Snuggly ball

The osculating sphere , or Schmiegekugel or in older literature osculating is in the mathematical branch of differential geometry is a sphere , which is a regular curve in three-dimensional space of intuition snugly in a given curve point as possible. It is about the generalization of the circle of curvature or osculating circle of a plane curve to space curves .

Definitions

Let it be a Frenet curve and let it be a point from the interior of the definition interval . It is said that a sphere with a center point and radius hugs the curve at the point or in optimally, if as many derivatives of the function as possible ${\displaystyle c:[a,b]\rightarrow \mathbb {R} ^{3}}$ ${\displaystyle s_{0}}$ ${\displaystyle [a,b]}$ ${\displaystyle m\in \mathbb {R} ^{3}}$ ${\displaystyle r>0}$ ${\displaystyle s_{0}}$ ${\displaystyle c(s_{0})}$

${\displaystyle s\mapsto \langle m-c(s),m-c(s)\rangle -r^{2}}$

in disappear. Disappearance of the first derivatives of what appropriate differentiability of demands, they say, the ball touching the point in th order. ${\displaystyle s_{0}}$ ${\displaystyle n}$ ${\displaystyle c}$ ${\displaystyle c}$ ${\displaystyle c(s_{0})}$ ${\displaystyle n}$

It is further the accompanying Frenet tripod and and had curvature or twist of the curve . ${\displaystyle s\mapsto (e_{1}(s),e_{2}(s),e_{3}(s))}$ ${\displaystyle \kappa (s)}$ ${\displaystyle \tau (s)}$ ${\displaystyle c}$

Clear existence of the snuggly ball

The osculating ball touches the curve in the third order.

The Frenet curve has a curvature and torsion different from 0 at the point . Then there is exactly one sphere that touches the curve at the point in the third order and with the above designations the following formulas apply for the center and radius: ${\displaystyle c:[a,b]\rightarrow \mathbb {R} ^{3}}$ ${\displaystyle s_{0}}$ ${\displaystyle c(s_{0})}$

${\displaystyle m=c(s_{0})+{\frac {1}{\kappa (s_{0})}}e_{2}(s_{0})-{\frac {\kappa '(s_{0})}{\tau (s_{0})\kappa (s_{0})^{2}}}e_{3}(s_{0})}$

${\displaystyle r={\sqrt {{\frac {1}{\kappa (s_{0})^{2}}}+\left({\frac {\kappa '(s_{0})}{\tau (s_{0})\kappa (s_{0})^{2}}}\right)^{2}}}}$

Since the Frenet tripod is an orthonormal system , the radius formula results directly from the center point formula. ${\displaystyle r^{2}=\langle m-c(s),m-c(s)\rangle }$

Note that this formula does not contain, that is, the center of the oscillating ball always lies in the normal plane, that is the plane through the point of the curve orthogonal to the tangent vector . ${\displaystyle e_{1}}$ ${\displaystyle c(s_{0})}$

Spherical curves

Frenet curves with non-vanishing torsion, the image of which lies in a solid spherical surface, are characterized by the fact that their flexible spheres coincide with this solid sphere at every point; such curves are called spherical. In particular is the center point

${\displaystyle m(s)=c(s)+{\frac {1}{\kappa (s)}}e_{2}(s)-{\frac {\kappa '(s)}{\tau (s)\kappa (s)^{2}}}e_{3}(s)}$

constant. So it must apply. If one evaluates this condition with reference to the Frenet formulas , one obtains: ${\displaystyle m'(s)=(0,0,0)}$

The image of a four times differentiable Frenet curve with non-vanishing torsion lies in a solid spherical surface if the curvature and torsion satisfy the following differential equation : ${\displaystyle \kappa }$ ${\displaystyle \tau }$

${\displaystyle {\frac {\tau }{\kappa }}=\left({\frac {\kappa '}{\tau \kappa ^{2}}}\right)'}$.

Such a characterization cannot of course apply to curves with vanishing torsion.Examples of such curves are circles contained in a spherical surface, because these flat curves have the torsion 0.

Because of the main theorem of curve theory , curvature and torsion cannot be independent for spherical curves. Since these are differential quantities, one must expect a relationship in the form of a differential equation between them. Note that the differential equation given above allows the spherical property to be checked without having to determine the sphere.

Individual evidence

1. ^ Wilhelm Schell : General theory of curves with double curvature Teubner-Verlag (1859), Cap. V: The oscillation ball and the straight oscillation cone
2. ^ David Hilbert , Stefan Cohn-Vossen : Illustrative Geometry , Springer-Verlag (1932), end of §27
3. ↑ Guido Walz (Ed.): Lexicon of Mathematics , Volume 4, page 466
4. ↑ Wolfgang Kühnel : Differentialgeometrie , Vieweg-Verlag (1999), ISBN 978-3-8348-0411-2 , sentence 2.10 (i)
5. ^ Wilhelm Blaschke , Kurt Leichtweiß : Elementare Differentialgeometrie , Grundlehren der Mathematischen Wissenschaften , Springer-Verlag (1973), ISBN 978-3-540-05889-2 , §14: Schmiegkugeln
6. ↑ Thomas Banchoff , Stephen Lovett : Differential Geometry of Curves and Surfaces , CRC Press (2016), ISBN 978-1-4822-4737-4 , Chapter 3.3: Osculating Plane and Osculating Sphere
7. ↑ Wolfgang Kühnel: Differentialgeometrie , Vieweg-Verlag (1999), ISBN 978-3-8348-0411-2 , sentence 2.10 (ii)