Compare Euclidean and Hyperbolic Geometry at a Calculus leve

Compare Euclidean and Hyperbolic Geometry at a Calculus level. Your explanation must include Calculus examples.

Solution

Euclidean geometry is the geometry which is related to geometrical studies with respect to flat planes in 3d and straight line axes in 2d while hyperbolic geometry is a geometry which related to study of geometrical constructions and studies in a hyperbolic plane instead of a flat plane. For example a straight line will appear straight in Euclidean geometry but in a hyperbolic geometry, it will appear as a semicircle in upper or lower half plane. Please note that in hyperbolic plane reference it will still appear straight.

The sum of the measures of the angles of any triangle is less than 180° if the geometry is hyperbolic, equal to 180° if the geometry is Euclidean,

Euclidean and non-Euclidean geometries naturally have many similar properties, namely those which do not depend upon the nature of parallelism.

In Euclidean geometry the lines remain at a constant distance from each other (meaning that a line drawn perpendicular to one line at any point will intersect the other line and the length of the line segment joining the points of intersection remains constant) and are known as parallels.

In hyperbolic geometry they \"curve away\" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels.