B1 - - - DEFINITIONS - - - 10/03/2004
When three planes intersect they form a trihedral angle that consists
of three edge angles and three face angles.
An edge angle is the angle between two of the three planes forming the
trihedral angle.
A face angle is the angle between two of the edges.
Though there are many applications where the relationship of three
planes may have to be considered, it is hard to visualize three
planes or three lines in three dimensional space, but if we consider
a sphere surrounding the trihedral angle and centered on the
intersection point (called the vertex of the trihedral angle), then
the three planes will form three great circles on the surface of
the sphere forming a spherical triangle.
PRINCIPLES FROM GEOMETRY
- The section of the surface of a sphere made by a plane is a
great circle if the plane passes through the center of the
sphere and a small circle if the plane does not pass through
the center of the sphere.
- A spherical triangle is the figure on the surface of a sphere
bounded by three arcs of great circles.
- Each side of a spherical triangle is less than the sum of the
other two.
- The sum of the three sides of a spherical triangle is less than
360 degrees.
- The sum of the angles of a spherical triangle is greater than
180 and less than 540 degrees.
- If two sides of a spherical triangle are equal, the angles opposite
them are equal; and conversely.
- If two sides of a spherical triangle are unequal, the angle opposite
these sides are unequal, and the greater side lies opposite the
greater angle; and conversely.
- The shortest distance on the surface of a sphere between two
points is an arc of the great circle joining them.
- The angle between two intersecting great circle arcs is
measured by the angle between the planes that created the
arcs (edge angle).
- The side of a spherical triangle has the same numeric
value as its central angle at the vertex of the trihedral
angle (face angle).
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