EUCLIDEAN VS NON-EUCLIDEAN GEOMETRY

Euclid
"The Father of Geometry"
C. 325-265 B.C.

Euclid is the most prominent mathematician of antiquity, and is commonly referred to as the "Father of Geometry". Though very little is known about his life except that he taught in Alexandria, Egypt, his treatise on geometry has endured the centuries and has formed the very basis for the geometry we use today.Euclid's text Elements is famously known as the first systematic discussion of geometry. Although many of Euclid's postulates had been developed by others prior to his book, Euclid is credited with developing the first comprehensive deductive system.

EUCLID'S 5 POSTULATES

• You can draw a straight line from any two points
• Straight lines can be extended forever in two directions.
• If given a line segment you can draw a circle having the segment as the radius and the endpoint as the center of the circle.
• Every right angle is congruent.
• Two lines are intersected to a third sum of the inner angle but there will be less than two right angles.

Euclidean lines:
You can draw a straight line between any two points.

Euclidean Perpendicular Lines:
Lines are perpendicular if they meet at right angles.

Euclidean Triangles:
The sum of the interior angles of any triangle will always add up to 180 degrees.

Spherical Geometry is the study of curved surfaces. Consider what would happen if instead of working on the Euclidean flat piece of paper, you work on a curved surface, such as a sphere. The study of spherical geometry has a direct connection to our daily lives since we live on a curved surface.

Spherical Lines:
Straight lines do not exist in Spherical Geometry, because as you start to draw a straight line on a spherical object, the line will immediately begin to curve, following the surface of the sphere.

SPHERICAL PERPENDICULAR LINES

Spherical Triangles:
3-sided figures that are enclosed by the sides of the circle, with angles that add up to more than 180 degrees.

Hyperbolic Geometry is the study of a saddle shaped space.

Hyperbolic Lines:
Hyperbolic lines are drawn in the Poincaré's Half-Plane Model which is a model for hyperbolic geometry in which a line is represented as an arc of a circle whose lines are perpendicular to its boundaries. Hyperbolic lines are the semicircumferences centered at a point of the boundary line and radius

Hyperbolic Perpendicular Lines:
Perpendicular Lines in the hyperbolic plane will either appear as lines perpendicular to the edge of the half-plane or as circles whose centers lie on the edge of the half-plane. Thus a hyperbolic line is either a circle perpendicular to the real axis or a vertical line.

A hyperbolic triangle is just three points connected by hyperbolic line segments.Since the hyperbolic line segments are typically curved, the angles of a hyperbolic triangle add up to less than 180 degrees. Furthermore, similar triangles do not exist in hyperbolic geometry.

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