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Concept
Dihedral angle
Summary
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.
When the two intersecting planes are described in terms of Cartesian coordinates by the two equations
the dihedral angle, between them is given by:
and satisfies
Alternatively, if nA and nB are normal vector to the planes, one has
where nA · nB is the dot product of the vectors and 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
and satisfies In this case, switching the two half-planes gives the same result, and so does replacing with In chemistry (see below), we define a dihedral angle such that replacing with changes the sign of the angle, which can be between −pi and pi.
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.
Official source
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