Editing Dihedral angle (section)

==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 {{math|'''r'''<sub>1</sub>}}, {{math|'''r'''<sub>2</sub>}}, {{math|'''r'''<sub>3</sub>}}, etc. then bond vectors are defined by {{math|'''u'''<sub>1</sub>}}={{math|'''r'''<sub>2</sub>}}−{{math|'''r'''<sub>1</sub>}}, {{math|'''u'''<sub>2</sub>}}={{math|'''r'''<sub>3</sub>}}−{{math|'''r'''<sub>2</sub>}}, and {{math|'''u'''<sub>i</sub>}}={{math|'''r'''<sub>i+1</sub>}}−{{math|'''r'''<sub>i</sub>}}, more generally.<ref name="kroger">{{cite book|last=Kröger|first=Martin|title=Models for polymeric and anisotropic liquids|year=2005|publisher=Springer|isbn=3540262105}}</ref> This is the case for [[kinematic chain]]s or [[amino acid]]s 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 {{math|'''u'''<sub>1</sub>}}, {{math|'''u'''<sub>2</sub>}} and {{math|'''u'''<sub>3</sub>}} are three consecutive bond vectors, the intersection of the half-planes is oriented, which allows defining a dihedral angle that belongs to the interval {{math|(−{{pi}}, {{pi}}]}}. This dihedral angle is defined by<ref>{{cite journal
|last1=Blondel |first1= Arnaud |last2=Karplus |first2=Martin |date= 7 Dec 1998 |title= New formulation for derivatives of torsion angles and improper torsion angles in molecular mechanics: Elimination of singularities |journal= Journal of Computational Chemistry
|volume= 17 |issue= 9 |pages= 1132–1141
|doi= 10.1002/(SICI)1096-987X(19960715)17:9<1132::AID-JCC5>3.0.CO;2-T
}}</ref>
:<math>\begin{align}
\cos \varphi&=\frac{ (\mathbf{u}_1 \times \mathbf{u}_2) \cdot (\mathbf{u}_2 \times \mathbf{u}_3)}{|\mathbf{u}_1 \times \mathbf{u}_2|\, |\mathbf{u}_2 \times \mathbf{u}_3|}\\
\sin \varphi&=\frac{ \mathbf{u}_2 \cdot((\mathbf{u}_1 \times \mathbf{u}_2) \times (\mathbf{u}_2 \times \mathbf{u}_3))}{|\mathbf{u}_2|\, |\mathbf{u}_1 \times \mathbf{u}_2|\, |\mathbf{u}_2 \times \mathbf{u}_3|},
\end{align}</math>
or, using the function [[atan2]],
:<math>\varphi=\operatorname{atan2}(\mathbf{u}_2 \cdot((\mathbf{u}_1 \times \mathbf{u}_2) \times (\mathbf{u}_2 \times \mathbf{u}_3)), |\mathbf{u}_2|\,(\mathbf{u}_1 \times \mathbf{u}_2) \cdot (\mathbf{u}_2 \times \mathbf{u}_3)).</math>

This dihedral angle does not depend on the orientation of the chain (order in which the point are considered) — reversing this ordering consists of replacing each vector by its opposite vector, and exchanging the indices 1 and 3. Both operations do not change the cosine, but change the sign of the sine. Thus, together, they do not change the angle.

A simpler formula for the same dihedral angle is the following (the proof is given below)
:<math>\begin{align}
\cos \varphi&=\frac{ (\mathbf{u}_1 \times \mathbf{u}_2) \cdot (\mathbf{u}_2 \times \mathbf{u}_3)}{|\mathbf{u}_1 \times \mathbf{u}_2|\, |\mathbf{u}_2 \times \mathbf{u}_3|}\\
\sin \varphi&=\frac{ |\mathbf{u}_2|\,\mathbf{u}_1 \cdot(\mathbf{u}_2 \times \mathbf{u}_3)}{|\mathbf{u}_1 \times \mathbf{u}_2|\, |\mathbf{u}_2 \times \mathbf{u}_3|},
\end{align}</math>
or equivalently,
:<math>\varphi=\operatorname{atan2}(
|\mathbf{u}_2|\,\mathbf{u}_1 \cdot(\mathbf{u}_2 \times \mathbf{u}_3) ,
(\mathbf{u}_1 \times \mathbf{u}_2) \cdot (\mathbf{u}_2 \times \mathbf{u}_3)).</math>

This can be deduced from previous formulas by using the [[vector quadruple product]] formula, and the fact that a [[scalar triple product]] is zero if it contains twice the same vector:
:<math>
 (\mathbf{u}_1\times\mathbf{u}_2)\times(\mathbf{u}_2\times\mathbf{u}_3) = [(\mathbf{u}_2\times\mathbf{u}_3)\cdot\mathbf{u}_1]\mathbf{u}_2 - [(\mathbf{u}_2\times\mathbf{u}_3)\cdot\mathbf{u}_2]\mathbf{u}_1
   = [(\mathbf{u}_2\times\mathbf{u}_3)\cdot\mathbf{u}_1]\mathbf{u}_2
</math>

Given the definition of the [[cross product]], this means that <math>\varphi</math> is the angle in the clockwise direction of the fourth atom compared to the first atom, while looking down the axis from the second atom to the third. Special cases (one may say the usual cases) are <math>\varphi = \pi</math>, <math>\varphi = +\pi/3</math> and <math>\varphi = -\pi/3</math>, which are called the ''trans'', ''gauche<sup>+</sup>'', and ''gauche<sup>−</sup>'' conformations.