论文标题
锥体与球体的交点的体积
Volume of Intersection of a Cone with a Sphere
论文作者
论文摘要
手稿提供了由固体锥和固体球的交点定义的体积的公式，该圆锥形的相交是球体半径的函数，锥顶点和球体中心之间的距离以及锥形孔角的距离。如果球体中心位于（扩展）锥轴上，则分析可能基于固定在锥轴上的气缸坐标，并且体积是有限锥和球形盖的众所周知体积的总和。在一般的几何形状上，球体中心不在（扩展）锥轴上。我们的方法通过切片空间垂直于锥形轴来计算体积，并通过整合由球体交点定义的透镜区域。这些音量积分是借助伯德·弗里德曼表（Byrd-Friedmann Tables）对第一，第二和第三类的椭圆形积分进行改造的。
The manuscript provides formulas for the volume of a body defined by the intersection of a solid cone and a solid sphere as a function of the sphere radius, of the distance between cone apex and sphere center, and of the cone aperture angle. If the sphere center lies on the (extended) cone axis the analysis may be based on cylinder coordinates fixed at the cone axis, and the volume is the sum of the well-known volumes of finite cones and sphere caps. At the general geometry the sphere center is not on the (extended) cone axis. Our approach calculates the volume by slicing space perpendicular to the cone axis and by integrating the lens areas defined by the sphere-cone intersection. These volume integrals are rephrased with the aid of the Byrd-Friedmann tables to Elliptic Integrals of the First, Second and Third Kind.
