It post can be called a continuation of the theme “Determination of the angles of the manipulator with the help of its mathematical model. Inverse  problem”.
Consider  the use of manipulators as multi-axis CNC machines.
Three-link manipulator with 5 degrees of freedom. In these examples  one of the restrictions on the movement of the manipulator links is that the position of the last link coincides with the normal to the surface along the entire trajectory of the working point movement.
That is, we, as it were, mathematically transform a system with many degrees of freedom to an analog of a lever mechanism with one degree of freedom, so that we can do the necessary work in a convenient to us way.
It seems that this approach is fully applicable directly to multi-axis CNC machines.

(In the texts of the programs, the normalization is carried out with respect to the coordinates of the last point, in order that the lengths of the integration interval coincide with the path length.)
MAN_3_5_for_MP.mw

MAN_3_5_for_MP_TR.mw

On the example of a manipulator with three degrees of freedom.
A mathematical model is created that takes into account degrees of freedom of the manipulator and the trajectory of the movement from the initial point to the final one (in the figure, the ends of the red curve). In the text of the program, these are the equations fi, i = 1..5.
Obviously, the straight line could be the simplest trajectory, but we will consider a slightly different variant. The solution of the system of equations is the coordinates of the points of the manipulator (x1, x2, x3) and (x4, x5, x6) in all trajectory. After that, knowing the lengths of the links and the coordinates of the points at each moment of time, any angles of the manipulator are calculated. The same selected trajectory is reproduced from these angles. The possible angles are displayed by black color.
All the work on creating a mathematical model and calculating the angles can be done without the manipulator itself, is sufficient to have only the instruction with technical characteristics.
To display some angles, the procedure created by vv is used.
MAN_2.mw

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.