WOLFRAM|DEMONSTRATIONS PROJECT

Construct a Dihedral Angle of a Tetrahedron Given Its Plane Angles at a Vertex

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α
1.
β
0.7
γ
0.5
d
1.2
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Given the tetrahedron
T=A'B'C'D'
, at the vertex
A'
let
α,β,γ
be the three plane angles
B'A'C'
,
B'A'D'
,
C'A'D'
. This Demonstration constructs the dihedral angle at the edge
A'D'
.
Construct the tetrahedron
ABCD
from the original tetrahedron
T
by modifying the lengths of the sides starting at
A=A'
: cut
T
off by the plane orthogonal to the edge
AD'
at the vertex
D=D'
. The dihedral angle along
AD
is the angle at vertex
D
of the triangle
BCD
.
Given the edge length
d
of
AB
and the angles
α,β,γ
, the other edges are given by:
a=dcos(β)
,
b=dsin(β)
,
f=a/cos(γ)
,
c=fsin(γ)
,
e=
2
d
+
2
f
-2dfcos(α)
. It helps to construct the net of
ABCD
.