论文标题
每个点都包含两个各向同性圆的表面
Surfaces containing two isotropic circles through each point
论文作者
论文摘要
我们证明（在某些技术假设下），$ \ mathbb r^3 $中的每个表面都包含两个平行于$ oz $的抛物线的弧形，每个点都有一个参数化$ \ left（\ frac {p（u，v）} {r（u，v）}，\ frac {q（u，v）} {r（u，v）}，\ frac {z（u，v）} {r^2（u^2（u^2（u，v）} \ right）对于某些$ p，q，q，q，q，q，q，q，q，q，q，q，q，q，q，math \ c。 $ p，q，r $最多具有$ u $和$ v $的最多学位，而$ z $最多具有2个$ u $和$ v $的学位。证明是基于这样的观察结果，即人们可以将垂直轴视为各向同性圆的抛物线。这使我们能够使用M. Skopenkov和R. Krasauskas最近工作的方法，其中将所有含有两个欧几里得圆圈的表面分类。这种方法还使我们能够在$ \ mathbb r^3 $中找到类似的参数化，其中包含两个任意各向同性圆（以相同的技术假设）。最后，我们获得了有关表面的顶部视图（沿$ oz $轴的投影）的一些结果。
We prove (under some technical assumptions) that each surface in $\mathbb R^3$ containing two arcs of parabolas with axes parallel to $Oz$ through each point has a parametrization $\left(\frac{P(u,v)}{R(u,v)},\frac{Q(u,v)}{R(u,v)},\frac{Z(u,v)}{R^2(u,v)}\right)$ for some $P,Q,R,Z\in\mathbb R[u,v]$ such that $P,Q,R$ have degree at most 1 in $u$ and $v$, and $Z$ has degree at most 2 in $u$ and $v$. The proof is based on the observation that one can consider a parabola with vertical axis as an isotropic circle; this allows us to use methods of the recent work by M. Skopenkov and R. Krasauskas in which all surfaces containing two Euclidean circles through each point are classified. Such approach also allows us to find a similar parametrization for surfaces in $\mathbb R^3$ containing two arbitrary isotropic circles through each point (under the same technical assumptions). Finally, we get some results concerning the top view (the projection along the $Oz$ axis) of the surfaces in question.
