But it has been very fruitful to consider what might happen if there were no such lines, or perhaps more than one such line. The first case is called hyperbolic and the second is called elliptic .
Great circles provide the shortest traveling distance from one point to another.
For a simple hyperbolic situation, let the points be all the points inside a circle, and let lines be the familiar straight-line segments forming the chords of a circle. Through every pair of points there is exactly one line. But given any point P not on a chord L , there are many chords through P that do not intersect L . Maurits C. Escher was a Dutch artist. He was influenced early in his career by the patterns made by tiles on ancient Spanish buildings.