Graphics Programs Reference
In-Depth Information
In prophetic utterances, time is often telescoped.
—Anonymous
The heights of the transformed telephone poles can be determined by a similar
expression. A pole located right at the viewer's location should maintain its height,
while poles that are moved closer should become taller but should remain smaller than
the nearest pole. If the nearest pole is l units tall, then the expression
l = l 1
zr
z + l
produces l values that range from l (for z =0)to(1
r ) l (for very large z ). The
Mathematica code
l=20.; r=0.1;
Table[l(1-(z r/(z+l))), {z,0,100,5}]
Table[%[[i]]-%[[i+1]], {i,1,20}]
Table[Line[{{i, 17}, {i, %%[[i]]}}], {i,1,21}]
Show[Graphics[%]]
selects l =20and r =0 . 1toobtain l values ranging from l to 0 . 9 l = 18. The results
are 20, 19.6, 19.33, 19.14, 19, 18.89, 18.8, 18.72, 18.67, 18.62, 18.57, 18.53, 18.5, 18.47,
18.44, 18.42, 18.40, 18.38, 18.363, 18.35, and 18.33. Figure 4.41b shows the top parts of
the poles to illustrate how the differences in height between consecutive poles diminish.
The third line of the code yields the 20 differences 0.4, 0.27, 0.19, 0.14, 0.11, 0.09, 0.073,
0.061, 0.051, 0.044, 0.038, 0.033, 0.029, 0.026, 0.0234, 0.021, 0.019, 0.017, 0.016, and
0.014. Thus, the height differences between consecutive telephone poles get smaller and
smaller.
After a three-dimensional scene has been telescoped point by point, we can use
perspective projection to display it in two dimensions.
Love looks through a telescope; envy, through a microscope.
—Josh Billings
4.12 Microscopic Projection
A sample observed through a microscope is normally thin. We can therefore assume that
points that go through a microscopic projection have the same (or similar) z coordinates.
In contrast to a telescope, which brings points closer to the observer, a microscope
“opensup”thepoints. Figure4.42showshowthisisdonebymovingpointsawayfrom
the z axis. If the view angle of a point P is θ , then the microscope places its projection
P such that its view angle is ,where m is the magnification power of the microscope.
Thus, the projection rule is
x
z + k
x
z + k
=tan θ
and
=tan( ) .
(4.5)
 
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