Dihedral Angle - Methods of Computation

Methods of Computation

The dihedral angle between two planes relies on being able to efficiently generate a normal vector to each of the planes. One approach is to use the cross product. If A₁, A₂, and A₃ are three non-collinear points on plane A, and B₁, B₂, and B₃ are three non-collinear points on plane B, then U_A = (A₂−A₁) × (A₃−A₁) is orthogonal to plane A and U_B = (B₂−B₁) × (B₃−B₁) is orthogonal to plane B. The (unsigned) dihedral angle can therefore be computed with either

Another approach to computing the dihedral angle is first to pick an arbitrary vector V that is not tangent to either of the two planes. Then applying the Gram–Schmidt process to the three vectors (A₂−A₁, A₃−A₁, V) produces an orthonormal basis of space, the third vector of which will be normal to plane A. Doing the same with the vectors (B₂−B₁, B₃−B₁, V) yields a vector normal to plane B. The angle between the two normal vectors can then be computed by any method desired. This approach generalizes to higher dimensions, but does not work with flats that have a codimension greater than 1.

To compute the dihedral angle between two flats, it is additionally necessary to ensure that each of the two normal vectors is selected to have a minimal projection onto the other flat. The Gram–Schmidt process does not guarantee this property, but it can be guaranteed with a simple eigenvector technique. If

is a matrix of orthonormal basis vectors for flat A, and

is a matrix of orthonormal basis vectors for flat B, and

the eigenvector with the smallest corresponding eigenvalue of, and

the eigenvector with the smallest corresponding eigenvalue of ,

then, the angle between and is the dihedral angle between A and B, even if A and B have a codimension greater than 1.