# Klein bottle

Structure of a three-dimensional Klein bottle In mathematics, the Klein bottle /ˈklaɪn/ is an example of a non-orientable surface; informally, it is a surface (a two-dimensional manifold) in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. The Klein bottle was first described in 1882 by the German mathematician Felix Klein. Construction This square is a fundamental polygon of the Klein bottle. Note that this is an "abstract" gluing in the sense that trying to realize this in three dimensions results in a self-intersecting Klein bottle. By adding a fourth dimension to the three-dimensional space, the self-intersection can be eliminated. This immersion is useful for visualizing many properties of the Klein bottle. A hand-blown Klein Bottle Dissecting the Klein bottle results in Möbius strips. Properties A mathematician named Klein Thought the Möbius band was divine. Notes Related:  MATH / GEOMETRYGeometric Topology

Polygon Triangulation - Graphics Programming and Theory Modern GPUs have an annoying habbit of only liking to draw triangles. Often when using 3D modeling programs or vector-based drawing applications to produce game art, you'll get instead a soup of arbitrary polygons. This is an issue if you plan to rasterize these polygons. Fortunately, polygons can be decomposed into triangles relatively easily.There are many different ways to decompose polygons into triangles. Typically though you only implement an algorithm that's advanced enough to suit your needs. For example, convex polygons are easier to triangulate than concave ones, and polygons with a hole in the middle of it are a little complicated to get right (and are also beyond the scope of this article: I'll be covering ear clipping at the most in this article. For Convex Polygons This one is by far the easiest. For this case, you can pick any vertex in the polygon and create a triangle fan outward. For Concave Polygons This algorithm is called "ear clipping." Here's the algorithm:

Boy's surface An animation of Boy's surface Boy's surface is discussed (and illustrated) in Jean-Pierre Petit's Le Topologicon.[1] Boy's surface was first parametrized explicitly by Bernard Morin in 1978. See below for another parametrization, discovered by Rob Kusner and Robert Bryant. Boy's surface is one of the two possible immersions of the real projective plane which have only a single triple point. [2] Construction To make a Boy's surface: Start with a sphere. Symmetry of the Boy's surface Model at Oberwolfach Model of a Boy's surface in Oberwolfach The Mathematical Research Institute of Oberwolfach has a large model of a Boy's surface outside the entrance, constructed and donated by Mercedes-Benz in January 1991. Applications Boy's surface can be used in sphere eversion, as a half-way model. Parametrization of Boy's surface A view of the parametrization described here ), let so that where x, y, and z are the desired Cartesian coordinates of a point on the Boy's surface. Let but

Real projective plane In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in R3 passing through the origin. (0, y) ~ (1, 1 − y) for 0 ≤ y ≤ 1 and (x, 0) ~ (1 − x, 1) for 0 ≤ x ≤ 1, as in the leftmost diagram on the right. Examples Projective geometry is not necessarily concerned with curvature and the real projective plane may be twisted up and placed in the Euclidean plane or 3-space in many different ways.[1] Some of the more important examples are described below. The projective sphere Consider a sphere, and let the great circles of the sphere be "lines", and let pairs of antipodal points be "points". The projective hemisphere Boy's surface – an immersion Roman surface Hemi polyhedra .