Editing Implicit surface (section)

== Formulas ==
Throughout the following considerations the implicit surface is represented by an equation
<math>F(x,y,z)=0</math>  where function <math>F</math> meets the necessary conditions of differentiability. The [[partial derivative]]s of
<math>F</math> are <math>F_x,F_y,F_z,F_{xx},\ldots</math>.

=== Tangent plane and normal vector ===
A surface point <math>(x_0, y_0,z_0)</math> is called '''regular''' [[if and only if]] the [[gradient]] of <math>F</math> at <math>(x_0, y_0,z_0)</math> is not the zero vector <math>(0, 0, 0)</math>, meaning
:<math> (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)</math>.
If the surface point <math>(x_0, y_0,z_0)</math> is ''not'' regular, it is called '''singular'''.

The equation of the tangent plane at a regular point <math>(x_0,y_0,z_0)</math> is
:<math>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,</math>
and a ''normal vector'' is 
:<math> \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.</math>

=== Normal curvature ===
In order to keep the formula simple the arguments <math>(x_0,y_0,z_0)</math> are omitted:

: <math>\kappa_n = \frac{\mathbf v^\top H_F\mathbf v}{\|\operatorname{grad} F\|}</math>

is the normal curvature of the surface at a regular point for the unit tangent direction <math> \mathbf v</math>. <math>H_F</math> is the [[Hessian matrix]] of <math>F</math> (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]].