Projects - Cengage Learning

Appendix C C-83
PROJECT
Parameterizations for Lines and Circles
In the previous project on lines, circles, and ray tracing we derived a vector
equation for a line in the x-y plane. You and your group might want to review
that derivation. In particular, let P 0 be a point in the x-y plane, P 0 be the position
vector with endpoint P 0 , and A be a nonzero vector. We showed that a vector
equation for the line in the x-y plane passing through the point P 0 with
direction vector A is
P P 0 tA for q t q (1)
where P is the position vector of a point P on the line (see Figure A).
y
P
P
0
P
tA
P
P= +tA
0 P 0
A
x
Figure A
Using equation (1) and vector components we can derive two useful scalar
equations for the same line. Let P 0 (x 0 , y 0 ) be the given point on the line,
A 8a, b9 Z 80, 09 be the nonzero direction vector of the line, and P (x, y)
be an arbitrary point on the line. The corresponding position vectors for the
points P 0 and P are P 0 8x 0 , y 0 9 and P 8x, y9, respectively. Substituting into
equation (1) we get
P P 0 tA for q t q
8x, y9 8x 0 , y 0 9 t8a, b9
8x, y9 8x 0 , y 0 9 8ta, tb9
8x, y9 8x 0 ta, y 0 tb9
So
x x
b 0 ta
for q t q
(2)
y y 0 tb
Equations (2) are called parametric equations for this line. If both a and b are
nonzero then we can solve each equation in (2) for t to get
or more simply,
x x 0
a
x x 0
a
t y y 0
b
y y 0
b
(3)