This Heron’s principle, concerning the reflection of
light, is a special case of Fermat’s principle in optics.
Proof. If and are on different sides of , then must be on the line , and the assertion is trivial since the vertical angles are equal. Thus, let the points and be on the same side of . Denote by and the points of the line where the normals of set through and intersect , respectively. Let be the intersection point of the lines and . Then, is the point of where the normal line of set through intersects .
which imply the equation
From this we can infer that also
Thus the corresponding angles and are equal.
We still state that the route is the shortest. If is another point of the line , then , and thus we obtain
- 1 Tero Harju: Geometria. Lyhyt kurssi. Matematiikan laitos. Turun yliopisto (University of Turku), Turku (2007).
|Date of creation||2014-09-15 15:38:36|
|Last modified on||2014-09-15 15:38:36|
|Last modified by||pahio (2872)|