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When, in the model, these lines are considered to be the points and the planes the lines of the projective plane PG(2, R), this association becomes a correlation (actually a polarity) of the projective plane. The sphere model is obtained by intersecting the lines and planes through the origin with a unit sphere centered at the origin.
Graph of the projective plane of order 7, having 57 points, 57 lines, 8 points on each line and 8 lines passing through each point, where each point is denoted by a rounded rectangle and each line by a combination of letter and number. Only lines with letter A and H are drawn. In the Dobble or Spot It! game, two points are removed.
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.
In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines , which either is one point (sometimes called a vertex ) or does not exist (if the lines are parallel ).
Desarguesian plane; Line at infinity; Point at infinity; Plane at infinity; Hyperplane at infinity; Projective line; Projective plane. Oval (projective plane) Roman surface; Projective space; Complex projective line; Complex projective plane; Fundamental theorem of projective geometry; Projective transformation. Möbius transformation; Cross ...
Dimension 0 (no lines): The space is a single point. Dimension 1 (exactly one line): All points lie on the unique line. Dimension 2: There are at least 2 lines, and any two lines meet. A projective space for n = 2 is equivalent to a projective plane. These are much harder to classify, as not all of them are isomorphic with a PG(d, K).
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A necessary condition for two lines to intersect is that they are in the same plane—that is, are not skew lines. Satisfaction of this condition is equivalent to the tetrahedron with vertices at two of the points on one line and two of the points on the other line being degenerate in the sense of having zero volume.
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