Implicit surface

Implicit surface torus (R=40, a=15).

Implicit surface of genus 2.

Implicit non-algebraic surface (wineglass).

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 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 $z=f(x,y)$ 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 x-, y- and z-coordinates of surface points are represented by three functions $x(s,t)\,,y(s,t)\,,z(s,t)$ depending on common parameters $s,t$ . The change of representations is usually 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:

plane $x+2y-3z+1=0$ .
sphere $x^{2}+y^{2}+z^{2}-4=0$ .
torus $(x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(x^{2}+y^{2})=0$ .
Surface of genus 2: $2y(y^{2}-3x^{2})(1-z^{2})+(x^{2}+y^{2})^{2}-(9z^{2}-1)(1-z^{2})=0$ (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 $F(x,y,z)=0$ can be solved (theoretically) for x, y or z. But in general the solution may not be feasible. This theorem is the key to the computation of essential geometric features of a surface: tangent planes, surface normals, curvatures (s. below). But they have an essential drawback: their visualization is difficult.

F(x,y,z)

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 (s. below) interesting surfaces.

Formulas

Throughout the following considerations the implicit surface is represented by an equation $F(x,y,z)=0$ where function $F$ meets the necessary conditions of differentiability. The partial derivatives of $F$ are $F_{x},...,F_{{xx}},...$ .

Tangent plane and normal vector

A surface point $(x_{0},y_{0},z_{0})$ is called regular, if

(F_{x}(x_{0},y_{0},z_{0}),F_{y}(x_{0},y_{0},z_{0}),F_{z}(x_{0},y_{0},z_{0}))\neq (0,0,0)

otherwise the point is singular.

The equation of the tangent plane at a regular point $(x_{0},y_{0},z_{0})$ is

F_x(x_0,y_0,z_0)(x-x_0)+F_y(x_0,y_0,z_0)(y-y_0)+F_z(x_0,y_0,z_0)(z-z_0)=0,

and a normal vector is

\mathbf n(x_0,y_0,z_0)=(F_x(x_0,y_0,z_0),F_y(x_0,y_0,z_0),F_z(x_0,y_0,z_0))^T.

Normal curvature

In order to keep the formula simple the arguments $(x_{0},y_{0},z_{0})$ are omitted:

$\kappa _{n}={\frac {{\mathbf v}^{\top }H_{F}{\mathbf v}}{\|\operatorname {grad}F\|}}$

is the normal curvature of the surface at a regular point for the unit tangent direction ${\mathbf v}$ . $H_{F}$ is the Hessian matrix of $F$ (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 4 point charges

Equipotential surface of point charges

The electrical potential of a point charge $q_{i}$ at point ${\mathbf p}_{i}=(x_{i},y_{i},z_{i})$ generates at point ${\mathbf p}=(x,y,z)$ the potential (omitting physical constants)

F_{i}(x,y,z)={\frac {q_{i}}{\|{\mathbf p}-{\mathbf p}_{i}\|}}

The equipotential surface for the potential value $c$ is the implicit surface $F_{i}(x,y,z)-c=0$ which is a sphere with center at point ${\mathbf p}_{i}$ .

The potential of $4$ point charges is represented by

F(x,y,z)={\frac {q_{1}}{\|{\mathbf p}-{\mathbf p}_{1}\|}}+{\frac {q_{2}}{\|{\mathbf p}-{\mathbf p}_{2}\|}}+{\frac {q_{3}}{\|{\mathbf p}-{\mathbf p}_{3}\|}}+{\frac {q_{4}}{\|{\mathbf p}-{\mathbf p}_{4}\|}}

For the picture the four charges equal 1 and are located at the points $(\pm 1,\pm 1,0)$ . The displayed surface is the equipotential surface (implicit surface) $F(x,y,z)-2.8=0$ .

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

F(x,y,z)={\sqrt {(x-1)^{2}+y^{2}+z^{2}}}\cdot {\sqrt {(x+1)^{2}+y^{2}+z^{2}}}\cdot {\sqrt {x^{2}+(y-1)^{2}+z^{2}}}\cdot {\sqrt {x^{2}+(y+1)^{2}+z^{2}}}

the constant distance product surface $F(x,y,z)-1.1=0$ is displayed.

Metamorphoses between two implicit surfaces: a torus and a 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 $F_{1}(x,y,z)=0,F_{2}(x,y,z)=0$ (in picture: a constant distance product surface and a torus) one defines new surfaces using the design parameter $\mu \in [0,1]$ :

F(x,y,z)=\mu F_{1}(x,y,z)+(1-\mu )F_{2}(x,y,z)=0

For the picture the design parameter is: $\mu =0,\,0.33,\,0.66,\,1$ .

Approximation of three tori (parallel projection)

POV-Ray image (central projection) of an approximation of three tori.

Smooth approximations of several implicit surfaces

Analogously to the smooth approximation with implicit curves the equation

F(x,y,z)=F_{1}(x,y,z)\cdot F_{2}(x,y,z)\cdot F_{3}(x,y,z)-c=0

represents for suitable parameters $c$ smooth approximations of three intersecting tori with equations

F_{1}=(x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(x^{2}+y^{2})=0

F_{2}=(x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(x^{2}+z^{2})=0

F_{3}=(x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(y^{2}+z^{2})=0

(For the picture the parameters are: $R=1,\,a=0.2,\,c=0.01$ )

POV-Ray image: metamorphoses between a sphere and a constant distance product surface (6 points).

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.

External links

References

Gomes, A., Voiculescu, I., Jorge, J., Wyvill, B., Galbraith, C.: Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms, 2009, Springer-Verlag London, ISBN 978-1-84882-405-8
Thorpe: Elementary Topics in Differential Geometry, Springer-Verlag, New York, 1979, ISBN 0-387-90357-7

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