A popular introductory description of a "tangent (in geometry)" is presented as
I like to find out whether and how this description might be expressed or translated in the setting of a given metric space (or generalizations, if possible); i.e. for a given set $\mathcal S$ together with (possibly generalized) distance values $$s : \mathcal S \times \mathcal S \rightarrow \mathbb R,$$ which may satisfy additional conditions.
It is certainly essential that the (image of the) plane curve under consideration and the (image of the) corresponding tangent line have at least the one point in common "at which they just touch", say point $\mathsf P$. But the property of "just touching" seems more subtle than just sharing one point.
Further, surely it is not relevant how the (images of the) curve or the tangent line are parametrized, in the strict sense of a curve as a function $\gamma : I \rightarrow \mathcal S$. It may not even be necessary to consider either of the two images as ordered sets at all. For further reference let's call set $K \subset \mathcal S$ the image of the curve under consideration, and set $L \subset \mathcal S$ the image of the tangent line (on $K$, at $\mathsf P$).
What does seem relevant, however, are values of circumcircle radii of triangles involving point $\mathsf P$:
the circumcircle radii of triangles consisting of point $\mathsf P$ and of any two points $\mathsf{A, B} \in K$ are apparently bounded (by a bound different from zero);
the tangent line image $L$ is straight, i.e. with Cayley-Menger determinants vanishing for any three points of $L$. The corresponding radii of circumcircles of any three points of $L$ are in this sense certainly bounded, too;
the circumcircle radii of triangles consisting of point $\mathsf P$, of any one point $\mathsf A \in K$ (on one "branch" of K) and of any one point $\mathsf M \in L$ (on the "branch" of L which is in a suitable sense "opposite" to that containing point $\mathsf A$) are apparently bounded (by a bound different from zero) as well.
My questions:
Are the two following properties (or definitions) already documented:
- two sets $K, L \subset \mathcal S$ "just touching each other in point $\mathsf P$" being defined as
$ (0): \qquad \mathsf P \in K, $
$ (1):$
$\forall \mathsf{A, B} \in K : \left( \begin{vmatrix}
0 & s^2_{\mathsf{P A}} & s^2_{\mathsf{P B}} & 1 \cr
s^2_{\mathsf{P A}} & 0 & s^2_{\mathsf{A B}} & 1 \cr
s^2_{\mathsf{P B}} & s^2_{\mathsf{A B}} & 0 & 1 \cr
1 & 1 & 1 & 0
\end{vmatrix} \gt 0 \right) \implies
\exists \kappa \in \mathbb R : \forall \mathsf{E, F} \in K :$
$\! \! \! \kappa \begin{vmatrix} 0 & \! s^2_{\mathsf{P A}} \! \! & \! s^2_{\mathsf{P B}} \! \! & 1 \cr s^2_{\mathsf{P A}} \! \! & 0 & \! s^2_{\mathsf{A B}} \! \! & 1 \cr s^2_{\mathsf{P B}} \! \! & \! s^2_{\mathsf{A B}} \! & 0 & 1 \cr 1 & 1 & 1 & 0 \end{vmatrix} (s^2_{\mathsf{P E}} ~ s^2_{\mathsf{P F}} ~ s^2_{\mathsf{E F}}) \ge \begin{vmatrix} 0 & \! s^2_{\mathsf{P E}} \! & \! s^2_{\mathsf{P F}} \! & 1 \cr s^2_{\mathsf{P E}} \! & 0 & \! s^2_{\mathsf{E F}} \! & 1 \cr s^2_{\mathsf{P F}} \! & \! s^2_{\mathsf{E F}} \! & 0 & 1 \cr 1 & 1 & 1 & 0 \end{vmatrix} (s^2_{\mathsf{P A}} ~ s^2_{\mathsf{P B}} ~ s^2_{\mathsf{A B}}); \! \!$
$ (2):$
$\forall \mathsf{H, J} \in L : \left( \begin{vmatrix}
0 & s^2_{\mathsf{P H}} & s^2_{\mathsf{P J}} & 1 \cr
s^2_{\mathsf{P H}} & 0 & s^2_{\mathsf{H J}} & 1 \cr
s^2_{\mathsf{P J}} & s^2_{\mathsf{H J}} & 0 & 1 \cr
1 & 1 & 1 & 0
\end{vmatrix} \gt 0 \right) \implies
\exists \lambda \in \mathbb R : \forall \mathsf{M, Q} \in L :$
$\! \! \! \lambda \begin{vmatrix} 0 & \! s^2_{\mathsf{P H}} \! & \! s^2_{\mathsf{P J}} \! & \! 1 \cr \! s^2_{\mathsf{P H}} \! \! & 0 & \! s^2_{\mathsf{H J}} \! & \! 1 \cr \! s^2_{\mathsf{P J}} \! \! & \! s^2_{\mathsf{H J}} \! & 0 & \! 1 \cr 1 & 1 & 1 & \! 0 \end{vmatrix} (s^2_{\mathsf{P M}} ~ s^2_{\mathsf{P Q}} ~ s^2_{\mathsf{M Q}}) \ge \begin{vmatrix} 0 & \! s^2_{\mathsf{P M}} \! & \! s^2_{\mathsf{P Q}} \! & \! 1 \cr \! s^2_{\mathsf{P M}} \! \! & 0 & \! s^2_{\mathsf{M Q}} \! & \! 1 \cr \! s^2_{\mathsf{P Q}} \! \! & \! s^2_{\mathsf{M Q}} \! & 0 & \! 1 \cr 1 & 1 & 1 & \! 0 \end{vmatrix} (s^2_{\mathsf{P H}} ~ s^2_{\mathsf{P J}} ~ s^2_{\mathsf{H J}}); \! \! \! $
$(3):$
$\forall \mathsf A \in K, \mathsf J \in L \text{ with } s^4_{\mathsf{A J}} \gt s^4_{\mathsf{P A}}; s^4_{\mathsf{A J}} \gt s^4_{\mathsf{P J}} :$
$ \left( \begin{vmatrix} 0 & s^2_{\mathsf{P A}} & s^2_{\mathsf{P J}} & 1 \cr s^2_{\mathsf{P A}} & 0 & s^2_{\mathsf{A J}} & 1 \cr s^2_{\mathsf{P J}} & s^2_{\mathsf{A J}} & 0 & 1 \cr 1 & 1 & 1 & 0 \end{vmatrix} \gt 0 \right) \implies \exists \mu \in \mathbb R : \forall \mathsf F \in K, \mathsf Q \in L \text{ with } s^4_{\mathsf{F Q}} \gt s^4_{\mathsf{P F}}; s^4_{\mathsf{F Q}} \gt s^4_{\mathsf{P Q}} :$
$ \! \! \! \mu \begin{vmatrix} 0 & \! s^2_{\mathsf{P A}} \! & \! s^2_{\mathsf{P J}} \! & \! 1 \cr s^2_{\mathsf{P A}} \! \! & 0 & \! s^2_{\mathsf{A J}} \! & \! 1 \cr s^2_{\mathsf{P J}} \! \! & \! s^2_{\mathsf{A J}} \! & 0 & \! 1 \cr 1 & 1 & 1 & \! 0 \end{vmatrix} (s^2_{\mathsf{P F}} ~ s^2_{\mathsf{P Q}} ~ s^2_{\mathsf{F Q}}) \ge \begin{vmatrix} 0 & \! s^2_{\mathsf{P F}} \! & \! s^2_{\mathsf{P Q}} \! & \! 1 \cr s^2_{\mathsf{P F}} \! \! & 0 & \! s^2_{\mathsf{F Q}} \! & \! 1 \cr s^2_{\mathsf{P Q}} \! \! & \! s^2_{\mathsf{F Q}} \! & 0 & \! 1 \cr 1 & 1 & 1 & \! 0 \end{vmatrix} (s^2_{\mathsf{P A}} ~ s^2_{\mathsf{P J}} ~ s^2_{\mathsf{A J}}); \! \! \! $
or in turn ("curves intersecting" meaning that arbitrarily small circumcircle radii can be found, regardless of "branches" being selected):
- two sets $\!K, L \subset \mathcal S\!$ which satisfy conditions $(0)$, $(1)$, $(2)$, but not $(3)$, being called "intersecting each other in point $\mathsf P$"
?
And if so:
Do these definitions also apply, or can they be adapted, to spaces in which Lorentzian distance can be defined?
