以吻曲线为中心的圆族的包络
Envelopes of Circles Centered on a Kiss Curve
AI总结：
本文借助动态几何系统与计算机代数系统，研究了以吻曲线及其偏移曲线为中心的圆族包络，通过参数与隐式方程转换分析了奇点。
摘要：
参数化平面曲线族的包络是一个重要课题，在数学及其应用方面均具意义。如今，该研究通常在技术丰富的环境中进行，自动化方法被开发并实现在软件中。这种探索涉及动态几何系统（主要用于交互式探索和猜想）与计算机代数系统（用于代数运算）之间的对话。我们研究了以所谓吻曲线及其偏移曲线为中心的圆族的包络，观察这些构造之间的差异。研究同时使用了参数表示和隐式方程，从参数表示到多项式表示的转换基于Gröbner基和消元法软件包。分析了包括尖点和自交点（crunodes）在内的奇点。
英文摘要：
Envelopes of parameterized families of plane curves is an important topic, both for the mathematics involved and for its applications. Nowadays, it is generally studied in a technology-rich environment, and automated methods are developed and implemented in software. The exploration involves a dialog between a Dynamic Geometry System (used mostly for interactive exploration and conjectures) and a Computer Algebra System (for algebraic manipulations). We study envelopes of families of circles centered on the so-called kiss curve and offsets of this curve, observing the differences between constructs. Both parametric presentations and implicit equations are used, switching from parametric to polynomial representation being based on packages for Gröbner bases and Elimination. Singular points, both cusps and points of self-intersection (crunodes), are analyzed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
