3D Concepts

UNIT 3

3-D Coordinate Spaces• Remember what we mean by a 3-D coordinate

space

x axis

y axis

z axis

P

y

zx

Right-Hand Reference System

Translations In 3-D• To translate a point in three dimensions by tx,

ty and tz simply calculate the new points as follows:• x’ = x + tx y’ = y + ty z’ = z + tz

(x’, y’, z’)

(x, y, z)

Translated Position

Scaling In 3-D

• To scale a point in three dimensions by sx, sy and sz simply calculate the new points as follows:• x’ = sx*x y’ = sy*y z’ = sz*z

(x, y, z)

Scaled Position

(x’, y’, z’)

Rotations In 3-D

x’ = x·cosθ - y·sinθ

y’ = x·sinθ + y·cosθ

z’ = z

x’ = x

y’ = y·cosθ - z·sinθ

z’ = y·sinθ + z·cosθ

x’ = z·sinθ + x·cosθ

y’ = y

z’ = z·cosθ - x·sinθ

• The equations for the three kinds of rotations in 3-D are as follows:

Homogeneous Coordinates In 3-D

• Similar to the 2-D situation we can use homogeneous coordinates for 3-D transformations - 4 coordinate column vector

• All transformations can then be represented as matrices

1

z

y

x

x axis

y axis

z axis

P

y

zxP(x, y, z) =

3D Transformation Matrices

1

0100

0010

0001

tztytx

1000

000

000

000

z

y

x

s

s

s

1000

0cos0sin

0010

0sin0cos

Translation bytx, ty, tz

Scaling by sx, sy, sz

1000

0cossin0

0sincos0

0001

Rotate About X-Axis

1000

0100

00cossin

00sincos

Rotate About Y-Axis Rotate About Z-Axis

Projections• Our 3-D scenes are all specified in 3-D world

coordinates• To display these we need to generate a 2-D

image - project objects onto a picture plane

• So how do we figure out these projections?

Picture Plane

Objects in World Space

Converting From 3D To 2D

• Projection is just one part of the process of converting from 3D world coordinates to a 2D image

Clip against view volume

Project onto projection

plane

Transform to 2-D device

coordinates

3-D world coordinate

output primitives

2-D device coordinates

3D Viewing• In 2D viewing we have 2D window & 2D viewport &

objects in the world coordinates.• The 3D viewing has an added dimension which makes it

complex as even though objects are 3D the display devices are only 2D.

• The mismatch between 3D objects & 2D displays is compensated by introducing projections. The projection transforms 3D objects into a 2D projection plane.

• View plane: It is nothing but the film plane in a camera which is positioned & oriented for a particular shot of the scene.

• World coordinates positions in the scene are transformed to viewing coordinates, then viewing coordinates are projected onto the view plane.

• View reference point: This point is the center of our viewing coordinate system.

• The production of a 2D image of higher dimensional object refers to graphical projection.

• A projection can be defined as a mapping of any point P[x,y,z] to its image P`[x`,y`,z`] onto the view plane, called as projection plane.

• Parallel & perspective projections are the two broad categories of projections.

Types of Projections• There are two broad classes of projection:– Parallel: Typically used for architectural and

engineering drawings.– Perspective: Realistic looking and used in

computer graphics.

Perspective ProjectionParallel Projection

Parallel Projections• Some examples of parallel projections

Orthographic Projection

Isometric Projection

Perspective Projections• Perspective projections are much more realistic

than parallel projections

Geometric projection

PerspectiveParallel

Orthographic

Three-point

Two-pointOblique

Top

One-point

Other

Other

Cavalier

Isometric

Cabinet

Front

Axonometric

Side elevatio

n

• Parallel projection:– If the direction of projection is perpendicular to the

projection plane, it is an orthographic projection.– If the direction of projection is not perpendicular to

the projection plane is called as oblique projection.– A multi-view projection displays a single face of a 3D

object.– Axonometric projections allow the user to place the

view-plane normal in any direction such that 3 adjacent faces of a cube like object are visible.

– Dimetric projections differ from isometric projections in the direction of the view-plane normal.

– Trimetric projections allow the viewer the most freedom in selecting the components of n.

• Perspective projection:– It is a type of projection where 3D objects are not

projected along parallel lines, but along lines emerging from a single point.

– A vanishing point is a point in a perspective drawing to which parallel lines appear to converge.

– One-point perspective exists when a painting plate is parallel to two axes of a rectilinear scene.

– Two point perspective

Assignment

OrthographicWireframe

Elevation

OrthographicWireframe

Plan

OrthographicWireframeEnd-Elevation

PerspectiveView