Trihedral angle

Fig. 1. Trihedral angle.

A trihedral angle is a part of space bounded by three flat angles with a common vertex and pairwise common sides that do not lie in the same plane. The common vertex O of these angles is called the vertex of the trihedral angle. The sides of the corners are called edges, the flat angles at the apex of a trihedral corner are called its faces. Each of the three pairs of faces of a trihedral angle forms a dihedral angle (bounded by a third face that is not included in the pair; if necessary, this restriction is naturally lifted, resulting in the necessary half-planes forming the entire dihedral angle without restriction). If you place the vertex of a trihedral corner in the center of a sphere, a spherical triangle bounded by it is formed on its surface, the sides of which are equal to the flat angles of the trihedral angle, and the corners to its dihedral angles.

Content

Triangle inequality for a triangular angle

Each flat angle of a trihedral angle is less than the sum of its other two flat angles.

Sum of the flat angles of a trihedral corner

The sum of the flat angles of a trihedral angle is less than 360 degrees.

Evidence

Let OABC be a given trihedral angle (see Fig. 1). Consider a trihedral angle with vertex A formed by the faces ABO, ACO and angle BAC. We write the inequality:

\angle BAC<\angle BAO+\angle CAO

{\ displaystyle \ angle BAC <\ angle BAO + \ angle CAO}

Similarly, for the remaining trihedral angles with vertices B and C:

\angle ABC<\angle ABO+\angle CBO

{\ displaystyle \ angle ABC <\ angle ABO + \ angle CBO}

\angle ACB<\angle ACO+\angle BCO

{\ displaystyle \ angle ACB <\ angle ACO + \ angle BCO}

Adding these inequalities and taking into account that the sum of the angles of triangle ABC is 180 °, we obtain

180<\angle BAO+\angle CAO+\angle ABO+\angle CBO+\angle BCO+\angle ACO=180-\angle AOB+180-\angle BOC+180-\angle AOC

{\ displaystyle 180 <\ angle BAO + \ angle CAO + \ angle ABO + \ angle CBO + \ angle BCO + \ angle ACO = 180- \ angle AOB + 180- \ angle BOC + 180- \ angle AOC}

Consequently : $\angle AOB+\angle BOC+\angle AOC<360$ ${\ displaystyle \ angle AOB + \ angle BOC + \ angle AOC <360}$

Fig. 2. Trihedral angle.

Cosine theorem for a trihedral angle

Let a trihedral angle be given (see Fig. 2), α, β, γ are its plane angles, A, B, C are dihedral angles made up by the planes of angles β and γ, α and γ, α and β.

The first cosine theorem for a trihedral angle: $\cos {\alpha }=\cos {\beta }\cos {\gamma }+\sin {\beta }\sin {\gamma }\cos {A}$ ${\ displaystyle \ cos {\ alpha} = \ cos {\ beta} \ cos {\ gamma} + \ sin {\ beta} \ sin {\ gamma} \ cos {A}}$

The second cosine theorem for a trihedral angle: $\cos {A}=-\cos {B}\cos {C}+\sin {B}\sin {C}\cos {\alpha }$ ${\ displaystyle \ cos {A} = - \ cos {B} \ cos {C} + \ sin {B} \ sin {C} \ cos {\ alpha}}$

Proof of the Second Cosine Theorem for a Trihedral Angle

Let OABC be a given trihedral angle. We drop the perpendiculars from the inner point of the trihedral angle on its face and get a new polar triangular angle (dual to this). The flat angles of one trihedral angle complement the dihedral angles of the other and the dihedral angles of one corner complement the flat angles of another to 180 degrees. That is, the flat angles of the polar angle are respectively equal: 180 - A; 180 - B; 180 - C , and dihedral - 180 - α; 180 - β; 180 - γ

We write the first cosine theorem for it

\cos({\pi -A})=\cos({\pi -\alpha })\sin({\pi -B})\sin({\pi -C})+

{\ displaystyle \ cos ({\ pi -A}) = \ cos ({\ pi - \ alpha}) \ sin ({\ pi -B}) \ sin ({\ pi -C}) +}

+\cos({\pi -B})\cos({\pi -C)}

{\ displaystyle + \ cos ({\ pi -B}) \ cos ({\ pi -C)}}

and after simplifications we get:

\cos {A}=\cos {\alpha }\sin {B}\sin {C}-\cos {B}\cos {C}

{\ displaystyle \ cos {A} = \ cos {\ alpha} \ sin {B} \ sin {C} - \ cos {B} \ cos {C}}

Sine theorem for a trihedral angle

${\sin {\alpha } \over \sin A}={\sin \beta \over \sin B}={\sin \gamma \over \sin C}$ ${\ displaystyle {\ sin {\ alpha} \ over \ sin A} = {\ sin \ beta \ over \ sin B} = {\ sin \ gamma \ over \ sin C}}$ where α, β, γ are the flat angles of the trihedral angle; A, B, C - opposite dihedral angles (see. Fig. 2).