# What is the Bi-affine plane

I want to know the definition of the Bi-affine plane. In an article it says that semi-symmetric plane is same as bi-affine plane. But I want to the exact definition and axioms. Also There are two types of Bi-affine planes Type I and Type II. It defines only in one paper, that is in German, but I don't know German and difficult to translate with math symbols (using google and other online translations).

Can you help me to find the definitions of the Bi-affine plane of type I and II, pls..

If there are any references, pls tell me...

Thanks.

• May I suggest you to provide a link to the paper in quetion? A few of our users (including me) are capable of German and might be able to help this way. Jan 28, 2014 at 0:18
• Geometriae Dedicata, Volume 4, Issue 2-4, 1975. Manfred Oehler's "Endliche biaffine Inzidenzebenen" pg:419-436.
– Ryan
Jan 28, 2014 at 0:22
• Can anyone help me....
– Ryan
Jan 28, 2014 at 9:19
• Unfortunately I don't have access to springerlink right now. Can you provide an excerpt containig the term (preferrably the first occurence)? The translation should be somewhat like "bi-affine incidence plane"... this might refer to the same topic in english (not sure though) Jan 28, 2014 at 11:25
• yes, that is the topic. But that is not same paper you given.
– Ryan
Jan 30, 2014 at 6:37

The very first page defines it:

Eine Inzidenzstruktur $\mathfrak A$, die (A1), (A2) und (A3,h) für eine natürliche Zahl $h$ erfüllt, heißt $h$-affine Inzidenzebene. Sind $\mathfrak P, \mathfrak G$ endlich, so heiß $\mathfrak A$ endlich. Eine $2$-affine Ebene heißt auch biaffin

In other words:
A bi-affine (incidence) plane is a structure satisfiying $(A1), (A2)$ and $(A3,2)$:
\begin{align*} (A1)\qquad & \text{for each } P, Q\in \mathfrak P \ \exists!\ g\in\mathfrak G \text{ such that } g \text{ incides with both } P,Q \\ (A2)\qquad & \text{There are } P,Q,R \in \mathfrak P \text{ such that no } g\in\mathfrak G \text{ incides with all three}\\ (A3,2)\qquad & \forall (P,g)\in\mathfrak P\times \mathfrak G: \ 1\le\pi(P,g) \le 2 \end{align*}

Definition 6 defines a $w$-line by $O(g) = n$ and a $v$-line by $O(g) = N$(where $N$ is the order and $n$ is the suborder)
Which is given in eqn. (5) / Definition 2 by $$N = \max_{g\in\mathfrak G} O(g), \qquad n=\min_{g\in\mathfrak G} O(G)$$

Definition 7 defines a Type I biaffine plane to be one such that every point has a $w$-line inciding it. Definition 8 defines a Type II biaffine plane to be one such that every point has a $v$-line inciding it.

• @Ryan happy to help. If this solved your question, please mark it as answered. If not, feel free to ask for further clarification. Jan 30, 2014 at 17:48
• I need to know about the Bi-affine plane Type II, it was in. Can you given me a little translation abt that. Thanks.
– Ryan
Jan 30, 2014 at 18:42
• @Ryan I've added all the details, you should be able to expand the definitions now to obtain the big picture. Jan 30, 2014 at 18:55
• Thanks AlexR, but I don't know how to say "answered".
– Ryan
Jan 30, 2014 at 19:08
• @Ryan To the left of the post where the up and down arrows are, there is also a checkmark. You click that and it's done ;) See here for some tutorial-like introduction texts to this forum. Jan 30, 2014 at 19:16

I read one paper that defined bi-affine as follows: We say $f(u,x)$ is bi-affine if the function $f(u,x)$ is affine for any $x$ and the function is affine for any fixed $u$.

Dimitris Bertsimas, Vishal Gupta, etc., "Data-driven robust optimization".