The representation of the tangent plane in the form of a square with a given length of the side at any point on the surface.

The equation of the tangent plane to the surface at a given point is obtained from the condition that the tangent plane is perpendicular to the normal vector. With the aid of any auxiliary point not lying on this normal to the surface, we define the direction on the tangent plane. From the given point in this direction, we lay off segments equal to half the length of the side of our square and with the help of these segments we construct the square itself, lying on the tangent plane with the center at a given point.

An examples of constructing tangent planes at points of the same intersection line for two surfaces.
Tangent_plane.mw

The distance from the point to the surface easily calculated using the NLPSolve of Optimization package. If the point is not special, we will find for it a point on the surface, the distance between these two points is the shortest between the selected point and the surface.
Two examples:  the implicit surface and the parametric surface.
To test, we restore the normals from the  calculated  points (red) by using analytical equations.
DISTANCE_TO_SURFACE.mw

The Möbius strip  Mobius_strip_rolling.mw

Variants :


The line and the curve on the surface.

Parametric equation of second-order curve in 3d. Draghilev method.
PLAN_CURVE_3d_1.mw
Examples:
x1^2+x1*x3+13*x2^2+x3-1=0;
x1+x2+x3=0;

 x1^2+0.1*x2^2+x3^2-9=0;
 x1+3*x3+1=0;

 x1^2-0.1*x2^2+x3^2-9=0;
 x1+3*x3+1=0;

Parametric equation of a circle in 3d by three points. Draghilev method.

CIRCLE_3_POINTS_geom3d_2.mw