Implicit surface

Article summary

In mathematics, an implicit surface is a surface in Euclidean space defined by an equation : F(x,y,z)=0. An implicit surface is the set of zeros of a function of three variables. Implicit means that the equation is not solved for or or . The graph of a function is usually described by an equation z=f(x,y) and is called an explicit representation. The third essential description of a surface is the parametric one: (x(s,t),y(s,t), z(s,t)) , where the -, - and -coordinates of surface points are represented by three functions x(s,t)\, , y(s,t)\, , z(s,t) depending on common parameters s,t . Generally the change of representations is simple only when the explicit representation z=f(x,y) is given: z-f(x,y)=0 (implicit), (s,t,f(s,t)) (parametric). Examples: #The plane x+2y-3z+1=0. #The sphere x^2+y^2+z^2-4=0. #The torus (x^2+y^2+z^2+R^2-a^2)^2-4R^2(x^2+y^2)=0. #A surface of genus 2: 2y(y^2-3x^2)(1-z^2)+(x^2+y^2)^2-(9z^2-1)(1-z^2)=0 (see diagram). #The surface of revolution x^2+y^2-(\ln(z+3.2))^2-0.02=0 (see diagram wineglass). For a plane, a sphere, and a torus there exist simple parametric representations. This is not true for the fourth example. The implicit function theorem describes conditions under which an equation F(x,y,z)=0 can be solved (at least implicitly) for , or . But in general the solution may not be made explicit.…

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