7.1 Axioms and basic properties of Elliptic Geometry

Elliptic geometry challenges our Euclidean assumptions. It's a world where parallel lines don't exist, and all lines eventually meet. Picture a sphere where great circles act as "straight" lines, always intersecting at two points.

In this geometric realm, triangles have angle sums exceeding 180°, and their areas relate directly to this excess. Rectangles and squares? They're impossible here. It's a mind-bending shift from our flat-plane thinking.

Axioms and Basic Properties of Elliptic Geometry

Axioms of elliptic geometry

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• Elliptic geometry founded on axioms differing from Euclidean geometry
  • Parallel postulate replaced by axiom stating given a line $l$ and a point $P$ not on $l$, no lines through $P$ parallel to $l$
  • Other axioms remain the same such as existence of unique line through two distinct points
• Sphere surface serves as model for elliptic geometry
  • Lines represented by great circles on the sphere (equator, meridians)
  • Two lines always intersect at exactly one point on the sphere

Proofs in elliptic geometry

• Triangle angle sum always exceeds $180^\circ$ in elliptic geometry
  • Proven using axioms and properties of great circles on a sphere
• Triangle area proportional to excess of angle sum over $180^\circ$
  • Gauss-Bonnet formula derived from axioms relates area to angle sum
• Rectangles and squares do not exist in elliptic geometry
  • Follows from triangle angle sum always greater than $180^\circ$

Properties of lines and angles

• Lines are great circles on a sphere
  • Great circles formed by intersection of sphere with plane through its center (slicing sphere in half)
  • Any two distinct points on sphere determine unique great circle
• Angles measured by dihedral angle between planes forming great circles
  • Angle between lines equals angle between tangent vectors at intersection point
• Parallel lines nonexistent in elliptic geometry
  • Any two distinct great circles on sphere intersect at exactly one point

Antipodal points on spheres

• Antipodal points are diametrically opposite pairs on a sphere
  • For point $P$ on sphere, antipodal point $P'$ obtained by drawing line through sphere center and $P$, finding intersection on opposite side
• Antipodal points identified as single point in elliptic geometry
  • Results in lines (great circles) closing back on themselves to form loops
  • Guarantees any two lines intersect at exactly one point
• Identifying antipodal points impacts properties of figures
  • Triangles have maximum area equal to one-eighth of sphere surface area