If one knows the points where a plane intercepts the coordinate axis one can easily determine the equation for the plane. Given the distances along the axes, x, y and z one can find the distance ρ of the proximal point, p, the closest point on the plane to the origin. This can then be used to find the direction, e, by finding the direction cosines, α, β and γ, and then we have the equation for the plane.

The intercepts for the plane are related to the Miller indices, (h k ℓ), used in crystallography where the planes are defined in terms of the unit cells of the crystal. A Euclidean coordinate system in which all directions can be measure using the same unit of length is analogous to a cubic crystal lattice. The formula for the distance of the plane is similar to our formula for the distance, ρ, of the plane from the origin if the lattice constant, a=1. So, given the Miller indices, we can find the distance and normal of a crystal plane.

We see that the Miller indices, (h k ℓ), are inversely proportional to distances of the intercepts from the origin.

Supplemental (Aug 16): The reason the crystallographers use Miller indices may be because they behave like wave vectors.

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