The geometry we all are familiar with is the Euclidean Space. There’s no curvature in this space— it is simple to be understood. This is the space we can see in most 3D games, or in a 3D modeling software.
Given a point in a Euclidean space at the coordinate and another point at , the distance between the two points, is easily found from the Pythagorean theorem:
Fig 03.01 Euclidean Distance
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In the case of Euclidean spaces with N dimensions, we can generalize the equation to:
Minkowski Space
In special relativity, we studied the Minkowski Space: a 4-Dimensional space with 3 spatial axes and 1 temporal axis. We learnt of the proper time , which is in essence, the distance between two points in Minkowski space. It can be found from the frame of reference in which the object of interest is stationary.
So we shall define the distance, , to be
Other Spaces
A spherical surface is a 2D non-Euclidean space. The distance between two points in differential form is given by:
Fig 03.02 Spherical Surface
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There is a way we can generalize this “distance-finding formula”. What we are looking for is known as [Differential Geometry](Differential Geometry Pt.1)