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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 ...
Green line has two intersections. Yellow line lies tangent to the cylinder, so has infinitely many points of intersection. Line-cylinder intersection is the calculation of any points of intersection, given an analytic geometry description of a line and a cylinder in 3d space. An arbitrary line and cylinder may have no intersection at all.
The ratio of the volume of a sphere to the volume of its circumscribed cylinder is 2:3, as was determined by Archimedes. The principal formulae derived in On the Sphere and Cylinder are those mentioned above: the surface area of the sphere, the volume of the contained ball, and surface area and volume of the cylinder.
In the 3rd century BC, Archimedes, using a method resembling Cavalieri's principle, [5] was able to find the volume of a sphere given the volumes of a cone and cylinder in his work The Method of Mechanical Theorems. In the 5th century AD, Zu Chongzhi and his son Zu Gengzhi established a similar method to find a sphere's volume. [2]
The intersection of two planes. 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 ...
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
intersection of two polygons: window test. If one wants to determine the intersection points of two polygons, one can check the intersection of any pair of line segments of the polygons (see above). For polygons with many segments this method is rather time-consuming. In practice one accelerates the intersection algorithm by using window tests ...
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