Point to point method of lighting calculation

The point-to-point method (also called the point-by-point or inverse-square-law method) calculates the illuminance at a particular point due to one or more lamps from their luminous intensity, using the inverse square law and Lambert's cosine law. It is used where the illumination at specific points matters: street lighting, flood lighting, local lighting and checking uniformity.

Principle

For a lamp of intensity I (cd) in the direction of point P, at distance d from P, with θ the angle between the light ray and the normal to the surface:

E = (I / d²) × cos θ        lux

On a horizontal plane, lamp at height h,
point at horizontal distance x:
  d = √(h² + x²),  cos θ = h/d
  E_H = I h / d³ = (I / h²) cos³θ
On a vertical plane:
  E_V = I x / d³ = (I / h²) cos²θ sin θ

   Lamp L
     |\
   h |  \ d
     |θ   \
     +-----P   (horizontal plane)
        x

Steps

1. Find the intensity I of each lamp towards P from the polar curve of the luminaire.
2. Find d and θ from the geometry.
3. Calculate E from each lamp; add the contributions of all lamps at that point.
4. Repeat for several points to check the minimum, maximum and uniformity.

Example: I = 400 cd, h = 2 m: directly below, E = 400/2² = 100 lux; at x = 2 m, d = 2.83 m, E = 400 × 2/2.83³ = 35.4 lux.

Limitation: it ignores light reflected from walls and ceiling, so the lumen method is used instead for general indoor lighting.