Implicit surface
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 three 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 formula_2 and is called an "explicit" representation. The third essential description of a surface is the "parametric" one: formula_3, where the "x"-, "y"- and "z"-coordinates of surface points are represented by three functions formula_4 depending on common parameters formula_5. Generally the change of representations  is simple only when the explicit representation formula_2 is given: formula_7 (implicit), formula_8 (parametric). "Examples": 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 formula_14 can be solved (at least implicitly) for "x", "y" or "z". But in general the solution may not be made explicit. This theorem is the key to the computation of essential geometric features of a surface: tangent planes, surface normals, curvatures (see below). But they have an essential drawback: their visualization is difficult . If formula_15 is polynomial in "x", "y" and "z", the surface is called algebraic. Example 5 is "non"-algebraic. Despite difficulty of visualization, implicit surfaces provide relatively simple techniques to generate theoretically (e.g. Steiner surface) and practically (see below) interesting surfaces. Formulas. Throughout the following considerations the implicit surface is represented by an equation formula_14 where function formula_17 meets the necessary conditions of differentiability. The partial derivatives of formula_17 are formula_19. Tangent plane and normal vector. A surface point formula_20 is called regular if and only if the gradient of formula_17 at formula_20 is not the zero vector formula_23, meaning If  the surface point formula_20 is "not" regular, it is called singular. The equation of the tangent plane at a regular point formula_26 is and a "normal vector" is Normal curvature. In order to keep the formula simple the arguments formula_26 are omitted: is the normal curvature of the surface at a regular point for the unit tangent direction formula_31. formula_32 is the Hessian matrix of formula_17 (matrix of the second derivatives). The proof of this formula relies (as in the case of an implicit curve) on the implicit function theorem and the formula for the normal curvature of a parametric  surface. Applications of implicit surfaces. As in the case of implicit curves it is an easy task to generate implicit surfaces with desired shapes by applying algebraic operations (addition, multiplication) on simple primitives. Equipotential surface of point charges. The electrical potential of a point charge formula_34 at point formula_35 generates at point formula_36 the potential (omitting physical constants) The equipotential surface for the potential value formula_38 is the implicit surface formula_39 which is a sphere with center at point formula_40. The potential of formula_41 point charges is represented by For the picture the four charges equal 1 and are located  at the points formula_43. The displayed surface is the equipotential surface (implicit surface) formula_44. Constant distance product surface. A Cassini oval can be defined as the point set for which the product of the distances to two given points is constant (in contrast, for an ellipse the "sum" is constant). In a similar way implicit surfaces can be defined by a constant distance product to several fixed points. In the diagram "metamorphoses" the upper left surface is generated by this rule: With the constant distance product surface formula_46 is displayed. Metamorphoses of implicit surfaces. A further simple method to generate  new implicit surfaces is called "metamorphosis" of implicit surfaces: For two implicit surfaces formula_47 (in the diagram: a constant distance product surface and a torus) one defines new surfaces using the design parameter formula_48: In the diagram the design parameter is successively formula_50 . Smooth approximations of several implicit surfaces. formula_51-surfaces can be used to approximate any given smooth and bounded object in formula_52 whose surface is defined by a single polynomial as a product of subsidiary polynomials. In other words, we can design any smooth object with a single algebraic surface. Let us denote the defining polynomials as formula_53 . Then, the approximating object is defined by the polynomial where formula_55 stands for the blending parameter that controls the approximating error. Analogously to the smooth approximation with implicit curves, the equation represents for suitable parameters formula_38 smooth approximations of three intersecting tori with equations Visualization of implicit surfaces. There are various algorithms for rendering implicit surfaces, including the marching cubes algorithm. Essentially there are two ideas for visualizing an implicit surface: One generates a net of polygons which is visualized (see surface triangulation) and the second relies on ray tracing which determines intersection points of rays with the surface. The  intersection points can be approximated by "sphere tracing", using a signed distance function to find the distance to the surface.