Implicit surface

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

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

Examples: For a plane, a sphere, and a torus there exist simple parametric representations.

The implicit function theorem describes conditions under which an equation

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.

is polynomial in x, y and z, the surface is called algebraic.

Throughout the following considerations the implicit surface is represented by an equation

The equation of the tangent plane at a regular point

is and a normal vector is In order to keep the formula simple the arguments

are omitted: is the normal curvature of the surface at a regular point for the unit tangent direction

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.

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.

The electrical potential of a point charge

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

(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

-surfaces [1] can be used to approximate any given smooth and bounded object in

In other words, we can design any smooth object with a single algebraic surface.

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

[3] 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.

[4] The intersection points can be approximated by sphere tracing, using a signed distance function to find the distance to the surface.

[5] Open-source or free software supporting algebraic implicit surface modelling: