2. Spherical Geometry
Spherical Geometry is one type of non-Euclidean geometry, which is based on a system of points, great
circles, and spheres.
•Much of spherical Geometry was developed by early Babylonians, Arabs, and Greeks.
•Their study was based in the astronomy of Earth and their need to be able to measure time accurately.
3. Great Circle
A great circle is the largest circle that can be
drawn on the surface of a sphere.
•An arc of a great circle is the shortest path
between 2 points
•There is a unique great circle passing through
any pair of non-polar points.
•A great circle is finite and returns to its original
point.
4. Geodesic
Geodesic is the shortest path between two points on the
surface of a sphere. It is analogous to a straight line in
Euclidean geometry.
Latitude and Longitude
Like coordinates on a flat plane, latitude and longitude are
used to specify the location of points on the surface of a
sphere. Latitude measures the north-south position, while
longitude measures the east-west position.
Antipodes These are diametrically opposite points on a sphere,
connected by a great circle. They have unique properties like having
the same latitude and longitude but opposite signs. Understanding
antipodes is essential in navigation and astronomy.
5. Spherical Geometry Parallel
Postulate
Through a point not on a line, there is no line
parallel to a given line.
•No parallel lines exist in spherical geometry
•If 3 points are collinear, any one if the three points
is between the other 2
• Two lines intersect at 2 points and form 8 angles
•Perpendicular lines intersect twice and form 8 right
angles
6. Spherical Triangle
A spherical triangle is formed by three great
circle arcs on the surface of a sphere, connecting
three non-collinear points.
7. Triangle sum is greater than 180 but less than 570.
A triangle can have more than 1 right or obtuse angle.
The angle measures of an equiangular triangle can vary
8. •The sum of its interior angle is greater than 180 but less than 570.
•A triangle can have more than 1 right or obtuse angle.
•The angle measures of an equiangular triangle can vary.
9. Stereographic
projection
Stereographic projection is a method of
mapping points on the surface of a sphere to
points on a plane. It is a conformal map that
preserves angles, making it useful in various
applications.
11. Spherical trigonometry
Spherical trigonometry is a branch of trigonometry that deals specifically with triangles on the surface of a
sphere. It includes the study of spherical angles, spherical Law of Sines, spherical Law of Cosines, and other
related formulas.
Spherical trigonometry is of great importance for calculations in astronomy, geodesy, and navigation.
