Triangulation

Introduction

Triangulation measures range with geometry. It is one of the two principles of Range Sensing.

How it works

• a narrow beam of light lights up an object and sweeps over its surface
• the sweep is in the plane made by the object, the detector and the source
• the detector is at a known distance from the source, and it looks at a small part of the surface
• when the detector sees the light spot, you can calculate the distance to that spot from the geometry

This gives one point measurement at a time.

Key equations

The source is at $P_1$, the detector is at $P_2$ and the light spot on the object is at $P_3$.

• $l_1$ is the known distance from the source to the detector
• $l_2$ is the distance from the detector to the object, which is the range
• $a$ is the angle at the source, between $l_1$ and the light beam
• $\theta$ is the angle at the detector, between $l_1$ and the line of sight to the spot

The angle at the object is $180^{\circ }-(\theta +a)$. From the sine rule:

$$\frac{l_1}{\sin (180^{\circ }-(\theta +a))}=\frac{l_2}{\sin a}$$ $$\boxed{\text{Range}=l_2=\frac{l_1\sin a}{\sin (\theta +a)}}$$