单叶旋转双曲面教学设计及其应用探究

Teaching Design and Application of the Hyperboloid of Revolution of One Sheet

摘要: 单叶旋转双曲面作为微积分教学中重要的旋转曲面，其“直纹面”属性常被学生忽视或难以理解。这一特性是连接其几何理论与实际应用的关键。本文结合教学实践，从三方面探讨单叶旋转双曲面的教学设计：通过具体问题推导引入其直纹面属性，借助Geogebra动图及外部视频实现直观演示，结合建筑、机械、交通等领域的应用案例，帮助学生建立对该曲面双重属性(旋转曲面与直纹面)的理解。研究表明，这种教学设计不仅能突破学生的认知难点，还能凸显数学理论与实际问题的关联，为微积分中几何内容的教学提供参考。

Abstract: The hyperboloid of revolution of one sheet, as an important surface of revolution in calculus teaching, has property of being a “ruled surface” that is often overlooked or difficult for students to understand. This characteristic is the key to connecting its geometric theory with practical applications. Based on teaching practice, this paper explores the teaching design of the hyperboloid of revolution of one sheet from three aspects: introducing its property of being a ruled surface through the derivation of specific problems, realizing intuitive demonstration with Geogebra animations and external video resources, and helping students establish an understanding of the dual properties (surface of revolution and ruled surface) of this surface by combining application cases in fields such as architecture, machinery, and transportation. The research shows that this teaching design can not only break through students’ cognitive difficulties but also highlight the connection between mathematical theories and practical problems, providing a reference for the teaching of geometric content in calculus.