The former is a curve in three-dimensions, while the latter is 'straight'.

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This is not the same as the shortest line-segment joining the two points in regular three-dimensional space. LinesĪs in plane geometry, a line-segment on a sphere is the shortest 'line' connecting two points - over the surface of the sphere. To emphasise this, we sometimes refer to 'the surface of the sphere'. Notice that by this definition, the sphere is just the 'surface' - it is a hollow sphere. For convenience, a radius of 1 is often used. The sphere itself now consists of all points satisfying the equation: The three variables are usually x, y and z, so a point in a 3-dimensional space would be notated ( x, y, z), for the relevant values of x, y and z.įor ease of calculations, we position our sphere so that it is centred on the 'origin' (the point where all the variables are zero - (0,0,0)). Many applications of spherical geometry will utilise a co-ordinate system - three variables which can uniquely identify any point in the given space. In plane geometry they are intrinsically obvious - not so on the sphere! The Sphereīefore anything else, we need to be certain we understand what is meant by the sphere. In order to make sure we understand each other, we need to define a few concepts. Things are not completely straightforward when we are dealing with spherical geometry. Although the Earth is not exactly spherical, geometry on a sphere is a good starting point for many real-world applications. However, the world we inhabit is not flat (although this theory was popular at some point in our history), but round. Most geometry is plane geometry - it takes place on a flat surface, and is therefore easy to solve using diagrams.

Spherical geometry khan academy