Non-Euclidean Geometry Introduction Cheat Sheet

Key Facts

• In Euclidean geometry, through a point $P$ not on a line $\ell$, there is exactly one line parallel to $\ell$.
• In spherical geometry, geodesics are great circles, and any two great circles intersect, so there are no parallel lines.
• In hyperbolic geometry, through a point $P$ not on a line $\ell$, there are infinitely many lines that do not meet $\ell$.
• Triangle angle sums compare as Euclidean: $\alpha + \beta + \gamma = \pi$, spherical: $\alpha + \beta + \gamma > \pi$, and hyperbolic: $\alpha + \beta + \gamma < \pi$.
• For a sphere of radius $R$, the area of a spherical triangle is $A = R^2(\alpha + \beta + \gamma - \pi)$, using radians.
• For a hyperbolic plane with curvature $K = -\frac{1}{R^2}$, the area of a hyperbolic triangle is $A = R^2(\pi - \alpha - \beta - \gamma)$, using radians.
• Curvature distinguishes the models: flat Euclidean geometry has $K = 0$, spherical geometry has $K > 0$, and hyperbolic geometry has $K < 0$.
• In the Poincaré disk model, hyperbolic geodesics appear as diameters or circular arcs that meet the boundary circle at right angles.

Vocabulary

Non-Euclidean geometry

A geometry that changes Euclid’s parallel postulate while keeping a logical system of points, lines, and measurements.

Geodesic

The shortest path between nearby points in a geometry, serving the role of a straight line.

Curvature

A measure of how a space bends, with $K = 0$ for flat geometry, $K > 0$ for spherical geometry, and $K < 0$ for hyperbolic geometry.

Spherical geometry

A geometry on the surface of a sphere where lines are great circles and triangle angle sums are greater than $\pi$.

Hyperbolic geometry

A geometry with negative curvature where triangle angle sums are less than $\pi$ and many parallels can pass through one point.

Angle defect

The amount $\pi - \alpha - \beta - \gamma$ by which a hyperbolic triangle’s angle sum is less than $\pi$.