINT PROJECTIONS

• One-point projection:

• The projection plane is

parallel to two of the

principle axes.

• Two-point projection:

• The projection plane is

parallel to only one of

the principle axes.

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RENAISSANCE ART

• Brunelleschi invented

method of determining

perspective projections in

early 1400’s.

• Dutch artist, Johannes

Vermeer, may used camera

obscura to create

perspective effect.

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Lady Standing at the Virginals

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Albrecht Dürer, Daraughsman Drawing a Recumbent Woman (1525) Woodcut illusion from 'The Teaching of Measurements

MATRIX REPRESENTATION I

• Configuration:

• The center of projection

is placed at (0,0)

• The function for

projection plane is z=f:

• For 3D point (x,y,z,1):

• w = z / f;

• u = x / w = x * f / z

• v = y / w = y * f / z

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Projectionplane

Z

X

u(x,z)

Center ofProjection

MATRIX REPRESENTATION II

• Configuration:

• The center of projection is

placed at (0,-f);

• The function for projection

plane is z=0:

• For 3D point (x,y,z,1):

• w = z / f + 1;

• u = x / w

• = x * f / (z + f)

• v = y / w

• = y * f / (z + f)

Friday, September 23, 2016 19f

Projectionplane

Z

X

u(x,z)

Center ofProjection