# Computer Graphics

## H&B Chapter 7

### Projections

Coordinate positions transformed to viewplane with parallel lines

• Perspective Projection

Coordinate positions transformed to viewplane with converging lines

#### Parallel Projections

• Orthographic Projection => used for front, side, back, top, bottom views

### Transformation equations for parallel projection

Orthographic Projection:

xp = x

yp = y

Oblique Projection

### Perspective Projections – general case

…where ( xprp, yprp, zprp ) is the perspective reference point

Begin with parametric equation for line…

xp = x - (x – xprp)u

yp = y - (y – yprp)u

zp = z - (z – zprp)u

u = 0 -> 1

solve for u where zvp = zp (viewplane)

substitute in xp, yp equations

multiplying through and rearranging…

Special Case -- zvp = zp = 0

Example: Special Case where zvp = zp = 0

 Perspective Projection Xprp Yprp Zprp 0.5 0.5 -1.5 Point x y z x' y' z' 1 0.30 0.30 0.30 0.33 0.33 0 2 0.30 0.70 0.30 0.34 0.67 0 3 0.30 0.30 0.70 0.36 0.36 0 4 0.30 0.70 0.70 0.37 0.64 0 5 0.70 0.70 0.70 0.64 0.64 0 6 0.70 0.30 0.70 0.64 0.36 0 7 0.70 0.70 0.30 0.67 0.67 0 8 0.70 0.30 0.30 0.67 0.33 0

### Example Too:

 Surface of Revolution x1 y1 z1 x2 y2 z2 0 0.750 0.250 0.000 0.25 0.750 0.000 60 0.375 0.250 -0.650 0.125 0.750 -0.217 120 -0.375 0.250 -0.650 -0.125 0.750 -0.217 180 -0.750 0.250 0.000 -0.25 0.750 0.000 240 -0.375 0.250 0.650 -0.125 0.750 0.217 300 0.375 0.250 0.650 0.125 0.750 0.217 360 0.750 0.250 0.000 0.25 0.750 0.000

xc

yc

zc

0.1

3

-10

X1p

Y1p

X2p

Y2p

0.750

0.250

0.250

0.750

0.387

0.059

0.123

0.700

-0.415

0.059

-0.132

0.700

-0.750

0.250

-0.250

0.750

-0.340

0.418

-0.118

0.798

0.364

0.418

0.127

0.798

0.750

0.250

0.250

0.750

### 12.4 View Volumes and General Projections

• View window or projection window used to set up telephoto or wide angle scene

### Focal Lengths and Angles of View

 35mm Camera Focal Length(mm) Angle of View (Degrees) Extreme Telephoto 800 3.5 400 6.0 200 12.5 Moderate Telephoto 135 18.0 85 29.0 50 46.0 Normal 43 53.0 Moderate Wide Angle 24 84.0 Wide Angle 18 94.0

## Viewing

• Edges of window parallel to x.y axes

a – parallelepiped                                            b – frustum

### Symmetric Perspective-Projection Frustum

Window dimensions specify field of view

• Second equation used by graphics libraries (eg OpenGL)
• These formulas are substituted into the perspective projection formulas above.
• The perspective equations map the symmetric frustum into a parallelepiped view volume