Elliptic geometry in 3 dimensions is a non-Euclidean geometry where the sum of the angles of a triangle is greater than 180 degrees. It is based on the geometry of a sphere, where lines are represented as great circles and parallel lines do not exist.
In Euclidean geometry, the sum of the angles of a triangle is always 180 degrees and parallel lines never meet. In elliptic geometry, the sum of the angles of a triangle is greater than 180 degrees and parallel lines intersect at the poles of the sphere.
Elliptic geometry in 3 dimensions has applications in navigation, map-making, and astronomy. It is also used in the study of spherical objects, such as planets and stars.
Some key concepts in elliptic geometry in 3 dimensions include great circles, antipodal points, and geodesics. Great circles are the equivalent of straight lines in Euclidean geometry, antipodal points are points on a sphere that are directly opposite each other, and geodesics are the shortest paths between two points on a sphere.
Elliptic geometry is one of three non-Euclidean geometries, along with hyperbolic geometry and Riemannian geometry. These geometries differ from Euclidean geometry in terms of their axioms and the properties of their shapes and spaces. Elliptic geometry and hyperbolic geometry are both curved geometries, while Riemannian geometry is a flat geometry.