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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.
A plane is tangent to the cylinder if it meets the cylinder in a single element. The right sections are circles and all other planes intersect the cylindrical surface in an ellipse. [6] If a plane intersects a base of the cylinder in exactly two points then the line segment joining these points is part of the cylindric section.
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 ...
Since is a quadratic function of , the projection of the intersection onto the xz-plane is the section of an orthogonal parabola; it is only a section due to the fact that < <. The vertex of the parabola lies at point ( − b , 0 , 0 ) {\displaystyle (-b,0,0)} , where
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.
If a cylinder is used in this sense, the above paragraph would read as follows: A plane section of a right circular cylinder of finite length [6] is a circle if the cutting plane is perpendicular to the cylinder's axis of symmetry, or an ellipse if it is neither parallel nor perpendicular to that axis. If the cutting plane is parallel to the ...
A plane has no curvature lines, because any normal curvature is zero. Hence, only the curvature lines of the cylinder are of interest: A horizontal plane intersects a cylinder at a circle and a vertical plane has lines with the cylinder in common. The idea of threefold orthogonal systems can be seen as a generalization of orthogonal trajectories.
Steinmetz solid (intersection of two cylinders) In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. Each of the curves of the intersection of two cylinders is an ellipse. The intersection of two cylinders is called a bicylinder.