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  2. Intersection (geometry) - Wikipedia

    en.wikipedia.org/wiki/Intersection_(geometry)

    Line–plane intersection. The intersection of a line and a plane in general position in three dimensions is a point. Commonly a line in space is represented parametrically ((), (), ()) and a plane by an equation + + =. Inserting the parameter representation into the equation yields the linear equation

  3. Line–plane intersection - Wikipedia

    en.wikipedia.org/wiki/Line–plane_intersection

    The three possible plane-line relationships in three dimensions. (Shown in each case is only a portion of the plane, which extends infinitely far.) 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 ...

  4. Intersection curve - Wikipedia

    en.wikipedia.org/wiki/Intersection_curve

    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 ...

  5. Conic section - Wikipedia

    en.wikipedia.org/wiki/Conic_section

    Conic sections visualized with torch light This diagram clarifies the different angles of the cutting planes that result in the different properties of the three types of conic section. A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane.

  6. Line–line intersection - Wikipedia

    en.wikipedia.org/wiki/Line–line_intersection

    In three or more dimensions, even two lines almost certainly do not intersect; pairs of non-parallel lines that do not intersect are called skew lines. But if an intersection does exist it can be found, as follows. In three dimensions a line is represented by the intersection of two planes, each of which has an equation of the form

  7. Plane–plane intersection - Wikipedia

    en.wikipedia.org/wiki/Planeplane_intersection

    Given two intersecting planes described by : + + + = and : + + + =, the dihedral angle between them is defined to be the angle between their normal directions:

  8. Transversal plane - Wikipedia

    en.wikipedia.org/wiki/Transversal_plane

    Transversal plane theorem for planes: Planes intersected by a transversal plane are parallel if and only if their alternate interior dihedral angles are congruent. Transversal line containment theorem: If a transversal line is contained in any plane other than the plane containing all the lines, then the plane is a transversal plane.

  9. Dupin's theorem - Wikipedia

    en.wikipedia.org/wiki/Dupin's_theorem

    A simple example with at least one pencil of curved surfaces: 1) all right circular cylinders with the z-axis as axis, 2) all planes, which contain the z-axis, 3) all horizontal planes (see diagram). A curvature line is a curve on a surface, which has at any point the direction of a principal curvature (maximal or minimal curvature).