Dihedral angle
Mathematical background

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

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

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

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

where n · n is the dot product of the vectors and | n | | n | 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 b, b and b such that P + b, P + b and P + b 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

and satisfies 0 ≤ φ < π. {\displaystyle 0\leq \varphi <\pi.} In this case, switching the two half-planes gives the same result, and so does replacing b 0 {\displaystyle \mathbf {b} _{0}} with − b 0. {\displaystyle -\mathbf {b} _{0}.} In chemistry (see below), we define a dihedral angle such that replacing b 0 {\displaystyle \mathbf {b} _{0}} with − b 0 {\displaystyle -\mathbf {b} _{0}} changes the sign of the angle, which can be between − π and π.
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