Non-Euclidean & Spherical Geometry Cheat Sheet

Key Facts

• In Euclidean geometry, exactly one line through a point not on a given line is parallel to the given line.
• In spherical geometry, there are no parallel lines because any two great circles intersect at two opposite points.
• In hyperbolic geometry, infinitely many lines through a point not on a given line do not intersect the given line.
• For a spherical triangle on a sphere of radius $R$, the area is $A = R^2E$, where $E = A_1 + A_2 + A_3 - \pi$ is the spherical excess in radians.
• For a spherical triangle, the angle sum satisfies $A_1 + A_2 + A_3 > \pi$, or greater than $180^{\circ}$.
• For a hyperbolic triangle with curvature $K = -\frac{1}{R^2}$, the area is $A = R^2D$, where $D = \pi - (A_1 + A_2 + A_3)$ is the angle defect in radians.
• For a hyperbolic triangle, the angle sum satisfies $A_1 + A_2 + A_3 < \pi$, or less than $180^{\circ}$.
• Curvature helps identify the geometry type: spherical geometry has positive curvature, Euclidean geometry has zero curvature, and hyperbolic geometry has negative curvature.

Vocabulary

Parallel Postulate

The rule describing how many lines through a point not on a given line can be parallel to the given line.

Great Circle

A circle on a sphere whose center is the center of the sphere, such as the equator or a line of longitude.

Geodesic

The shortest path between nearby points on a surface, acting like a straight line within that geometry.

Spherical Excess

The amount $E = A_1 + A_2 + A_3 - \pi$ by which a spherical triangle’s angle sum exceeds $180^{\circ}$.

Angle Defect

The amount $D = \pi - (A_1 + A_2 + A_3)$ by which a hyperbolic triangle’s angle sum is less than $180^{\circ}$.

Curvature

A measure of how a space bends, with positive, zero, or negative curvature leading to spherical, Euclidean, or hyperbolic geometry.