The standard approach to finding the envelope of a family of curves or a family of surfaces is shown to be a parameter optimization problem. This statement is first verified by discussing the envelope of a one-parameter family of plane curves. The standard approach is given, and the corresponding optimization problem is established. Extensions of the envelope problem to multiple parameters and families of surfaces are considered, as well as envelope problems that cannot be solved by the standard approach, for example, problems involving multiple parameters subject to constraints on the parameters.

中文翻译：

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找到一个包络是一个优化问题

找到曲线族或曲面族的包络的标准方法显示为参数优化问题。该陈述首先通过讨论平面曲线的单参数族的包络来验证。给出了标准方法，并建立了相应的优化问题。考虑将包络问题扩展到多个参数和表面族，以及标准方法无法解决的包络问题，例如，涉及受参数约束的多个参数的问题。