Dihedral angle

A dihedral angle is the angle between two intersecting planes or half-planes. It is a plane angle formed on a third plane, perpendicular to the line of intersection between the two planes or the common edge between the two half-planes. In higher dimensions, a dihedral angle represents the angle between two hyperplanes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common.

When the two intersecting planes are described in terms of Cartesian coordinates by the two equations

the dihedral angle, ${\displaystyle \varphi }$ between them is given by:

and satisfies ${\displaystyle 0\leq \varphi \leq \pi /2.}$ It can easily be observed that the angle is independent of ${\displaystyle d_{1}}$ and ${\displaystyle d_{2}}$.

Alternatively, if n_A and n_B are normal vector to the planes, one has

where n_A · n_B is the dot product of the vectors and |n_A| |n_B| is the product of their lengths.

The absolute value is required in above formulas, as the planes are not changed when changing all coefficient signs in one equation, or replacing one normal vector by its opposite.

However the absolute values can be and should be avoided when considering the dihedral angle of two half planes whose boundaries are the same line. In this case, the half planes can be described by a point P of their intersection, and three vectors b₀, b₁ and b₂ such that P + b₀, P + b₁ and P + b₂ belong respectively to the intersection line, the first half plane, and the second half plane. The dihedral angle of these two half planes is defined by