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#1 2019-01-08 15:07:17

Βεν Γ. Κυθισ
Registered: 2018-10-09
Posts: 17

Possibly cool new geometry

I have come up with a geometry that is similar to spherical geometry. I defined it as being like Euclidean geometry except for that parallel lines must also intersect, which makes all straight lines except for perpendicular lines intersect in at least two places, perpendicular lines intersect an unknown amount of times. The only differences from spherical geometry I know for sure are:

  • The plane is infinitely extending and not on a sphere.

  • Antipodal points don't exist.

  • Lines have at least one point at infinity.

  • The natural units of length and area are fixed, but currently unknown; it is unknown if a natural unit of angle measurement exists.

  • Polar lines don't exist.

  • The area of a triangle is unbounded.

A representation I came up with is where lines seem to bend in parabolic shapes away from some chosen reference point, with the bending getting more extreme the farther the line is from the point; lines that go through the reference point appear straight, and how extremely a line bends away from the point is defined by the natural unit of length. There are problems with this representation though:

  • If you have two distinct lines, there will be two points at which they will intersect, but if you move the reference point the intersection points will also move.

  • A pair of perpendicular lines with at least one of the lines on the reference point intersect in one place while perpendicular lines off the reference point intersect in two places.

  • Line segments move around in unknown ways when the reference point moves.

  • The angle of intersection points might change with the position of the reference point but I don't know for sure.


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