

The
trajectory of light as an example of
the shortest path
The minimal problem
Light is known to propagate along straight lines in a uniform medium.
Going from one point to another, it chooses the shortest path.
In this lab, you will answer the question:
What is the trajectory of light that starts at point P and
passes through point Q after being reflected at some unknown point
R on mirror M in between?
To specify the point
R, we apply the principle of the shortest distance. This principle
requires that the sum
PR + RQ
be minimal. This minimum requirement alone will specify the location
of R on the mirror.
First solution
We know from daily experience that an observer at Q sees the
reflected light coming from an imiginary reflected source P'
seemingly located behind the mirror. The trajectory of light connecting
P' with Q should be the shortest possible path between these
points, i.e. it should be the straight line P'Q. This line
intersects with the mirror M at point R_{*}. Since
PR_{*} = P'R_{*}, we have
PR_{*} + R_{*}Q =
P'R_{*} + R_{*}Q = P'Q.
This is the shortest distance because P'Q is a straight line.
To prove this, assume the contrary, i.e. that the reflection occurs at
some point
R rather than at R_{*}. For the triangle
P'QR we have the inequality
P'R + RQ > P'Q = P'R_{*} + R_{*}Q.
This inequality shows that the straight line P'Q yields the
shortest possible distance.
An elementary geometric analysis shows that if N_{*}R_{*} is the normal to M at R_{*}, then the angle of incidence
PR_{*}N_{*} is equal to the angle of the reflection N_{*}R_{*}Q
PR_{*}N_{*} = N_{*}R_{*}Q,
and this equality of angles fixes the choice R_{*} of
R. For any other point R on the mirror, the equality of the relevant
angles will not occur
PRN NRQ.
Second solution
Introduce the function
(T) = PT + TQ
defined as the sum of distances from two fixed points P and Q to a variable point T. Let us fix a
positive number d > PQ and consider the locus of points
T having the same value of (T)
equal to d:
(T) = d.
All such points belong to an ellipse with foci P and Q. If
we choose a number d_{1} > d, then the equation
(T) = d_{1} will define a larger ellipse, confocal with the previous one.
If we picked other values of d and plotted the corresponding
ellipses, we would come up with a series of level curves
(T) = const; this chart being a
family of confocal ellipses with focal points P and Q.
Some of these ellipses intersect with the mirror M at two points,
some do not intersect it at
all, and only one ellipse has one common point with the mirror. This selected point is the required
point R_{*} of reflection, and the relevant ellipse specifies the minimum distance PR_{*} + R_{*}Q.
Problems
 Let (x_{P}, y_{P}) be the coordinates of
P, (x_{Q}, y_{Q}) be the coordinates
of
Q, and (x, y) be the coordinates of R.
Without lack of generality, we may assume that the mirror lies along the
xaxis (y = 0). The sum PR + RQ can be
computed
PR + RQ = sqrt((x_{P} 
x)^{2} + y_{P}^{2}) +
sqrt((x_{Q} 
x)^{2} + y_{Q}^{2}).
This sum (which is a function of x) should be minimal. Let
x_{*} be the value of x minimizing this sum and
R_{*}(x_{*}, 0) be the corresponding
point on the mirror.
Show by direct calculation that
min(PR + RQ)  =  (PR +
RQ)_{x = x*} 
 =  PR_{*} +
R_{*}Q 
 =  P'R_{*} + R_{*}Q
= P'Q 
 =  sqrt((x_{Q} 
x_{P})^{2} +
(y_{Q} + y_{P})^{2}) 
 Comparing the first and second solutions, demonstrate the following property of an ellipse: Two straight lines connecting the focal points of an
ellipse with an arbitrary point R on its periphery make equal
angles with the tangent to the ellipse at point R. Your
solution does not need to contain any formulas. A wellworded
description will suffice.
 Based on the second solution, suggest, in words, a procedure for
finding the point
of light reflection from a curvilinear mirror of arbitrary shape.
 Three towns A, B, and C want to build three roads
towards a common transportation center. They want to choose the location
R
of this center so as to minimize the total length of the roads and, hence,
the construction costs. Formulate this minima problem and solve it
by applying
a procedure similar to the one described above in the second solution.
Use the result of Problem 2 to give a complete descriptive
characterization of the desired location R.
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Created by Henry Fink
Last updated: Sunday, September 28, 1997

