Implicit surface

implicit surface torus (R=40, a=15)

implicit surface of genus 2

implicit non algebraic surface (wineglas)

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 unsually 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:

a plane $x+2y-3z+1=0$ ,
a sphere $x^2+y^2+z^2-4=0$ ,
a 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 wine glas).

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 for the computation of essential geometric features of the 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 the visualization of an implicit surface is difficult, they provide rather 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

$(F_x(x_0,y_0,z_0),F_y(x_0,y_0,z_0),F_z(x_0,y_0,z_0))\ne (0,0,0)$ ,

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

\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

is a normal vector.

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 premitives.

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 (e.g.) $4$ 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 are 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 displayd.

metamorphoses between two implicit surfaces (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)

PovRay-image (central projection) of an approximation of three tori

Smooth approximationen 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: $R=1, \, a=0.2, \, c=0.01$ )

PovRay-Bild: metamorphoses between a sphere and a constant distance product surface (6 points)

Visualization of implicit surfaces

The visualization of implicit surfaces needs great efforts. Essentially there are two ideas 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

Thorpe: Elementary Topics in Differential Geometry, Springer-Verlag, New York, 1979, ISBN 0-387-90357-7