Computer Graphics

3D Viewing

H&B Chapter 7

Projections

• Parallel Projection

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:

x_p = x

y_p = y

Oblique Projection

Perspective Projections – general case

…where ( x_prp, y_prp, z_prp ) is the perspective reference point

Begin with parametric equation for line…

x_p = x - (x – x_prp)u

y_p = y - (y – y_prp)u

z_p = z - (z – z_prp)u

u = 0 -> 1

solve for u where z_vp = z_p (viewplane)

substitute in x_p, y_p equations

multiplying through and rearranging…

Special Case -- z_vp = z_p = 0

Example: Special Case where z_vp = z_p = 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
Perspective Projection		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

Normalized Perspective Projection View Coordfinates

Three Dimensional Clipping

3D Clipping Planes for Symmetric Cube :