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The linear maps (or linear functions) of vector spaces, viewed as geometric maps, map lines to lines; that is, they map collinear point sets to collinear point sets and so, are collineations. In projective geometry these linear mappings are called homographies and are just one type of collineation.
Tangential – intersecting a curve at a point and parallel to the curve at that point. Collinear – in the same line; Parallel – in the same direction. Transverse – intersecting at any angle, i.e. not parallel. Orthogonal (or perpendicular) – at a right angle (at the point of intersection).
For each pair of lines, there can be only one cell where the two lines meet at the bottom vertex, so the number of downward-bounded cells is at most the number of pairs of lines, () /. Adding the unbounded and bounded cells, the total number of cells in an arrangement can be at most n ( n + 1 ) / 2 + 1 {\displaystyle n(n+1)/2+1} . [ 5 ]
There exist four points such that no three are collinear (points not on a single line). In an affine plane, two lines are called parallel if they are equal or disjoint. Using this definition, Playfair's axiom above can be replaced by: [2] Given a point and a line, there is a unique line which contains the point and is parallel to the line.
One also says "two generic lines intersect in a point", which is formalized by the notion of a generic point. Similarly, three generic points in the plane are not collinear ; if three points are collinear (even stronger, if two coincide), this is a degenerate case .
In Euclidean and projective geometry, five points determine a conic (a degree-2 plane curve), just as two (distinct) points determine a line (a degree-1 plane curve).There are additional subtleties for conics that do not exist for lines, and thus the statement and its proof for conics are both more technical than for lines.
Simply, a collineation is a one-to-one map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. One may formalize this using various ways of presenting a projective space. Also, the case of the projective line is special, and hence generally treated ...
This occurs if the lines are parallel, or if they intersect each other. Two lines that are not coplanar are called skew lines . Distance geometry provides a solution technique for the problem of determining whether a set of points is coplanar, knowing only the distances between them.