Editing Implicit surface (section)

=== 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).]]