 For example, on a closed curve formula_1 of length , for each point of the curve there is a unique other point of the curve (called the antipodal of ) at distance from .A compact surface fills a closed curve if its border (also called boundary, denoted ) is the curve .The filling is said isometric if for any two points of the boundary curve , the distance between them along is the same (not less) than the distance along the boundary.