An equidistant hypersurface or parallel hypersurface is a hypersurface in geometry that runs around a hypersurface at a constant distance . This generalizes the concept of the parallel curve, which runs around a reference line at a constant distance. 2-dimensional equidistant hypersurfaces are also referred to as parallel surfaces .
Let the function be continuously differentiable and have a regular set of zeros (this is then a -dimensional submanifold of ). For sufficiently small the equidistant hypersurface is spaced from the set of zeros of the envelope of the spheres flock .
The set of spheres is given by the equations
described.
The envelope has in each point with one of the spheres (parameterized by ) the tangent space in the point in common. The tangent vectors to the sphere at the point satisfy the equations
In the case of the envelope, the coulter parameters also generally change . In addition to equations (1) and (2) , the equations also
result for the tangential vectors of the envelope
From (3) and (4) it follows
for all that correspond to tangential vectors in the tangent space of the envelope, therefore for all because of (5)
According to (6) this results
from which it follows that with
gives.
With (1), (2) and (7) one has scalar equations for the unknowns . Under the assumptions made, these equations thus define a -dimensional manifold, which is then precisely the envelope of the set of spheres - i.e. the equidistant hypersurface.
Alternative geometric interpretation: The vectors form a maximum set of linearly independent normal vectors on the manifold in the point . Equation (7) says that the vector from the point to the corresponding point on the equidistant is exactly perpendicular to the manifold and equation (2) that the point should have the distance from .
Application examples
Rotary engine
Application in mold construction: Casting processes (injection molding, die casting, permanent mold casting, etc.) are often used for the production of mass-produced articles. The inside of the mold has the negative contour of the article to be manufactured as a cavity. A constant wall thickness is crucial for many articles. Therefore, mastering the equidistant is of great importance in this area.
The equidistant function is offered by most CAD systems. For example, in AUTOCAD, the German command is “relocate”. With this you can, for example, very quickly define a drainage with a specified distance around a house.
In a rotary engine , the envelope curve of the trochoid (wheel curve) of the rotor at distance d is an equidistant hypersurface.