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One possibility to determine a polygon of points of the intersection curve of two surfaces is the marching method (see section References). It consists of two essential parts: The first part is the curve point algorithm, which determines to a starting point in the vicinity of the two surfaces a point on the intersection curve. The algorithm ...
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.
The normal section of a surface at a particular point is the curve produced by the intersection of that surface with a normal plane. [1] [2] [3] The curvature of the normal section is called the normal curvature. If the surface is bow or cylinder shaped, the maximum and the minimum of these curvatures are the principal curvatures. If the ...
The surface-to-surface intersection (SSI) problem is a basic workflow in computer-aided geometric design: Given two intersecting surfaces in R 3, compute all parts of the intersection curve. If two surfaces intersect, the result will be a set of isolated points, a set of curves, a set of overlapping surfaces, or any combination of these cases. [1]
At any point on a surface, we can find a normal vector that is at right angles to the surface; planes containing the normal vector are called normal planes. The intersection of a normal plane and the surface will form a curve called a normal section and the curvature of this curve is the normal curvature. For most points on most “smooth ...
More generally, any algebraic curve of genus one, for example the intersection of two quadric surfaces embedded in three-dimensional projective space, is called an elliptic curve, provided that it is equipped with a marked point to act as the identity.
This curve is called the characteristic of the family at a. As a varies the locus of these characteristic curves defines a surface called the envelope of the family of surfaces. The envelope of a family of surfaces is tangent to each surface in the family along the characteristic curve in that surface.
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 ...