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 at point generates at point the potential (omitting physical constants) The equipotential surface for the potential value is the implicit surface which is a sphere with center at point . The potential of point charges is represented by For the picture the four charges equal 1 and are located at the points . The displayed surface is the equipotential surface (implicit surface) . 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 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 (in the diagram: a constant distance product surface and a torus) one defines new surfaces using the design parameter : In the diagram the design parameter is successively . Smooth approximations of several implicit surfaces -surfaces can be used to approximate any given smooth and bounded object in 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 . Then, the approximating object is defined by the polynomial where stands for the blending parameter that controls the approximating error. Analogously to the smooth approximation with implicit curves, the equation represents for suitable parameters smooth approximations of three intersecting tori with equations (In the diagram the parameters are ) 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,