Straight away on Curved Spaces

Abstract

Two points of a Euclidean space can always be connected by the simplest of the curves, the straight line. This curve is unique (it s required), with a constant direction and a minimal length between the two points. The straight line is known to be fundamental in the building of the Euclidean geometry, for the production of all imaginable figures, from the simple triangle to the most complex curved shapes. And when the figures have to be immersed in curved shapes, out of a Euclidean space and without its useful straight line tool, it seems natural to search for a similar fundamental curve possessing its basic properties : uniqueness of the curve, constant direction, minimal length. This curve, known to be a geodesic line, is defined by a differential system whose solutions cannot generally be expressed in a simple way, (let s say as a finite degree polynomial expression), and so leads to a great complexity in the definition of figures belonging to curved spaces. But other approachs exist... The present contribution, based on a conceptual tool for creating and managing curved shapes, the Pascalian Forms, or pForms, (published in Editions de l Esperou / Montpellier / 2004), an attempt to generalize the de Casteljau algorithm, focuses on three cases of "straight lines" drawn on curved shapes : 1) the first case shows how a geodesic can be the solution to a very practical problem : applying a long thin plank on a toroidal roof upon a swimming pool ; 2) the second case shows how an apparently complex spatial curve (the Threefoil Knot Curve) followed by a staircase in the MC Escher style can be simplified by using immersed pSegments ; 3) the third case shows how the natural/organic shapes in the Sagrada Familia Temple dreamed by the catalan architect A. Gaudi have been mastered using ruled surfaces (pSurfaces).