Heron's Principle of Reflection

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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.



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.



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.