Description
Problem 1
(a) Derive the combined rotation and translation needed to transform world coordinate W into camera
coordinate C as illustrated in figure 1. Notice that Cz and Cx belong to the plane defined by Wz and
Wx.
(b) Consider a square in the world coordinate system defined by the points a,b,c,d. Assume such a square
has unit area. Show that the same square in the camera reference system has still unit area.
(c) Are parallel lines in the world reference system still parallel in the camera reference system? Justify
your answer.
(d) Does the vector defined by a and b have the same orientation in both reference system? Justify your
answer.
Figure 1
1
Problem 2
Consider a perspective projection where a point
P =
x
y
z
is projected onto an image plane Π′
represented by k = f
′ as shown in figure 2. The first, second and third
coordinate axes are denoted by i,j, and k, respectively. Consider the projection of an infinitely long line
Q =
1
1
0
+ t
0
0
1
in the world coordinate system where −∞ ≤ t ≤ −1. Calculate its two endpoints.
Figure 2
Problem 3
Two points x1 = (1, 3)⊤, x2 = (3, 1)⊤ in R
2 are transformed to x
′
1
, x
′
2 by a planar projective transformation
H
H =
1.520 −1.902 1.000
3.300 23.490 3.000
1.000 3.000 1.000
(a) Find the line l that passes through x1 and x2.
(b) Find the line l
′
that passes through x
′
1 and x
′
2
. You can use MATLAB to help with your computation.
(c) Derive an analytical expression that relates l with l
′
through H.
2
