The $\mu$-basis is a developing algebraic tool to study the expressions of rational curves and surfaces. It can play a bridge role between the parametric forms and implicit forms and show some advantages in implicitization, inversion formulas and singularity computation. However, it is difficult and there are few works to compute the $\mu$-basis from an implicit form. In this paper, we derive the explicit forms of $\mu$-basis for implicit monoid curves and surfaces, including the conics and quadrics which are particular cases of these entities. Additionally, we also provide the explicit form of $\mu$-basis for monoid curves and surfaces defined by any rational parametrization (not necessarily in standard proper form). Our technique is simply based on the linear coordinate transformation and standard forms of these curves and surfaces. As a practical application in numerical situation, if an exact multiple point can not be computed, we can consider the problem of computing “approximate $\mu$-basis” as well as the error estimation.

这 $\mu$-basis是研究有理曲线和曲面的表达式的一种发展中的代数工具。它可以在参数形式和隐式形式之间起到桥梁作用，并在隐式化，反演公式和奇异性计算方面显示出一些优势。但是，这很困难，几乎没有什么工作可以计算出$\mu$-基于隐式形式。在本文中，我们推导了的显式形式$\mu$-用于隐式等分面曲线和曲面的基础，包括圆锥形和二次曲面，这是这些实体的特殊情况。此外，我们还提供了以下形式的显式形式$\mu$-由任何有理参数化（不一定采用标准适当形式）定义的类半体曲线和曲面的基础。我们的技术仅基于这些曲线和曲面的线性坐标变换和标准形式。作为在数值情况下的实际应用，如果无法计算出精确的多点，我们可以考虑计算“近似值”的问题。$\mu$-基础”以及误差估计。