Equidistant hypersurface

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 . ${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{r}}$ ${\displaystyle n-r}$ ${\displaystyle R^{n}}$ ${\displaystyle d>0}$ ${\displaystyle A}$ ${\displaystyle d}$ ${\displaystyle f^{-1}(0)}$ ${\displaystyle f}$ ${\displaystyle {\big (}S_{d}({\bar {x}}){\big )}_{{\bar {x}}\in f^{-1}(0)}}$

The set of spheres is given by the equations

${\displaystyle {\begin{matrix}\|x-{\bar {x}}\|^{2}&=&d^{2}\\f({\bar {x}})&=&0\end{matrix}}}$

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 ${\displaystyle x\in A}$ ${\displaystyle {\bar {x}}}$ ${\displaystyle x}$ ${\displaystyle dx\in \mathbb {R} ^{n}}$ ${\displaystyle S_{d}({\bar {x}})}$ ${\displaystyle x}$

${\displaystyle {\begin{matrix}(1)&&f({\bar {x}})&=&0,\\(2)&&\|x-{\bar {x}}\|^{2}&=&d^{2},\\(3)&\quad &(x-{\bar {x}})^{T}dx&=&0.\end{matrix}}}$

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  ${\displaystyle {\bar {x}}}$ ${\displaystyle d{\bar {x}}\in \mathbb {R} ^{n}}$

${\displaystyle {\begin{matrix}(4)&\quad &(x-{\bar {x}})^{T}(dx-d{\bar {x}})&=&0,\\(5)&&f'({\bar {x}})\;d{\bar {x}}&=&0.\end{matrix}}}$

From (3) and (4) it follows

${\displaystyle (6)\quad (x-{\bar {x}})^{T}d{\bar {x}}=0}$

for all that correspond to tangential vectors in the tangent space of the envelope, therefore for all because of (5) ${\displaystyle d{\bar {x}}}$ ${\displaystyle dx}$

${\displaystyle d{\bar {x}}\in \ker f'({\bar {x}})=\left(\operatorname {image} \left(f'({\bar {x}})^{T}\right)\right)^{\perp }.}$

According to (6) this results

${\displaystyle (x-{\bar {x}})\in \left(\left(\operatorname {image} (f'({\bar {x}})^{T}\right)^{\perp }\right)^{\perp }=\operatorname {image} (f'({\bar {x}})^{T}),}$

from which it follows that  with ${\displaystyle \lambda \in \mathbb {R} ^{r}}$

${\displaystyle (7)\quad (x-{\bar {x}})=f'({\bar {x}})^{T}\lambda }$

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. ${\displaystyle n+r+1}$ ${\displaystyle 2n+r}$ ${\displaystyle x,{\bar {x}},\lambda }$ ${\displaystyle n-1}$

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  . ${\displaystyle \left(f'_{k}({\bar {x}})\right)^{T}}$  ${\displaystyle (k=1,\ldots ,r)}$ ${\displaystyle f^{-1}(0)}$ ${\displaystyle {\bar {x}}}$ ${\displaystyle (x-{\bar {x}})}$ ${\displaystyle {\bar {x}}}$ ${\displaystyle x}$ ${\displaystyle f^{-1}(0)}$ ${\displaystyle x}$ ${\displaystyle d}$ ${\displaystyle f^{-1}(0)}$

Application examples

Rotary engine

1. 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.
2. 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.
3. In a rotary engine , the envelope curve of the trochoid (wheel curve) of the rotor at distance d is an equidistant hypersurface.