| 引用本文: | 曲焱炎,高岱,宫娜.圆锥内定点至锥表面垂点轨迹[J].哈尔滨工业大学学报,2016,48(7):73.DOI:10.11918/j.issn.0367-6234.2016.07.011 |
| Qü Yanyan,GAO Dai,GONG Na.Research on the pedals trajectory on cone surface of inside point to element line[J].Journal of Harbin Institute of Technology,2016,48(7):73.DOI:10.11918/j.issn.0367-6234.2016.07.011 |
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| 摘要: |
| 为解决实际工程中遇到的圆锥曲线问题，研究圆锥内任意点至所有素线垂足轨迹的方程. 根据圆锥形成的性质和向量几何的理论，如果点至素线的方向向量垂直素线方向向量，那么二者向量的点积等于零，由此求出垂足参数表达式. 推算并化简垂足轨迹点参数方程，通过极坐标与直角坐标的转换，得出垂足点轨迹为圆锥与球偏交得的两曲面交线以及交线的笛卡尔坐标表达式. 结果表明，若以圆锥内一点与圆锥顶点连线为直径作球，则球面与圆锥素线交点和圆锥内定点连线垂直于素线；研究轨迹线的投影性质表明，其正面投影为抛物线，水平投影为闭合的二次曲线.
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| 关键词: 圆锥 素线 方向向量 垂足轨迹 球偏交圆锥 交线 |
| DOI：10.11918/j.issn.0367-6234.2016.07.011 |
| 分类号:TH126 |
| 文献标识码:A |
| 基金项目:哈工大实验室工程图学建设项目; 哈工大教学研究项目; 黑龙江省高等教育教学改革项目 |
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| Research on the pedals trajectory on cone surface of inside point to element line |
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Qü Yanyan, GAO Dai, GONG Na
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（School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China）
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| Abstract: |
| This paper studies the trajectory equations of pedals of an inside arbitrary point to all element lines of a cone for solving the problem of cone curve in actual engineering. Based on the feature of a cone and the theory of differential geometry, when the vector from an inside arbitrary point to an element line is perpendicular to the vector of that element line, their scalar product has to be zero, thus the parameter expression of the pedals is derived. After calculated and simplified, the parameter equation of track points shows an intersection track line of the cone and an offset sphere. And the expression in the Descartes coordinate is given. Furthermore, a conclusion is obtained that if there defines a sphere by a diameter from the summit to an arbitrary point in the cone, then lines from the given point to the intersection of that sphere and cone are perpendicular to the intersected conical element lines. Meanwhile, the projection property of the trajectory is studied, and the frontal projection is a parabola, and the horizontal projection is a quadratic curve.
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| Key words: cone element line vector the trajectory of perpendicular point cone intersecting oblique sphere intersecting line |
