Weighted path minimum (Snell-type)

Overview

Minimize AP + k·PB for k < 1 (different “costs” on two segments).

Explanation

Point A off a line, B on the line; choose P on the line to minimize AP + k·PB (0 < k < 1).
k weights the second segment (slow ground vs fast road).
Optimum: sin θ = k, where θ is the angle between AP and the normal — akin to refraction.

How to use

Drag roadside P; the green mark is optimal where sin θ = k.

Knowledge points

• Shortest paths
• Trigonometry