The three types of geometry revolve around Euclid's fifth postulate, the parallel postulate. This postulate states that, within a two-dimensional plane, for any given lineℓa point Am which is not onℓ, there is exactly one line through A that does not intersect ℓ.
TYPE ONE:
Euclidean- Every line has only one other line parallel to it. Euclidean is the most used and commonly taught in school.
TYPE TWO:
Elliptic- When given a line L and a point P outside of a line, there are no lines parallel to the line L that crosses through point P.
TYPE THREE:
Hyperbolic- Every line has many distinct parallel lines. There is always atleast one parallel line.
Key Differences:
- In Elliptic there are no parallel lines while Euclidean has one and Hyperbolic has many.
- The sum of the angles of a triangle equal 180 in Euclidean geometry, but in Elliptic the angles equal to slightly more than 180.
- In Euclidean, the lines stay at a constant distance away from each other even when extended to infinity, and are known as parallels.
- In Hyperbolic they "curve away" from each other, increasing in distance as you move farther from the point of intersection with the common perpendicular. These are sometimes called ultraparallels.
- In Elliptic, they "curve towards" each other and intersect eventually.
- In Elliptic, the Pythagorean Theorem does not work.
