EECS101: HOMEWORK #1

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Description

For this assignment, your TA will supply a C program to help generate image files that you
will display. For each case you only need to change the indicated two assignment statements
for the image coordinates xprime and yprime. You will also need to add declarations for any
other variables that you may need, for example, to define constants. The supplied function
plot logical point maps the (x

, y′
) image plane over the range −4 ≤ x
′ ≤ 4 and −4 ≤ y
′ ≤ 4 to
image pixel coordinates.
Consider the imaging system of Figure 2-1 in the Horn textbook (page 20) having a pinhole
at the origin with the z-axis horizontal (positive z to the left), the y-axis vertical (positive y up)
and the x-axis forward (positive x out). The image plane is located at z = f

(f
′ > 0). All
events in 3-D are assumed to take place in the space z < 0. The coordinates (x

, y′
) define points
in the image plane and the coordinates (x, y, z) define points in the world. Curves in space can
be defined parametrically using the parameter t.
1. For a line in 3-D the parametric equations are:
x = x0 + ta
y = y0 + tb
z = z0 + tc
for constants x0, y0, z0, a, b, and c. As t takes on values in the range (−∞, ∞) we get a line.
a) Write expressions for the image coordinates x
′ and y

in terms of these constants and f
′ and t
for both the perspective and orthographic projection of the line.
b) Is the perspective projection of the line also a line in the image? Explain.
c) Is the orthographic projection of the line also a line in the image? Explain.
d) Using the constant settings:
x0 = .5
y0 = −1
z0 = 0
a = 0
b = 1
c = −1
f
′ = 1
use a C program to generate images of both the perspective and orthographic projections of the
line. Since ±∞ is difficult to represent in the computer we can use values of t for a smaller
range like between 0.01 and 10000. The sample C program will only display regions in the range
−4 ≤ x
′ ≤ 4, −4 ≤ y
′ ≤ 4 so you may want to further limit the range of t so that the values of
x
′ and y
′ will fall within this range.
e) What happens in the perspective projection case as t goes to ∞? Is this consistent with the
2. For two parallel lines in a plane that is parallel to the image plane the value for z is a
constant z0. The parametric equations for these lines are
x = x1 + ta
xˆ = x2 + ta
y = y1 + tb
yˆ = y2 + tb
z = z0
zˆ = z0
where Line 1 is defined by (x, y, z) and Line 2 is defined by (ˆx, y, ˆ zˆ) and t is in the range (−∞, ∞).
a) Write expressions for the image coordinates x

and y

in terms of the constants above and f

and t for both the perspective and orthographic projection of the lines.
b) Using the following constant settings:
x1 = 0.5
x2 = −0.5
y1 = −1
y2 = −1
z0 = −1, −2, −3
a = 1
b = 1
f
′ = 1
use a C program to generate images of both the perspective and orthographic projections of the
lines by letting t range from 0.01 to 10000.
c) Will the projections of the lines in the image be parallel for i) perspective projection and ii)
orthographic projection? Explain.
e) Is orthographic projection a good approximation to perspective projection for the geometry
for this problem? Explain.
f) What occurs if z0 = |f

|?
3. Consider two parallel lines that lie in a plane that is not parallel to the image plane. Assume
that these lines are defined by
x = x1
xˆ = x2
y = y0 + tb
yˆ = y0 + tb
z = z0 + tc
zˆ = z0 + tc
a) Write expressions for the image coordinates x
′ and y

in terms of the constants above and f

and t for both the perspective and orthographic projection of the lines.
b) Using the following constant settings,
x1 = −1
x2 = 1
y0 = −1
z0 = 0
b = 0, 1, −1
c = 1, −1
f
′ = 1
use a C program to generate images of both the perspective and orthographic projections of the
lines by letting t range from 0.01 to 10000.
c) Will the projections of the lines be parallel for i) perspective projection and 2) orthographic
projection? Explain.