Tangent plane and normal vector

A surface point ${\displaystyle (x_{0},y_{0},z_{0})}$ is called regular if and only if the gradient of ${\displaystyle F}$ at ${\displaystyle (x_{0},y_{0},z_{0})}$ is not the zero vector ${\displaystyle (0,0,0)}$, meaning

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

If the surface point ${\displaystyle (x_{0},y_{0},z_{0})}$ is not regular, it is called singular.

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

${\displaystyle F_{x}(x_{0},y_{0},z_{0})(x-x_{0})+F_{y}(x_{0},y_{0},z_{0})(y-y_{0})+F_{z}(x_{0},y_{0},z_{0})(z-z_{0})=0,}$

and a normal vector is

${\displaystyle \mathbf {n} (x_{0},y_{0},z_{0})=(F_{x}(x_{0},y_{0},z_{0}),F_{y}(x_{0},y_{0},z_{0}),F_{z}(x_{0},y_{0},z_{0}))^{T}.}$

Normal curvature

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

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

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

Equipotential surface of point charges

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

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

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

The potential of ${\displaystyle 4}$ point charges is represented by

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

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

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 (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

${\displaystyle {\begin{aligned}F(x,y,z)={}&{\Big (}{\sqrt {(x-1)^{2}+y^{2}+z^{2}}}\cdot {\sqrt {(x+1)^{2}+y^{2}+z^{2}}}\\&\qquad \cdot {\sqrt {x^{2}+(y-1)^{2}+z^{2}}}\cdot {\sqrt {x^{2}+(y+1)^{2}+z^{2}}}{\Big )}\end{aligned}}}$

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

Metamorphoses of implicit surfaces

A further simple method to generate new implicit surfaces is called metamorphosis of implicit surfaces:

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

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

In the diagram the design parameter is successively ${\displaystyle \mu =0,\,0.33,\,0.66,\,1}$ .

Smooth approximations of several implicit surfaces

${\displaystyle \Pi }$-surfaces ^[1] can be used to approximate any given smooth and bounded object in ${\displaystyle R^{3}}$ 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 ${\displaystyle f_{i}\in \mathbb {R} [x_{1},\ldots ,x_{n}](i=1,\ldots ,k)}$. Then, the approximating object is defined by the polynomial

${\displaystyle F(x,y,z)=\prod _{i}f_{i}(x,y,z)-r}$^[1]

where ${\displaystyle r\in \mathbb {R} }$ stands for the blending parameter that controls the approximating error.

Analogously to the smooth approximation with implicit curves, the equation

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

represents for suitable parameters ${\displaystyle c}$ smooth approximations of three intersecting tori with equations

${\displaystyle {\begin{aligned}F_{1}=(x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(x^{2}+y^{2})=0,\\[3pt]F_{2}=(x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(x^{2}+z^{2})=0,\\[3pt]F_{3}=(x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(y^{2}+z^{2})=0.\end{aligned}}}$

(In the diagram the parameters are ${\displaystyle R=1,\,a=0.2,\,r=0.01.}$)