论文标题
表达曲线
Expressive curves
论文作者
论文摘要
我们启动对我们称为表达式的一类真实平面代数曲线的研究。这些是曲线定义多项式的曲线，其曲线真实点集的拓扑允许的临界点数量最少。这个概念可以看作是孤立平面曲线奇点的真实倍地化概念的全球版本。我们证明，如果（a）$ c $的每个不可约组件可以通过真实的多项式（普通或三角法）参数化，（b）仿射平面中$ c $的所有单数均为普通的双胞胎节点，以及（c）（c）现实点$ c $的$ c $。相反，除非在无穷大时表现出一些外来行为，否则具有实际不可还原组件的表达曲线必须满足条件（a） - （c）。我们描述了几种产生表达曲线的结构，并讨论了大量示例，包括：线条，抛物线和圆圈的布置； Chebyshev和Lissajous曲线；低核酸和表体choids;还有更多。
We initiate the study of a class of real plane algebraic curves which we call expressive. These are the curves whose defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of a curve. This concept can be viewed as a global version of the notion of a real morsification of an isolated plane curve singularity. We prove that a plane curve $C$ is expressive if (a) each irreducible component of $C$ can be parametrized by real polynomials (either ordinary or trigonometric), (b) all singular points of $C$ in the affine plane are ordinary hyperbolic nodes, and (c) the set of real points of $C$ in the affine plane is connected. Conversely, an expressive curve with real irreducible components must satisfy conditions (a)-(c), unless it exhibits some exotic behaviour at infinity. We describe several constructions that produce expressive curves, and discuss a large number of examples, including: arrangements of lines, parabolas, and circles; Chebyshev and Lissajous curves; hypotrochoids and epitrochoids; and much more.
