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In analytic geometry, the intersection of a line and a plane in three-dimensional space can be the empty set, a point, or a line. It is the entire line if that line is embedded in the plane, and is the empty set if the line is parallel to the plane but outside it. Otherwise, the line cuts through the plane at a single point.
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 ...
We wish to find a point which is on both planes (i.e. on their intersection), so insert this equation into each of the equations of the planes to get two simultaneous equations which can be solved for and .
Secant-, chord-theorem. For the intersecting secants theorem and chord theorem the power of a point plays the role of an invariant: . Intersecting secants theorem: For a point outside a circle and the intersection points , of a secant line with the following statement is true: | | | | = (), hence the product is independent of line .
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.
Intersection lineations are measured in relation to the two structures which intersect to form them. For instance, according to the measurement conventions of structural geology , original bedding, S 0 intersected by a fold's axial plane foliation, forms an intersection lineation L 0-1 , with an azimuth and plunge defined by the fold.
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.
An oriented plane segment or bivector generalizes the directed line segment. Beyond Euclidean geometry, geodesic segments play the role of line segments. A line segment is a one-dimensional simplex ; a two-dimensional simplex is a triangle.
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