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3D Projection Transformations. Soon Tee Teoh CS 147. 3D Projections. Rays converge on eye position. Rays parallel to view plane. Perspective. Parallel. Orthographic. Oblique. Cabinet. Cavalier. Elevations. Axonometric. Isometric. Perspective and Parallel Projections. View plane.

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3d projections
3D Projections

Rays converge on eye position

Rays parallel to view plane

Perspective

Parallel

Orthographic

Oblique

Cabinet

Cavalier

Elevations

Axonometric

Isometric

perspective and parallel projections
Perspective and Parallel Projections

View plane

Perspective

Parallel

3d projections1
3D Projections

Rays converge on eye position

Rays parallel

Perspective

Parallel

Rays at angle to view plane

Rays perpendicular to view plane

Orthographic

Oblique

Cabinet

Cavalier

Elevations

Axonometric

Isometric

principal axes
Principal Axes
  • Man-made objects often have “cube-like” shape. These objects have 3 principal axes.

From www.loc.gov/ jefftour/cutaway.html

one point two point three point perspective
One point, two point, three point perspective
  • Depends on how many principal axes intersect with view plane.
  • Parallel lines not parallel to view plane have the same vanishing point.

One point perspective: One principal axis intersects view plane

one point two point three point perspective1
One point, two point, three point perspective

Two point perspective: two principal axes intersect view plane

one point two point three point perspective2
One point, two point, three point perspective

Three point perspective: Three principal axes intersect view plane

one point two point three point perspective3
One point, two point, three point perspective

View Plane

Three point

Two point

One point

3d projections2
3D Projections

Rays converge on eye position

Rays parallel

Perspective

Parallel

Rays at angle to view plane

Rays perpendicular to view plane

Orthographic

Oblique

View plane aligned with principal axes

View plane not aligned with principal axes

Cabinet

Cavalier

Elevations

Axonometric

Isometric

Trimetric

Dimetric

front elevation
Front Elevation
  • Parallel Orthogonal Elevation

Front elevation of tallest buildings in the world

From members.iinet.net.au/ ~paulkoh

isometric view
Isometric View
  • In isometric view, the three principal axes of the object intersect the view plane at equal distance. Therefore, when projected, they are 120o apart.

http://sucod.shef.ac.uk/sucod/gallery/arc320/2003/p2/Rachel/images/colour.JPG

3d projections3
3D Projections

Rays converge on eye position

Rays parallel

Perspective

Parallel

Rays at angle to view plane

Rays perpendicular to view plane

Orthographic

Oblique

Cabinet

Cavalier

Elevations

Axonometric

Isometric

oblique projections
Oblique projections
  • Projection lines are at an angle to the view plane.
  • Let the angle be a be the angle the projection line makes with the view plane.
  • tan a = 1 (or, a = 45o) called cavalier projection
  • tan a = 2 (or, a = 63.4o) called cabinet projection

1

2

a

a

1

1

cavalier

cabinet

1

1/2

1

1

orthographic parallel projection matrix
Orthographic Parallel Projection Matrix
  • Transform each vertex from Viewing Coordinates into Normalized Coordinates using orthographic projection
  • Suppose that a point is (x,y,z) in Viewing Coordinates, what’s the transformation necessary to transform it to (x’,y’,z’) in Normalized Coordinates?
  • Given: the dimensions of the view window: xwmin, xwmax, ywmin, ywmax
  • Orthogonal Projection Matrix on p. 362.
    • Basically Translate center of view to origin and then Scale to (-1,1) cube
    • Translate by -(min+max)/2, then scale by 2/(max-min).

xwmax+ xwmin

xwmax - xwmin

2

xwmax - xwmin

0 0 -

2

ywmax - ywmin

ywmax+ ywmin

ywmax - ywmin

0 -

0

M =

-2

znear - zfar

znear + zfar

znear - zfar

0 0

0 0 0 1

perspective projection matrix
Perspective Projection Matrix
  • Suppose camera position is at origin (0,0,0)
  • Suppose view plane is at distance d from origin
  • Consider top view

Original point in viewing coordinates

View plane

x

Point projected to view plane

x

x’

(0,0,0)

z

d

z

The projected coordinate, x’ = dx/z

Similarly, y’ = dy/z

perspective projection matrix1
Perspective Projection Matrix

1 0 0 0

0 1 0 0

0 0 1 0

0 0 1/d 0

Perspective Projection Matrix M =

In homogeneous coordinates,

1 0 0 0 x x

0 1 0 0 y y

0 0 1 0 z z

0 0 1/d 0 1 z/d

=

Point in viewing coordinates

Point projected to view plane

In normal coordinates, (x’,y’,z’) = (dx/z,dy/z,d)