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The analytic determination of the intersection curve of two surfaces is easy only in simple cases; for example: a) the intersection of two planes, b) plane section of a quadric (sphere, cylinder, cone, etc.), c) intersection of two quadrics in special cases. For the general case, literature provides algorithms, in order to calculate points of ...
A simple example with at least one pencil of curved surfaces: 1) all right circular cylinders with the z-axis as axis, 2) all planes, which contain the z-axis, 3) all horizontal planes (see diagram). A curvature line is a curve on a surface, which has at any point the direction of a principal curvature (maximal or minimal curvature). The set of ...
The intersection points are: (−0.8587, 0.7374, −0.6332), (0.8587, 0.7374, 0.6332). A line–sphere intersection is a simple special case. Like the case of a line and a plane, the intersection of a curve and a surface in general position consists of discrete points, but a curve may be partly or totally contained in a surface.
Viviani's curve as intersection of a sphere and a cylinder. In the case = +, the cylinder and sphere are tangential to each other at point (,,). The intersection resembles a figure eight: it is a closed curve which intersects itself. The above parametrization becomes
a = 1, b = 2, c = 0, 0.8, 1. Start with the usual equation for the torus: (+ + +) = (+).Interchanging y and z so that the axis of revolution is now on the xy-plane, and setting z=c to find the curve of intersection gives
Viviani's curve is a special Clelia curve. For a Clelia curve the relation between the angles is =. Viviani's curve (red) as intersection of the sphere and a cone (pink) Subtracting 2× the cylinder equation from the sphere's equation and applying completing the square leads to the equation
Ellipsoid with lines of curvature as intersection curves with confocal hyperboloids =, =, = Applications: Due to Dupin's theorem on threefold orthogonal systems of surfaces, the intersection curve of any two confocal quadrics is a line of curvature.
The generation of a bicylinder Calculating the volume of a bicylinder. A bicylinder generated by two cylinders with radius r has the volume =, and the surface area [1] [6] =.. The upper half of a bicylinder is the square case of a domical vault, a dome-shaped solid based on any convex polygon whose cross-sections are similar copies of the polygon, and analogous formulas calculating the volume ...