Computer Graphics
|
3D Viewing
|
Coordinate
positions transformed to viewplane with parallel lines
Coordinate
positions transformed to viewplane with converging lines
Orthographic
Projection:
xp = x
yp = y
Oblique Projection
…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 |
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0.5 |
0.5 |
-1.5 |
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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 |
Surface of Revolution |
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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 |
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|
xc |
yc |
zc |
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Perspective Projection
|
0.1 |
3 |
-10 |
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X1p |
Y1p |
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|
X2p |
Y2p |
|
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0.750 |
0.250 |
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|
0.250 |
0.750 |
|
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0.387 |
0.059 |
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|
0.123 |
0.700 |
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-0.415 |
0.059 |
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-0.132 |
0.700 |
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-0.750 |
0.250 |
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-0.250 |
0.750 |
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-0.340 |
0.418 |
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-0.118 |
0.798 |
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0.364 |
0.418 |
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|
0.127 |
0.798 |
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0.750 |
0.250 |
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0.250 |
0.750 |
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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 |
a –
parallelepiped b
– frustum
Window dimensions specify
field of view