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Two intersecting lines. In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or another line.Distinguishing these cases and finding the intersection have uses, for example, in computer graphics, motion planning, and collision detection.
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 force has a point of application and a line of action, therefore it defines the Plücker coordinates of a line in space and has zero pitch. A torque, on the other hand, is a pure moment that is not bound to a line in space and is an infinite pitch screw. The ratio of these two magnitudes defines the pitch of the screw.
A more descriptive term is force couple or pure moment. Its effect is to impart angular momentum but no linear momentum. In rigid body dynamics, force couples are free vectors, meaning their effects on a body are independent of the point of application. The resultant moment of a couple is a special case of moment.
Moments are usually defined with respect to a fixed reference point and refer to physical quantities located some distance from the reference point. For example, the moment of force, often called torque, is the product of a force on an object and the distance from the reference point to the object. In principle, any physical quantity can be ...
Alternatively, a line can be described as the intersection of two planes. Let L be a line contained in distinct planes a and b with homogeneous coefficients (a 0 : a 1 : a 2 : a 3) and (b 0 : b 1 : b 2 : b 3), respectively. (The first plane equation is =, for example.)
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.
In detail, the number of points required to determine a curve of degree d is the number of monomials of degree d, minus 1 from projectivization. For the first few d these yield: d = 1: 2 and 1: two points determine a line, two lines intersect in a point, d = 2: 5 and 4: five points determine a conic, two conics intersect in four points,