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