Editing Implicit surface (section)

== 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. 
 
[[File:Epotfl-4ladung.svg|thumb|Equipotential surface of 4 point charges]]

=== Equipotential surface of point charges ===
{{Main article|Equipotential}}

The electrical potential of a point charge  <math>q_i</math> at point <math>\mathbf p_i=(x_i,y_i,z_i)</math> generates at point <math> \mathbf p=(x,y,z)</math> the potential (omitting physical constants)
: <math>F_i(x,y,z)=\frac{q_i}{\|\mathbf p -\mathbf p_i\|}.</math>
The equipotential surface for the potential value <math>c</math> is the implicit surface  <math> F_i(x,y,z)-c=0 </math> which is a sphere with center at point <math>\mathbf p_i</math>.

The potential of <math>4</math> point charges is represented by 
: <math>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\|}.</math>

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

=== 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

: <math>
\begin{align}
F(x,y,z) = {} &  \sqrt{(x-1)^2+y^2+z^2}\cdot \sqrt{(x+1)^2+y^2+z^2} \\
& \quad \cdot \sqrt{x^2+(y-1)^2+z^2}\cdot\sqrt{x^2+(y+1)^2+z^2}
\end{align}
</math>

the constant distance product surface  <math>F(x,y,z)-1.1=0</math> is displayed.
[[File:Metamorphose-torus-4ppfl.svg|400px|thumb|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 ''metamorphosis'' or ''[[homotopy]]'' of implicit surfaces:

For two implicit surfaces <math>F_0(x,y,z)=0, F_1(x,y,z)=0</math> (in the diagram: a constant distance product surface and a torus) one defines new surfaces using the design parameter <math> \mu \in [0,1]</math>:
:<math>F_\mu (x,y,z)=\mu F_1(x,y,z)+(1-\mu)\,F_0(x,y,z)=0</math> 
In the diagram the design parameter is successively <math>\mu=0, \, 0.33, \, 0.66, \, 1</math> .

[[File:Approx-3tori.svg|240px|thumb|Approximation of three tori ([[parallel projection]])]]
[[File:Approx-3tori-pov.png|280px|thumb|[[POV-Ray]] image (central projection) of an approximation of three tori.]]

=== Smooth approximations of several implicit surfaces ===

<math>\Pi</math>-surfaces <ref name="RaposoGomes2019">{{cite news|author1=Adriano N. Raposo|author2=Abel J.P. Gomes|title=Pi-surfaces: products of implicit surfaces towards constructive composition of 3D objects|date=2019|publisher=WSCG 2019 27. International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision|arxiv=1906.06751}}</ref> can be used to approximate any given smooth and bounded object in <math>R^3</math> 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 <math>f_i\in\mathbb{R}[x_1,\ldots,x_n](i=1,\ldots,k)</math>. Then, the approximating object is defined by the polynomial
:<math>F(x,y,z) = \prod_i f_i(x,y,z) - r</math><ref name="RaposoGomes2019"/>
where <math>r\in\mathbb{R}</math> stands for the blending parameter that controls the approximating error.

Analogously to the smooth approximation with implicit curves, the equation
:<math>F(x,y,z)=F_1(x,y,z)\cdot F_2(x,y,z)\cdot F_3(x,y,z) -r= 0</math>
represents for suitable parameters <math>c</math> smooth approximations of three intersecting tori with equations

: <math>
\begin{align}
F_1=(x^2+y^2+z^2+R^2-a^2)^2-4R^2(x^2+y^2)=0, \\[3pt]
F_2=(x^2+y^2+z^2+R^2-a^2)^2-4R^2(x^2+z^2)=0, \\[3pt]
F_3=(x^2+y^2+z^2+R^2-a^2)^2-4R^2(y^2+z^2)=0.
\end{align}
</math>

(In the diagram the parameters are <math> R=1, \, a=0.2, \, r=0.01.</math>)
[[File:Metamorphose-kugel-6pfl.png|400px|thumb|POV-Ray image: metamorphoses between a sphere and a constant distance product surface (6 points).]]