![]() | ||
In mathematics an implicit surface is a surface in Euclidean space defined by an equation
Contents
- Formulas
- Tangent plane and normal vector
- Normal curvature
- Applications of implicit surfaces
- Equipotential surface of point charges
- Constant distance product surface
- Metamorphoses of implicit surfaces
- Smooth approximations of several implicit surfaces
- Visualization of implicit surfaces
- References
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
Examples:
- plane
x + 2 y − 3 z + 1 = 0 . - sphere
x 2 + y 2 + z 2 − 4 = 0 . - torus
( x 2 + y 2 + z 2 + R 2 − a 2 ) 2 − 4 R 2 ( x 2 + y 2 ) = 0 . - Surface of genus 2:
2 y ( y 2 − 3 x 2 ) ( 1 − z 2 ) + ( x 2 + y 2 ) 2 − ( 9 z 2 − 1 ) ( 1 − z 2 ) = 0 (s. picture). - Surface of revolution
x 2 + y 2 − ( ln ( z + 3.2 ) ) 2 − 0.02 = 0 (s. picture wineglas).
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
Despite difficulty of visualization, implicit surfaces provide relatively 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
Tangent plane and normal vector
A surface point
otherwise the point is singular.
The equation of the tangent plane at a regular point
and a normal vector is
Normal curvature
In order to keep the formula simple the arguments
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.
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
The equipotential surface for the potential value
The potential of
For the picture the four charges equal 1 and are located at the points
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 (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 picture metamorphoses the upper left surface is generated by this rule: With
the constant distance product surface
Metamorphoses of implicit surfaces
A further simple method to generate new implicit surfaces is called metamorphoses of implicit surfaces:
For two implicit surfaces
For the picture the design parameter is:
Smooth approximations of several implicit surfaces
Analogously to the smooth approximation with implicit curves the equation
represents for suitable parameters
(For the picture the parameters are:
Visualization of implicit surfaces
The visualization of implicit surfaces requires great effort. 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.