# Heron's Principle of Reflection

## Physical Law

The angle of incidence of a ray of light is equal to the angle of reflecion when that ray is reflected in a (plane) mirror.

## Proof

Let a ray of light $L$ travel from $A$ to $B$ by way of a mirror $M$.

Let $B'$ be the mirror image of $B$ so that $M$ is the perpendicular bisector of $BB'$.

Let $P$ be the point where $L$ is reflected from $M$.

The total length of the path of $L$ is $AP + PB = AP + PB'$.

From Fermat's Principle of Least Time, this length is required to be a minimum.

It is to be demonstrated that $P$ is the point on $M$ such that $APB'$ is a straight line.

Aiming for a contradiction, suppose $L$ went through any point $P'$, for example, which is not on the straight line $AB'$.

Then $AP'B'$ is a triangle.

But $AP' + P'B$ is longer than $APB$.

Hence $L$ does not pass along the line $AP'B'$.

The result follows by Proof by Contradiction.

$\blacksquare$

## Examples

### Journey to the River

Mary, who is standing at a point $S$, wants to walk to the point $T$.

However, first she wants to take a drink of water from a river.

She wants to walk as short a distance as possible.

To what point on the riverbank should she walk?

## Source of Name

This entry was named for Heron of Alexandria.

## Historical Note

**Heron's Principle of Reflection** was reported in Heron of Alexandria's *Catoptrica* of $\text c. 75 \text { CE}$.

While this was merely a deduction of his based on observation, he was completely correct.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.7$: Heron (first century A.D.) - 1992: David Wells:
*Curious and Interesting Puzzles*... (previous) ... (next): Light Reflected off a Mirror