A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the union of a line and two half-planes that have this line as a common edge. In higher dimensions, a dihedral angle represents the angle between two hyperplanes. The planes of a flying machine are said to be at positive dihedral angle when both starboard and port main planes (commonly called "wings") are upwardly inclined to the lateral axis; when downwardly inclined they are said to be at a negative dihedral angle.

Mathematical background

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

${\displaystyle a_{1}x+b_{1}y+c_{1}z+d_{1}=0}$

${\displaystyle a_{2}x+b_{2}y+c_{2}z+d_{2}=0}$

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

${\displaystyle \cos \varphi ={\frac {\left\vert a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}\right\vert }{{\sqrt {a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}}{\sqrt {a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}}}}$

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

${\displaystyle \cos \varphi ={\frac {\left\vert \mathbf {n} _{\mathrm {A} }\cdot \mathbf {n} _{\mathrm {B} }\right\vert }{|\mathbf {n} _{\mathrm {A} }||\mathbf {n} _{\mathrm {B} }|}}}$

where n_A · n_B is the dot product of the vectors and |n_A| |n_B| is the product of their lengths.^[1]

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

${\displaystyle \cos \varphi ={\frac {(\mathbf {b} _{0}\times \mathbf {b} _{1})\cdot (\mathbf {b} _{0}\times \mathbf {b} _{2})}{|\mathbf {b} _{0}\times \mathbf {b} _{1}||\mathbf {b} _{0}\times \mathbf {b} _{2}|}}}$,

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

In polymer physics

In some scientific areas such as polymer physics, one may consider a chain of points and links between consecutive points. If the points are sequentially numbered and located at positions r₁, r₂, r₃, etc. then bond vectors are defined by u₁=r₂−r₁, u₂=r₃−r₂, and u_i=r_i+1−r_i, more generally.^[2] This is the case for kinematic chains or amino acids in a protein structure. In these cases, one is often interested in the half-planes defined by three consecutive points, and the dihedral angle between two consecutive such half-planes. If u₁, u₂ and u₃ are three consecutive bond vectors, the intersection of the half-planes is oriented, which allows defining a dihedral angle that belongs to the interval (−π, π. This dihedral angle is defined by^[3]

Zdroj:https://en.wikipedia.org?pojem=Dihedral_angle
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