01K = 0
Euclidean
Flat space. One parallel through an external point. Geodesic triangles sum to 180°.
Curvature · geodesics · parallelism · global structure
Non-Euclidean Geometry
Euclidean geometry is one possible geometry, not the definition of geometry itself. Change the curvature or the parallel postulate and familiar statements about lines, triangles, distance, and global shape change with it.
The fifth-postulate fork
Geometry changes when “straight” and “parallel” are allowed to live on curved spaces.
In Euclidean geometry, through a point outside a line there is exactly one parallel line. Hyperbolic geometry allows infinitely many nonintersecting geodesics through that point, while elliptic geometry has no global parallels because its geodesics eventually meet.
Core distinction
A geodesic is the locally straightest path permitted by the geometry. On a sphere, great circles play that role; in hyperbolic space, geodesics follow the metric of negatively curved space rather than ordinary straight lines drawn on a flat sheet.
local straightness ≠ global Euclidean behavior
Curvature comparator
One local question, three different global geometries.
Pick a constant-curvature model and compare geodesics, parallel behavior, and triangle angle sums. Euclidean geometry appears as the zero-curvature middle case.
Constant-curvature relation
α + β + γ − π = K · A
For a geodesic triangle on a constant-curvature surface, the angle excess or deficit tracks curvature and area. Positive K adds angle; negative K removes it.
K > 0 · Positive curvature
α + β + γ > 180°
Geodesics bend toward one another on a sphere-like surface.
Parallel behavior
No globally parallel geodesics in the complete elliptic model
Triangle test
α + β + γ > 180°
schematic geodesic comparison · not to scale
Three constant-curvature models
The sign of curvature changes familiar geometric rules.
These are idealized model geometries. More general surfaces can have curvature that varies from point to point.
02K > 0
Elliptic / spherical
Positive curvature. Great-circle geodesics eventually meet. Geodesic triangles have angle excess.
03K < 0
Hyperbolic
Negative curvature. Infinitely many nonintersecting geodesics can pass through the external point. Triangles have angle deficit.