Example 7-1. Reflection property for the parabola.
The parabola y^2 = 4 px with focus F having coordinates (p,0) can be represented
parametrically. One parametric representation for the position vector is
r =r (t) = t
2
4 p
ˆe 1 +tˆe 2 , −∞ < t < ∞ (7 .3)
and the resulting parabola is illustrated in the figure 7-1. In this figure assume the
surface of the parabola is a mirrored surface.
Figure 7-1. Light ray PB gets reflected to ray through focus.
The derivative vector
dr
dt
=r ′(t) = t
2 p
eˆ 1 +ˆe 2
produces a tangent vector to the curve and the vector
ˆet=∣∣dr^1
dt
∣∣dr
dt =
t
√^2 peˆ^1 +ˆe^2
1 + t^2 / 4 p^2
=
t√ˆe 1 + 2 pˆe 2
4 p^2 +t^2
is a unit tangent vector to the curve.
Consider a general point P on the parabola where a light ray P B parallel to the
x−axis hits the parabola. Construct the normal to the parabola and label the angle
∠BP C the angle θ 1 and then label the angle ∠F P C the angle θ 2. The angle θ 1 is
called the angle of incidence and the angle θ 2 is called the angle of reflection. Also
