论文标题
表面家庭的等距
Equidistants for families of surfaces
论文作者
论文摘要
对于$ \ mathbb {r}^3 $中的平滑表面，本文包含某些仿射等距的本地研究，该研究是点的位点 平行切线平面的接触点（但不包括等距包含一个或另一个接触点的比率为0和1）。研究的情况通常发生在1参数家族中，其中两个抛物线的表面具有平行的切线平面，在该平面上，独特的渐近方向也平行。奇异性是通过将等距分类为2参数从$ \ Mathbb {r}^4 $到$ \ MATHBB {r}^3 $的2参数展开的临界值的分类。特别是，在所谓的“超级屠杀和弦”附近发生的奇异事物加入了两个特殊的抛物线点。对于沿该和弦的给定比率，确定了一个或三个特殊点，在该比例中，等距的奇异性变得更加特别。许多由此产生的奇异性在文献中以抽象的分类发生在文献之前，因此本文还为这些奇异性提供了自然的几何环境，与它们得出的表面的几何形状有关。
For a smooth surface in $\mathbb{R}^3$ this article contains local study of certain affine equidistants, that is loci of points at a fixed ratio between points of contact of parallel tangent planes (but excluding ratios 0 and 1 where the equidistant contains one or other point of contact). The situation studied occurs generically in a 1-parameter family, where two parabolic points of the surface have parallel tangent planes at which the unique asymptotic directions are also parallel. The singularities are classified by regarding the equidistants as critical values of a 2-parameter unfolding of maps from $\mathbb{R}^4$ to $\mathbb{R}^3$. In particular, the singularities that occur near the so-called `supercaustic chord', joining the two special parabolic points, are classified. For a given ratio along this chord either one or three special points are identified at which singularities of the equidistant become more special. Many of the resulting singularities have occurred before in the literature in abstract classifications, so the article also provides a natural geometric setting for these singularities, relating back to the geometry of the surfaces from which they are derived.
