Define the unit vectors ˆeA,eˆB,ˆeCfrom the center of the unit sphere to the points
A, B, C on the surface of the sphere and observe that by using the definition of a cross
product and dot product one obtains
|ˆeA׈eC|= sin b,
eˆA·ˆeC= cos b,
|ˆeA׈eB|= sin c,
ˆeA·eˆB= cosc,
|ˆeC׈eB|= sin a
ˆeC·ˆeB= cos a
(9 .8)
Note that since the sphere is a unit sphere the angles a, b, c are given respectively by
the arcs
BC ,
AC and
AB or arcs opposite the vertices A, B, C.
The angle between two intersecting planes is called a dihedral angle. The dihe-
dral angle can be calculated from the unit normal vectors to the intersecting planes.
In figure 9-1 , let
ˆeB׈eC= sin aˆe 0 BC , ˆeA׈eC= sin beˆ 0 AC , ˆeA׈eB= sin cˆe 0 AB (9 .9)
define the unit vectors ˆe 0 BC ,ˆe 0 AC ,ˆe 0 AB which are perpendicular to the planes defin-
ing the dihedral angles α, β, γ. The cross product relations given by the equation
(9.8) together with the unit normal vectors can be used to calculate the cosines
associated with the angle α, β, γ. One finds that
ˆe 0 BC ·eˆ 0 AB = cos β, ˆe 0 BC ·eˆ 0 AC = cos γ, ˆe 0 AC ·ˆe 0 AB = cos α (9 .10)
and with the aid of equations (9.9) one can write
cos γ=
|(ˆeB×eˆC)·(ˆeA׈eB)|
|ˆeB׈eC||ˆeA׈eC|
(9 .11)
with similar expressions for the representation of cos αand cos β. The relation (9.11)
can be simplified using the dot product relation (6.32) which is repeated here
(A×B)·(C×D) = (A·C)(B·D)−(A·D)(B·C) (9 .12)
The numerator in equation (9.11) can then be expressed
(ˆeB׈eC)·(ˆeA׈eC) =( ˆeB·ˆeA)( ˆeC·eˆC)−(ˆeB·eˆC)( ˆeC·ˆeA)
= cos c−cos acos b
(9 .13)
The results from equations (9.8) and (9.13) show that the equation (9.11) can be
expressed in the form
cos c= cos acos b+ sin asin bcos γ (9 .14)
