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The trigonometry of a tetrahedron explains the relationships between the lengths and various types of angles of a general tetrahedron.
Contents
Classical trigonometric quantities
The following are trigonometric quantities generally associated to a general tetrahedron:
Let
Furthermore, let
where
Define the following quantities:
Area and volume
Let
or by the following formula (if an angle and two corresponding edges are known):
Let
where
Affine triangle
Take the face
The usual laws for planar trigonometry of a triangle hold for this triangle.
Projective triangle
Consider the projective (spherical) triangle at the point
The usual laws for spherical trigonometry hold for this projective triangle.
Alternating sines theorem
Take the tetrahedron
Putting any of the four vertices in the role of O yields four such identities, but at most three of them are independent; if the "clockwise" sides of three of the four identities are multiplied and the product is inferred to be equal to the product of the "counterclockwise" sides of the same three identities, and then common factors are cancelled from both sides, the result is the fourth identity.
Three angles are the angles of some triangle if and only if their sum is 180° (π radians). What condition on 12 angles is necessary and sufficient for them to be the 12 angles of some tetrahedron? Clearly the sum of the angles of any side of the tetrahedron must be 180°. Since there are four such triangles, there are four such constraints on sums of angles, and the number of degrees of freedom is thereby reduced from 12 to 8. The four relations given by the sine law further reduce the number of degrees of freedom, from 8 down to not 4 but 5, since the fourth constraint is not independent of the first three. Thus the space of all shapes of tetrahedra is 5-dimensional.
Law of sines for the tetrahedron
See: Law of sines#Higher dimensions
Law of cosines for the tetrahedron
The law of cosines for the tetrahedron relates the areas of each face of the tetrahedron and the dihedral angles about a point. It is given by the following identity:
Relationship between dihedral angles of tetrahedron
Take the general tetrahedron
Then the area of the face
Skew distances between edges of tetrahedron
Take the general tetrahedron
To find
First, construct a line through
As a consequence, the quantity
By the volume formula, the tetrahedron
- Take the spherical triangle of the tetrahedron
X at the pointP i α i , j , α i , k , α i , l θ i j , θ i k , θ i l - Take the spherical triangle of the tetrahedron
X at the pointP i α i , l , α k , j , λ and the only known opposite angle is that ofλ , given byπ − θ i k
Combining the two equations gives the following result:
Making
Note that the denominator is a re-formulation of the Bretschneider-von Staudt formula, which evaluates the area of a general convex quadrilateral.