|
In mathematics an implicit surface is a surface in Euclidean space defined by an equation : An implicit surface is the set of zeros of a function of 3 variables. ''Implicit'' means, that the equation is not solved for x or y or z. The graph of a function is usually described by an equation and is called an ''explicit'' representation. The third essential description of a surface is the ''parametric'' one: , where the x-, y- and z-coordinates of surface points are represented by three functions depending on common parameters . The change of representations is unsually simple only, when the explicit representation is given: (implicit), (parametric). ''Examples'': #a plane , #a sphere , #a torus , #surface of genus 2: (s. picture), # surface of revolution (s. picture ''wine glas''). For a plane, a sphere and a torus there exist simple parametric representations. This is not true for the 4. example. The implicit function theorem describes conditions, under which an equation can be solved (theoretically) for x, y or z. But in general the solution may not be feasible. This theorem is the key for the computation of essential geometric features of the surface: tangent planes, surface normals, curvatures (s. below). But they have an essential drawback: their visualization is difficult. :If is polynomial in x,y and z, the surface is called algebraic. :Example 5. is ''non'' algebraic. Despite the visualization of an implicit surface is difficult, they provide rather simple techniques to generate theoretically (e.g. Steiner surface) and practically (s. below) interesting surfaces. == Formulas == Throughout the following considerations the implicit surface is represented by an equation where function meets the necessary conditions of differentiability. The partial derivatives of are . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「implicit surface」の詳細全文を読む スポンサード リンク
|