Google Image Result for. Euclidean space. This article is about Euclidean spaces of all dimensions. For 3-dimensional Euclidean space, see 3-dimensional space. A sphere, the most perfect spatial shape according to Pythagoreans, also is an important concept in modern understanding of Euclidean spaces Every point in three-dimensional Euclidean space is determined by three coordinates. Intuitive overview[edit] In order to make all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation, and rotation for a mathematically described space. Even when used in physical theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments, and so on. A purely mathematical definition of Euclidean space also ignores questions of units of length and other physical dimensions: the distance in a "mathematical" space is a number, not something expressed in inches or metres.
Euclidean structure[edit] Distance[edit] Angle[edit] Parametric surface. A parametric surface is a surface in the Euclidean space R3 which is defined by a parametric equation with two parameters Parametric representation is the most general way to specify a surface. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization. Examples[edit] The simplest type of parametric surfaces is given by the graphs of functions of two variables: Surfaces of revolution give another important class of surfaces that can be easily parametrized. If the graph z = f(x), a ≤ x ≤ b is rotated about the z-axis then the resulting surface has a parametrization Using the spherical coordinates, the unit sphere can be parameterized by is invertible. where and.
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