query: two lines in the same plane that do not intersect are called

pos: Parallel (geometry) Parallel (geometry) In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel; otherwise they are called skew lines. Parallel planes are planes in the same three-dimensional space that never

pos: Parallel (geometry) Parallel (geometry) In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel; otherwise they are called skew lines. Parallel planes are planes in the same three-dimensional space that never

pos: Projective plane points at infinity. They constitute a line in "K"P — namely, the line arising from the plane in "K" — called the line at infinity. The points at infinity are the "extra" points where parallel lines intersect in the construction of the extended real plane; the point (0, "x", "x") is where all lines of slope "x" / "x" intersect. Consider for example the two lines in the affine plane "K". These lines have slope 0 and do not intersect. They can be regarded as subsets of "KP via the embedding above, but these subsets are not lines in "KP.

pos: Projective plane Projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus "any" two distinct lines in a projective plane intersect in one and only one point. A projective plane is a 2-dimensional projective space, but not all projective planes can be embedded in 3-dimensional

pos: Line (geometry) and, in the special case where the conic is a pair of lines, we have the Pappus line. Parallel lines are lines in the same plane that never cross. Intersecting lines share a single point in common. Coincidental lines coincide with each other—every point that is on either one of them is also on the other. Perpendicular lines are lines that intersect at right angles. In three-dimensional space, skew lines are lines that are not in the same plane and thus do not intersect each other. In many models of projective geometry, the representation of a line rarely conforms to

pos: Stereographic projection projection point. Parallel lines, which do not intersect in the plane, are transformed to circles tangent at projection point. Intersecting lines are transformed to circles that intersect transversally at two points in the sphere, one of which is the projection point. (Similar remarks hold about the real projective plane, but the intersection relationships are different there.) The loxodromes of the sphere map to curves on the plane of the form where the parameter measures the "tightness" of the loxodrome. Thus loxodromes correspond to logarithmic spirals. These spirals intersect radial lines in the plane at equal angles, just as the loxodromes

pos: Three-dimensional space though there are an infinite number of possible methods. See Euclidean space. Below are images of the above-mentioned systems. Two distinct points always determine a (straight) line. Three distinct points are either collinear or determine a unique plane. Four distinct points can either be collinear, coplanar or determine the entire space. Two distinct lines can either intersect, be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a unique plane, so skew lines are lines that do not meet and do not lie in a common plane. Two distinct planes can either meet in a common

pos: Problem of Apollonius be tangent. If the angle between lines or circles at an intersection point is zero, they are said to be "tangent"; the intersection point is called a "tangent point" or a "point of tangency". (The word "tangent" derives from the Latin present participle, "tangens", meaning "touching".) In practice, two distinct circles are tangent if they intersect at only one point; if they intersect at zero or two points, they are not tangent. The same holds true for a line and a circle. Two distinct lines cannot be tangent in the plane, although two parallel lines can be considered as tangent

pos: Parallel (geometry) do not intersect. Equivalently, they are parallel if and only if the distance from a point "P" on line "m" to the nearest point in plane "q" is independent of the location of "P" on line "m". Similar to the fact that parallel lines must be located in the same plane, parallel planes must be situated in the same three-dimensional space and contain no point in common. Two distinct planes "q" and "r" are parallel if and only if the distance from a point "P" in plane "q" to the nearest point in plane "r" is independent of the location

pos: Line at infinity Line at infinity In geometry and topology, the line at infinity is a projective line that is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at infinity is also called the ideal line. In projective geometry, any pair of lines always intersects at some point, but parallel lines do not intersect in the real plane. The line at infinity is added to the real plane. This completes the plane, because now parallel lines intersect at a point which lies on

pos: Finite geometry projective planes that are not isomorphic to Galois geometries, namely the non-Desarguesian planes. Similar results hold for other kinds of finite geometries. The following remarks apply only to finite "planes". There are two main kinds of finite plane geometry: affine and projective. In an affine plane, the normal sense of parallel lines applies. In a projective plane, by contrast, any two lines intersect at a unique point, so parallel lines do not exist. Both finite affine plane geometry and finite projective plane geometry may be described by fairly simple axioms. An affine plane geometry is a nonempty set "X" (whose

pos: Real projective line Real projective line In geometry, a real projective line is an extension of the usual concept of line that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not intersect but seem to intersect "at infinity". For solving this problem, points at infinity have been introduced, in such a way that in a real projective plane, two distinct projective lines meet in exactly one point. The set of these points at infinity, the "horizon" of the visual perspective in the plane, is a real projective line. It is the circle of directions emanating

pos: Line–line intersection of them); if they are distinct but have the same slope they are said to be parallel and have no points in common; otherwise they have a single point of intersection. The distinguishing features of non-Euclidean geometry are the number and locations of possible intersections between two lines and the number of possible lines with no intersections (parallel lines) with a given line. 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

pos: Line at infinity the line at infinity. Also, if any pair of lines intersect at a point on the line at infinity, then the pair of lines are parallel. Every line intersects the line at infinity at some point. The point at which the parallel lines intersect depends only on the slope of the lines, not at all on their y-intercept. In the affine plane, a line extends in two opposite directions. In the projective plane, the two opposite directions of a line meet each other at a point on the line at infinity. Therefore, lines in the projective plane are closed curves,

pos: Line–line intersection 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 formula_51 Thus a set of "n" lines can be represented by 2"n" equations in the 3-dimensional coordinate vector "w" = ("x", "y", "z"): where now "A" is 2"n" × 3 and "b" is 2"n" × 1. As before there is a unique intersection point if and only if "A" has

pos: Projective space are identified in the projective plane. Lines (blue) in the plane are mapped to great circles if one also includes one pair of antipodal points on the equator. Any two great circles intersect precisely in two antipodal points (identified in the projective plane). Great circles corresponding to parallel lines intersect on the equator. So "any" two lines have exactly one intersection point inside P(R). This phenomenon is axiomatized in projective geometry. The "real projective space" of dimension "n" or "projective" "n"-"space", P(R), is roughly the set of the lines in R passing through the origin. For defining it as a

pos: Absolute geometry that if lines a and b are cut by a transversal t such that there is a pair of congruent alternate interior angles, then a and b are parallel.) The foregoing construction, and the alternate interior angle theorem, do not depend on the parallel postulate and are therefore valid in absolute geometry. In absolute geometry it is also provable that two lines perpendicular to the same line cannot intersect (which makes the two lines parallel by definition of parallel lines), proving that the summit angles of an Saccheri quadrilateral cannot be obtuse, and that spherical geometry is not an absolute

pos: Non-Euclidean geometry between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line "ℓ" and a point "A", which is not on "ℓ", there is exactly one line through "A" that does not intersect "ℓ". In hyperbolic geometry, by contrast, there are infinitely many lines through "A" not intersecting "ℓ", while in elliptic geometry, any line through "A" intersects "ℓ". Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane

pos: Perspective projection distortion object would appear rotated but still with minimal projection distortion (provided that the projection plane was also rotated). The difference between the images of the same object produced by "artificial" perspective projection and by "natural" perspective projection is called "perspective distortion." It should be noted that both "artificial" and "natural" projection (the image beheld by the eye) foreshorten real objects and that parallel lines appear to intersect, so that is not the difference between them. The distortion is produced by the difference between the concave retina and the flat projection plane. If an artificial perspective would be projected onto a

pos: Degenerate conic consists thus of two complex conjugate lines that intersect in the unique real point, formula_9, of the conic. The pencil of ellipses of equations formula_10 degenerates, for formula_5, into two parallel lines and, for formula_3, into a double line. The pencil of circles of equations formula_13 degenerates for formula_14 into two lines, the line at infinity and the line of equation formula_15. Over the complex projective plane there are only two types of degenerate conics – two different lines, which necessarily intersect in one point, or one double line. Any degenerate conic may be transformed by a projective transformation into

pos: Perspective (graphical) with one-point perspective. These parallel lines converge at the vanishing point. One-point perspective exists when the picture plane is parallel to two axes of a rectilinear (or Cartesian) scene – a scene which is composed entirely of linear elements that intersect only at right angles. If one axis is parallel with the picture plane, then all elements are either parallel to the picture plane (either horizontally or vertically) or perpendicular to it. All elements that are parallel to the picture plane are drawn as parallel lines. All elements that are perpendicular to the picture plane converge at a single point

pos: Prismatic surface by a symmetry operator. Prismatic surface A prismatic surface is a surface generated by all the lines that are parallel to a given line and intersect a broken line that is not in the same plane as the given line. The broken line is the "directrix" of the surface; the parallel lines are its "generators" (or "elements"). If the broken line is closed (i.e., a closed polygon), then the surface is a "closed prismatic surface". With regards to crystallography, a prismatic surface is a single face of a prismatic form, which is an open form consisting of three, four, or

pos: Prismatic surface Prismatic surface A prismatic surface is a surface generated by all the lines that are parallel to a given line and intersect a broken line that is not in the same plane as the given line. The broken line is the "directrix" of the surface; the parallel lines are its "generators" (or "elements"). If the broken line is closed (i.e., a closed polygon), then the surface is a "closed prismatic surface". With regards to crystallography, a prismatic surface is a single face of a prismatic form, which is an open form consisting of three, four, or six identical faces related

pos: Real projective plane plane from an infinite distance, a line on the "z" = 0 plane is where parallel lines intersect. In the equation xℓ = 0 there are two column vectors. You can keep either constant and vary the other. If we keep the point x constant and vary the coefficients ℓ we create new lines that go through the point. If we keep the coefficients constant and vary the points that satisfy the equation we create a line. We look upon x as a point, because the axes we are using are "x", "y", and "z". If we instead plotted the

pos: Line–line intersection is required (parallel lines/coincident lines, overlapping/non-overlapping intervals). The formula_20 and formula_21 coordinates of the point of intersection of two non-vertical lines can easily be found using the following substitutions and rearrangements. Suppose that two lines have the equations formula_22 and formula_23 where formula_24 and formula_25 are the slopes (gradients) of the lines and where formula_26 and formula_27 are the "y"-intercepts of the lines. At the point where the two lines intersect (if they do), both formula_21 coordinates will be the same, hence the following equality: We can rearrange this expression in order to extract the value of formula_20, and so,

pos: Parallel postulate fifth postulate. In the list above, it is always taken to refer to non-intersecting lines. For example, if the word "parallel" in Playfair's axiom is taken to mean 'constant separation' or 'same angles where crossed by any third line', then it is no longer equivalent to Euclid's fifth postulate, and is provable from the first four (the axiom says 'There is at most one line...', which is consistent with there being no such lines). However, if the definition is taken so that parallel lines are lines that do not intersect, or that have some line intersecting them in the same

pos: Homogeneous coordinates are called points at infinity, and are considered to lie on a new line, the line at infinity. There is a point at infinity corresponding to each direction (numerically given by the slope of a line), informally defined as the limit of a point that moves in that direction away from the origin. Parallel lines in the Euclidean plane are said to intersect at a point at infinity corresponding to their common direction. Given a point on the Euclidean plane, for any non-zero real number "Z", the triple is called a "set of homogeneous coordinates" for the point. By this

pos: Projective space way than is possible without homogeneous coordinates. Furthermore, various statements in geometry can be made more consistent and without exceptions. For example, in the standard Euclidean geometry for the plane, two lines always intersect at a point "except" when parallel. In a projective representation of lines and points, however, such an intersection point exists even for parallel lines, and it can be computed in the same way as other intersection points. Other mathematical fields where projective spaces play a significant role are topology, the theory of Lie groups and algebraic groups, and their representation theories. As outlined above, projective space

pos: Projective space is a geometric object that formalizes statements like "Parallel lines intersect at infinity." For concreteness, we give the construction of the real projective plane P(R) in some detail. There are three equivalent definitions: The last formula goes under the name of homogeneous coordinates. In homogeneous coordinates, any point with is equivalent to . So there are two disjoint subsets of the projective plane: that consisting of the points for , and that consisting of the remaining points . The latter set can be subdivided similarly into two disjoint subsets, with points and . In the last case, "x" is necessarily

neg: Parallel (geometry) same three-dimensional space that do not intersect need not be parallel. Only if they are in a common plane are they called parallel; otherwise they are called skew lines. Two distinct lines "l" and "m" in three-dimensional space are parallel if and only if the distance from a point "P" on line "m" to the nearest point on line "l" is independent of the location of "P" on line "m". This never holds for skew lines. A line "m" and a plane "q" in three-dimensional space, the line not lying in that plane, are parallel if and only if they

neg: Line (geometry) are called parallel. In higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. Any collection of finitely many lines partitions the plane into convex polygons (possibly unbounded); this partition is known as an arrangement of lines. Lines in a Cartesian plane or, more generally, in affine coordinates, can be described algebraically by linear equations. In two dimensions, the equation for non-vertical lines is often given in the "slope-intercept form": where: The slope of the line through points formula_2 and formula_3, when formula_4, is given by formula_5

neg: Skew lines Skew lines In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. Two lines that both lie in the same plane must either cross each other or be parallel, so skew lines can exist only in three or more dimensions. Two lines are skew if and only if they are not coplanar. If four points are chosen at random uniformly within a unit cube, they will almost surely define a pair of skew

neg: Desargues's theorem the lines and belong to the same plane and must intersect. Further, if the two triangles lie on different planes, then the point belongs to both planes. By a symmetric argument, the points and also exist and belong to the planes of both triangles. Since these two planes intersect in more than one point, their intersection is a line that contains all three points. This proves Desargues's theorem if the two triangles are not contained in the same plane. If they are in the same plane, Desargues's theorem can be proved by choosing a point not in the plane, using

neg: Coplanarity Coplanarity In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. However, a set of four or more distinct points will, in general, not lie in a single plane. Two lines in three-dimensional space are coplanar if there is a plane that includes them both. This occurs if the lines are parallel, or if they intersect each other. Two lines that are not coplanar are called skew lines. Distance

neg: Angles between flats Angles between flats The concept of angles between lines in the plane and between pairs of two lines, two planes or a line and a plane in space can be generalized to arbitrary dimension. This generalization was first discussed by Jordan. For any pair of flats in a Euclidean space of arbitrary dimension one can define a set of mutual angles which are invariant under isometric transformation of the Euclidean space. If the flats do not intersect, their shortest distance is one more invariant. These angles are called canonical or principal. The concept of angles can be generalized to pairs

neg: Angles between flats be naturally extended to arbitrary inner products and subspaces with infinite dimensions. Angles between flats The concept of angles between lines in the plane and between pairs of two lines, two planes or a line and a plane in space can be generalized to arbitrary dimension. This generalization was first discussed by Jordan. For any pair of flats in a Euclidean space of arbitrary dimension one can define a set of mutual angles which are invariant under isometric transformation of the Euclidean space. If the flats do not intersect, their shortest distance is one more invariant. These angles are called

neg: Vanishing point to the axis which is cut by this image plane will have images that meet at a single vanishing point. Lines parallel to the other two axes will not form vanishing points as they are parallel to the image plane. This is one-point perspective. Similarly, when the image plane intersects two world-coordinate axes, lines parallel to those planes will meet form two vanishing points in the picture plane. This is called two-point perspective. In three-point perspective the image plane intersects the , , and axes and therefore lines parallel to these axes intersect, resulting in three different vanishing points. The

neg: Piloting (navigation) a fix is where two or more position lines intersect at any given time. If three position lines can be obtained, the resulting "cocked hat", where the 3 lines do not intersect at the same point, but create a triangle where the vessel is inside, gives the navigator an indication of the accuracy in the three separate position lines. If two geographic features are visually aligned (the edge of an island aligned with the edge of an island behind, a flag pole and a building, etc.), the extension of the line joining the features is called a "transit". A transit

neg: Incidence (geometry) point are said to be "concurrent". The set of all lines in a plane incident with the same point is called a "pencil of lines" centered at that point. The computation of the intersection of two lines shows that the entire pencil of lines centered at a point is determined by any two of the lines that intersect at that point. It immediately follows that the algebraic condition for three lines, to be concurrent is that the determinant, Incidence (geometry) In geometry, an incidence relation is a heterogeneous relation that captures the idea being expressed when phrases such as "a

neg: Transversal plane the lines, then the plane is a transversal plane. Transversal plane In geometry, a transversal plane is a plane that intersects (not contains) two or more lines or planes. A transversal plane may also form dihedral angles. Transversal plane theorem for lines: Lines that intersect a transversal plane are parallel if and only if their alternate interior angles formed by the points of intersection are congruent. 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

neg: Enumerative geometry in the plane, since lines in the projective plane intersect, with multiplicity two because it is doubled, and thus satisfies the same intersection condition (intersection of multiplicity two) as a nondegenerate conic that is "tangent" to the line. The general Bézout theorem says 5 general quadrics in 5-space will intersect in 32 = 2 points. But the relevant quadrics here are not in general position. From 32, 31 must be subtracted and attributed to the Veronese, to leave the correct answer (from the point of view of geometry), namely 1. This process of attributing intersections to 'degenerate' cases is a

neg: Non-Euclidean geometry in a second paper in the same year, defined the Klein model which models the entirety of hyperbolic space, and used this to show that Euclidean geometry and hyperbolic geometry were equiconsistent so that hyperbolic geometry was logically consistent if and only if Euclidean geometry was. (The reverse implication follows from the horosphere model of Euclidean geometry.) In the hyperbolic model, within a two-dimensional plane, for any given line "ℓ" and a point "A", which is not on "ℓ", there are infinitely many lines through "A" that do not intersect "ℓ". In these models the concepts of non-Euclidean geometries are

neg: Line (geometry) but is sometimes defined as a set of points obeying a linear relationship when some other fundamental concept is left undefined. In an axiomatic formulation of Euclidean geometry, such as that of Hilbert (Euclid's original axioms contained various flaws which have been corrected by modern mathematicians), a line is stated to have certain properties which relate it to other lines and points. For example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point. In two dimensions, i.e., the Euclidean plane, two lines which do not intersect

neg: Transversal plane Transversal plane In geometry, a transversal plane is a plane that intersects (not contains) two or more lines or planes. A transversal plane may also form dihedral angles. Transversal plane theorem for lines: Lines that intersect a transversal plane are parallel if and only if their alternate interior angles formed by the points of intersection are congruent. 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

neg: Three-dimensional space line or are parallel (do not meet). Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel. A line can lie in a given plane, intersect that plane in a unique point or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line. A hyperplane is a subspace of one dimension less

neg: Hypercycle (geometry) calculated from the radius "r" and the distance between the points where the normals intersect with the axis "d" using the formula . In the Poincaré disk model of the hyperbolic plane, hypercycles are represented by lines and circle arcs that intersect the boundary circle at non-right angles. The representation of the axis intersects the boundary circle in the same points, but at right angles. In the Poincaré half-plane model of the hyperbolic plane, hypercycles are represented by lines and circle arcs that intersect the boundary line at non-right angles. The representation of the axis intersects the boundary line in

neg: Translation plane as follows: Embed as a hyperplane of . Define an incidence structure with "points," the points of not on and "lines" the planes of meeting in a line of . Then is a translation affine plane of order . Let be the projective completion of . In a set of mutually skew lines with the property that any line intersecting three lines of must intersect all the lines of is called a regulus. The lines intersecting all the lines of are called transversals of . Any three mutually skew lines of lie in precisely one regulus. A spread of is

neg: Hyperbolic geometry in Euclidean geometry. For example, given 2 intersecting lines there are infinitely many lines that do not intersect either of the given lines. These properties all are independent of the model used, even if the lines may look radically different. Non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in Euclidean geometry: This implies that there are through "P" an infinite number of coplanar lines that do not intersect "R". These non-intersecting lines are divided into two classes: Some geometers simply use "parallel" lines instead of "limiting parallel" lines, with "ultraparallel" lines being just "non-intersecting". These

neg: Cramer's paradox same time, Euler published examples showing a cubic curve which was not uniquely defined by 9 points and discussed the problem in his book "Introductio in analysin infinitorum". The result was publicized by James Stirling and explained by Julius Plücker. For first order curves (that is lines) the paradox does not occur, because "n" = 1 so "n" = 1 < "n"("n" + 3) / 2 = 2. In general two distinct lines "L" and "L" intersect at a single point "P" unless the lines are of equal gradient (slope), in which case they do not intersect at all. A

neg: Streamlines, streaklines, and pathlines Streamlines, streaklines, and pathlines Streamlines, streaklines and pathlines are field lines in a fluid flow. They differ only when the flow changes with time, that is, when the flow is not steady. Considering a velocity vector field in three-dimensional space in the framework of continuum mechanics, we have that: By definition, different streamlines at the same instant in a flow do not intersect, because a fluid particle cannot have two different velocities at the same point. Similarly, streaklines cannot intersect themselves or other streaklines, because two particles cannot be present at the same location at the same instant of time;

neg: Projective plane lines that do not intersect at a point of the quadrangle meet. The seventh line consists of all the diagonal points (usually drawn as a circle or semicircle). The name "Fano" for this subplane is really a misnomer. Gino Fano (1871–1952), in developing a new set of axioms for Euclidean geometry, took as an axiom that the diagonal points of any quadrangle are never collinear. This is called "Fano's Axiom". A Fano subplane however violates Fano's Axiom. They really should be called "Anti-Fano subplanes", but this name change has not had many supporters. In finite desarguesian planes, PG(2,"q"), Fano subplanes

neg: Fix (position) bearings recorded. Bearing lines are then plotted on a chart through the locations of the sighted items. The intersection of these lines is the current position of the vessel. Usually, a fix is where two or more position lines intersect at any given time. If three position lines can be obtained, the resulting "cocked hat", where the three lines do not intersect at the same point, but create a triangle, gives the navigator an indication of the accuracy. The most accurate fixes occur when the position lines are at right angles to each other. Fixes are a necessary part of

neg: Conic section source of interesting and beautiful results in Euclidean geometry. A conic is the curve obtained as the intersection of a plane, called the "cutting plane", with the surface of a double cone (a cone with two "nappes"). It shall be assumed that the cone is a right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice. Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines. These are called degenerate conics and some

neg: Cramer's paradox single point is not sufficient to define a line (two are needed); through the point "P" there pass not only the two given lines but an infinite number of other lines as well. Similarly two nondegenerate conics intersect at most at 4 finite points in the real plane, which is fewer than the 3 = 9 given as a maximum by Bézout's theorem, and 5 points are needed to define a nondegenerate conic. In a letter to Euler, Cramer pointed out that the cubic curves "x" − "x" = 0 and "y" − "y" = 0 intersect in precisely 9

neg: Skew lines exist; two flats must either intersect or be skew. Let be the set of points on an "i"-flat, and let be the set of points on a "j"-flat. In projective "d"-space, if then the intersection of and must contain a ("i"+"j"−"d")-flat. (A "0"-flat is a point.) In either geometry, if and intersect at a "k"-flat, for , then the points of determine a ("i"+"j"−"k")-flat. Skew lines In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of

neg: Kirkman's schoolgirl problem quadric. For each of the 28 points off "S" there are 6 lines through it which do not intersect "S". As there are seven days in a week, the heptad is an important part of the solution: A heptad is determined by any two of its points. Each of the 28 points off "S" lies in two heptads. There are 8 heptads. The projective linear group PGL(3,2) is isomorphic the alternating group on the 8 heptads. The schoolgirl problem consists in finding seven lines in the 5-space which do not intersect and such that any two lines always have a

neg: Parallel (geometry) intersect l ) since it involves no measurements is useful in non-Euclidean geometries. In general geometry the three properties above give three different types of curves, equidistant curves, parallel geodesics and geodesics sharing a common perpendicular, respectively. While in Euclidean geometry two geodesics can either intersect or be parallel, in hyperbolic geometry, there are three possibilities. Two geodesics belonging to the same plane can either be: In the literature "ultra parallel" geodesics are often called "non-intersecting". "Geodesics intersecting at infinity" are called "limiting parallel". As in the illustration through a point "a" not on line "l" there are two limiting

neg: General position or equivalently that points chosen at random will almost surely (with probability 1) be in general position. General position In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the "general case" situation, as opposed to some more special or coincidental cases that are possible, which is referred to as special position. Its precise meaning differs in different settings. For example, generically, two lines in the plane intersect in a single point (they are not parallel or coincident). One also says "two generic lines intersect in

neg: Projective plane Add the point (0, 1, 0) to each subset; that is, let These are lines in "K"P; ū arises from the plane in "K", while ȳ arises from the plane The projective lines ū and ȳ intersect at (0, 1, 0). In fact, all lines in "K" of slope 0, when projectivized in this manner, intersect at (0, 1, 0) in "K"P. The embedding of "K" into "K"P given above is not unique. Each embedding produces its own notion of points at infinity. For example, the embedding has as its complement those points of the form ("x", 0, "x"), which

neg: General position General position In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the "general case" situation, as opposed to some more special or coincidental cases that are possible, which is referred to as special position. Its precise meaning differs in different settings. For example, generically, two lines in the plane intersect in a single point (they are not parallel or coincident). 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

neg: Stellation diagram Stellation diagram In geometry, a stellation diagram or stellation pattern is a two-dimensional diagram in the plane of some face of a polyhedron, showing lines where other face planes intersect with this one. The lines cause 2D space to be divided up into regions. Regions not intersected by any further lines are called elementary regions. Usually infinite regions are excluded from the diagram, along with any infinite portions of the lines. Each elementary region represents a top face of one cell, and a bottom face of another. A collection of these diagrams, one for each face type, can be used

neg: Asymptote that are tangent to the projectivized curve through a point at infinity. For example, one may identify the asymptotes to the unit hyperbola in this manner. Asymptotes are often considered only for real curves, although they also make sense when defined in this way for curves over an arbitrary field. A plane curve of degree "n" intersects its asymptote at most at "n"−2 other points, by Bézout's theorem, as the intersection at infinity is of multiplicity at least two. For a conic, there are a pair of lines that do not intersect the conic at any complex point: these are

neg: Graphical projection Perspective projection is a linear projection where three dimensional objects are projected on a "picture plane". This has the effect that distant objects appear smaller than nearer objects. It also means that lines which are parallel in nature (that is, meet at the point at infinity) appear to intersect in the projected image, for example if railways are pictured with perspective projection, they appear to converge towards a single point, called vanishing point. Photographic lenses and the human eye work in the same way, therefore perspective projection looks most realistic. Perspective projection is usually categorized into "one-point", "two-point" and "three-point

neg: Apollonius of Perga elongation destroying the perpendicularity in all other cases. Conjugates are defined for the two branches of a hyperbola resulting from the cutting of a double cone by a single plane. They are called conjugate branches. They have the same diameter. Its centroid bisects the segment between vertices. There is room for one more diameter-like line: let a grid of lines parallel to the diameter cut both branches of the hyperbola. These lines are chord-like except that they do not terminate on the same continuous curve. A conjugate diameter can be drawn from the centroid to bisect the chord-like lines. These

neg: Line–line intersection the ordered triple formula_35. The mapping from 3D to 2D coordinates is formula_36. We can convert 2D points to homogeneous coordinates by defining them as formula_37. Assume that we want to find intersection of two infinite lines in 2-dimensional space, defined as formula_38 and formula_39. We can represent these two lines in line coordinates as formula_40 and formula_41, The intersection formula_42 of two lines is then simply given by, formula_43 If formula_44 the lines do not intersect. In two dimensions, more than two lines almost certainly do not intersect at a single point. To determine if they do and, if

neg: Multivector an affine component of "P", which is the intersection of the lines through the origin of R with a selected hyperplane, such as . Lines through the origin of R intersect the plane to define an affine version of the projective plane that only lacks the points , called the points at infinity. Points in the affine component of the projective plane have coordinates . A linear combination of two points and defines a plane in R that intersects E in the line joining p and q. The multivector defines a parallelogram in R given by Notice that substitution of

neg: Symmedian The green lines are the angle bisectors; the symmedians and medians are symmetric about the angle bisectors (hence the name "symmedian.") The concept of a symmedian point extends to (irregular) tetrahedra. Given a tetrahedron ABCD two planes P and Q through AB are isogonal conjugates if they form equal angles with the planes ABC and ABD. Let M be the midpoint of the side CD. The plane containing the side AB that is isogonal to the plane ABM is called a symmedian plane of the tetrahedron. The symmedian planes can be shown to intersect at a point, the symmedian point.

neg: Line (geometry) point on the line between its end points. Depending on how the line segment is defined, either of the two end points may or may not be part of the line segment. Two or more line segments may have some of the same relationships as lines, such as being parallel, intersecting, or skew, but unlike lines they may be none of these, if they are coplanar and either do not intersect or are collinear. The "shortness" and "straightness" of a line, interpreted as the property that the distance along the line between any two of its points is minimized (see

neg: Smooth projective plane joined by a unique line. Then formula_25 is homeomorphic to the point space of the real plane formula_1, any two distinct lines intersect in a unique point, and the geometric operations are continuous (apply to the complement of a line). A familiar family of examples was given by Moulton in 1902. These planes are characterized by the fact that they have a 4-dimensional automorphism group. They are not isomorphic to a smooth plane. More generally, all non-classical compact 2-dimensional planes formula_7 such that formula_30 are known explicitly; none of these is smooth: Theorem. "If formula_7 is a smooth 2-dimensional plane

neg: Secret sharing points, there is sufficient information to fit a th degree polynomial to them, the first coefficient being the secret. Two nonparallel lines in the same plane intersect at exactly one point. Three nonparallel planes in space intersect at exactly one point. More generally, any "n" nonparallel -dimensional hyperplanes intersect at a specific point. The secret may be encoded as any single coordinate of the point of intersection. If the secret is encoded using all the coordinates, even if they are random, then an insider (someone in possession of one or more of the -dimensional hyperplanes) gains information about the secret

neg: Conic section five points determine a conic, a circle (which may be degenerate) is determined by three points. To obtain the extended Euclidean plane, the absolute line is chosen to be the line at infinity of the Euclidean plane and the absolute points are two special points on that line called the circular points at infinity. Lines containing two points with real coordinates do not pass through the circular points at infinity, so in the Euclidean plane a circle, under this definition, is determined by three points that are not collinear. It has been mentioned that circles in the Euclidean plane can

neg: Fano plane plane a set of four points, no three of which are collinear, and the six lines joining pairs of these points is a configuration known as a complete quadrangle. The lines are called "sides" and pairs of sides that do not meet at one of the four points are called "opposite sides". The points at which opposite sides meet are called "diagonal points" and there are three of them. If this configuration lies in a projective plane and the three diagonal points are collinear, then the seven points and seven lines of the expanded configuration form a subplane of the

neg: Conic section use of coordinates) approach to defining the conic sections in a projective plane was given by Jakob Steiner in 1867. A "perspective" mapping formula_37 of a pencil formula_38 onto a pencil formula_39 is a bijection (1-1 correspondence) such that corresponding lines intersect on a fixed line formula_43, which is called the "axis" of the perspectivity formula_37. A "projective" mapping is a finite sequence of perspective mappings. As a projective mapping in a projective plane over a field (pappian plane) is uniquely determined by prescribing the images of three lines, for the Steiner generation of a conic section, besides two points

neg: Vertical and horizontal robust sophisticated instruments and work on the same fundamental principle. In the flat earth scenario, where the earth is notionally a large (infinite) flat surface with gravitational field at a right angle to the surface, the earth surface is horizontal and any plane parallel to the earth surface is also horizontal. Vertical planes, e.g., walls, may be parallel to each other or they may intersect at a vertical line. Horizontal surfaces do not intersect. Furthermore, a plane cannot both be a horizontal plane at one place and a vertical plane somewhere else. When the curvature of the earth is taken

neg: Tangent lines to circles lines intersect a circle at two points, whereas another line may not intersect a circle at all. This property of tangent lines is preserved under many geometrical transformations, such as scalings, rotation, translations, inversions, and map projections. In technical language, these transformations do not change the incidence structure of the tangent line and circle, even though the line and circle may be deformed. The radius of a circle is perpendicular to the tangent line through its endpoint on the circle's circumference. Conversely, the perpendicular to a radius through the same endpoint is a tangent line. The resulting geometrical figure of

neg: Perpendicular a segment that is perpendicular to the plane, meaning that it is perpendicular to all lines in the plane that pass through the nearest point in the plane to the given point. In the two-dimensional plane, right angles can be formed by two intersected lines if the product of their slopes equals −1. Thus defining two linear functions: and , the graphs of the functions will be perpendicular and will make four right angles where the lines intersect if . However, this method cannot be used if the slope is zero or undefined (the line is parallel to an axis).

neg: Euler diagram possible relationships between any 2 sets; completely inclusive, partially inclusive, and exclusive. This is also referred to as containment, overlap or neither or, especially in mathematics, it may be referred to as subset, intersection and disjoint. Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the elements of the set, and the exterior, which represents all elements that are not members of the set. Curves whose interior zones do not intersect represent disjoint sets. Two curves whose interior zones intersect represent sets that have common elements; the zone inside both curves represents the

neg: Intersection theory no component in common), they intersect in some set of points, which for instance one can count, obtaining an "intersection number", and we may wish to do the same for a given curve: the analogy is that intersecting distinct curves is like multiplying two numbers: , while self-intersection is like squaring a single number: . Formally, the analogy is stated as a symmetric bilinear form (multiplication) and a quadratic form (squaring). A geometric solution to this is to intersect the curve not with itself, but with a slightly pushed off version of itself. In the plane, this just means translating

neg: Rational mapping but not isomorphic. Indeed, any two lines in formula_19 intersect, but the lines in formula_20 defined by formula_26 and formula_27 cannot intersect since their intersection would have all coordinates zero. To compute the function field of formula_20 we pass to an affine subset (which does not change the field, a manifestation of the fact that a rational map depends only on its behavior in any open subset of its domain) in which formula_29; in projective space this means we may take formula_30 and therefore identify this subset with the affine formula_31-plane. There, the coordinate ring of formula_20 is via the

neg: Transversality (mathematics) intersection. (The intersection points can be counted modulo 2, ignoring the signs, to obtain a coarser invariant.) This descends to a bilinear intersection product on homology classes of any dimension, which is Poincaré dual to the cup product on cohomology. Like the cup product, the intersection product is graded-commutative. The simplest non-trivial example of transversality is of arcs in a surface. An intersection point between two arcs is transverse if and only if it is not a tangency, i.e., their tangent lines inside the tangent plane to the surface are distinct. In a three-dimensional space, transverse curves do not intersect.

neg: Line–line intersection Line–line intersection In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or a line. Distinguishing these cases and finding the intersection point have use, for example, in computer graphics, motion planning, and collision detection. In three-dimensional Euclidean geometry, if two lines are not in the same plane they are called skew lines and have no point of intersection. If they are in the same plane there are three possibilities: if they coincide (are not distinct lines) they have an infinitude of points in common (namely all of the points on either

neg: Projective linear group is contained in two lines, while any two lines intersect in two points; this is similar to a finite projective plane, except that rather than two points determining one line (and two lines determining one point), they determine two lines (respectively, points). In this case (the Paley biplane, obtained from the Paley digraph of order 11), the points are the affine line (the finite field) F, where the first line is defined to be the five non-zero quadratic residues (points which are squares: 1, 3, 4, 5, 9), and the other lines are the affine translates of this (add a

neg: Mohr's circle Mohr circle (Figure 9). To confirm the location of the Pole, we could draw a line through point B on the Mohr circle parallel to the plane B where formula_179 acts. This line would also intersect the Mohr circle at the Pole (Figure 9). From the Pole, we draw lines to different points on the Mohr circle. The coordinates of the points where these lines intersect the Mohr circle indicate the stress components acting on a plane in the physical space having the same inclination as the line. For instance, the line from the Pole to point C in the

neg: Affine plane (incidence geometry) orthogonal Latin squares of order . For an arbitrary field , let be a set of -dimensional subspaces of the vector space , any two of which intersect only in {0} (called a partial spread). The members of , and their cosets in , form the lines of a translation net on the points of . If this is a -net of order . Starting with an affine translation plane, any subset of the parallel classes will form a translation net. Given a translation net, it is not always possible to add parallel classes to the net to form an

neg: Geometry studied as an affine space, where collinearity and ratios can be studied but not distances; it can be studied as the complex plane using techniques of complex analysis; and so on. Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. In modern terms, an angle is the figure formed by two rays, called the "sides" of the angle, sharing a common endpoint, called the "vertex" of the angle. In Euclidean geometry, angles are used to study polygons and

neg: Tangent lines to circles an inscribed circle, whereas a general parallelogram does not. For two circles, there are generally four distinct lines that are tangent to both (bitangent) – if the two circles are outside each other – but in degenerate cases there may be any number between zero and four bitangent lines; these are addressed below. For two of these, the external tangent lines, the circles fall on the same side of the line; for the two others, the internal tangent lines, the circles fall on opposite sides of the line. The external tangent lines intersect in the external homothetic center, whereas the

neg: Incidence geometry of lines. Adding four new points, each being added to all the lines of a single parallel class (so all of these lines now intersect), and one new line containing just these four new points produces the projective plane of order three, a configuration. Conversely, starting with the projective plane of order three (it is unique) and removing any single line and all the points on that line produces this affine plane of order three (it is also unique). Removing one point and the four lines that pass through that point (but not the other points on them) produces the

neg: Plücker coordinates the plane. This corresponds to a two-dimensional linear subspace belonging to the Klein quadric. If three distinct and non-coplanar lines intersect in a point, their linear combinations generate a two-parameter family of lines, all the lines through the point. This also corresponds to a two-dimensional linear subspace belonging to the Klein quadric. A ruled surface is a family of lines that is not necessarily linear. It corresponds to a curve on the Klein quadric. For example, a hyperboloid of one sheet is a quadric surface in P ruled by two different families of lines, one line of each passing through

neg: Graphical projection vanishing points of all horizontal lines lie on the horizon line. If, as is often the case, the picture plane is vertical, all vertical lines are drawn vertically, and have no finite vanishing point on the picture plane. Various graphical methods can be easily envisaged for projecting geometrical scenes. For example, lines traced from the eye point at 45° to the picture plane intersect the latter along a circle whose radius is the distance of the eye point from the plane, thus tracing that circle aids the construction of all the vanishing points of 45° lines; in particular, the intersection

neg: Concave polygon Concave polygon A simple polygon that is not convex is called concave, non-convex or reentrant. A concave polygon will always have at least one reflex interior angle—that is, an angle with a measure that is between 180 degrees and 360 degrees exclusive. Some lines containing interior points of a concave polygon intersect its boundary at more than two points. Some diagonals of a concave polygon lie partly or wholly outside the polygon. Some sidelines of a concave polygon fail to divide the plane into two half-planes one of which entirely contains the polygon. None of these three statements holds for

neg: Elliptic geometry ordinary line of σ corresponds to this plane; instead a line at infinity is appended to σ. As any line in this extension of σ corresponds to a plane through O, and since any pair of such planes intersects in a line through O, one can conclude that any pair of lines in the extension intersect: the point of intersection lies where the plane intersection meets σ or the line at infinity. Thus the axiom of projective geometry, requiring all pairs of lines in a plane to intersect, is confirmed. Given "P" and "Q" in σ, the elliptic distance between

neg: Latitude constant latitude and constant longitude, which are constructed with reference to the rotation axis of the Earth. The primary reference points are the poles where the axis of rotation of the Earth intersects the reference surface. Planes which contain the rotation axis intersect the surface at the meridians; and the angle between any one meridian plane and that through Greenwich (the Prime Meridian) defines the longitude: meridians are lines of constant longitude. The plane through the centre of the Earth and perpendicular to the rotation axis intersects the surface at a great circle called the Equator. Planes parallel to the

neg: Elliptic geometry Elliptic geometry Elliptic geometry is a geometry in which Euclid's parallel postulate does not hold. Elliptic geometry is studied in two, three, or more dimensions. The appearance of this geometry in the nineteenth century stimulated the development of non-Euclidean geometry generally, including hyperbolic geometry. Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the interior angles of any triangle is always greater than 180°. In elliptic geometry, two lines perpendicular to a given line must intersect. In fact, the perpendiculars on one side all intersect at the "absolute

neg: Circles of Apollonius varying ratios of the two speeds, are disjoint from each other and form a continuous family that cover the entire plane; this family of circles is known as a "hyperbolic pencil". Another family of circles, the circles that pass through both A and B, are also called a pencil, or more specifically an "elliptic pencil". These two pencils of Apollonian circles intersect each other at right angles and form the basis of the bipolar coordinate system. Within each pencil, any two circles have the same radical axis; the two radical axes of the two pencils are perpendicular, and the centers

neg: Poincaré disk model two geodesics intersect, while in the case of translations, they do not. One result of this is that if hyperbolic space is translated such that the origin in the unit Poincaré disk is translated to formula_58, formula_59 is translated to This also applies for higher dimensions. M. C. Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter around 1956 inspired Escher's interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane. Escher's wood engravings "Circle Limit I–IV" demonstrate this concept between 1958 and 1960, the final one being "Circle Limit

neg: Ford circle the circles in the Apollonian gasket generated by the lines formula_29 and formula_11 and the circle formula_31 By interpreting the upper half of the complex plane as a model of the hyperbolic plane (the Poincaré half-plane model) Ford circles can be interpreted as horocycles. In hyperbolic geometry any two horocycles are congruent . When these horocycles are circumscribed by apeirogons they tile the hyperbolic plane with an order-3 apeirogonal tiling. There is a link between the area of Ford circles, Euler's totient function formula_32 the Riemann zeta function formula_33 and Apéry's constant formula_34 As no two Ford circles intersect, it

neg: Ideal point Ideal point In hyperbolic geometry, an ideal point, omega point or point at infinity is a well defined point outside the hyperbolic plane or space. Given a line "l" and a point "P" not on "l", right- and left-limiting parallels to "l" through "P" converge to "l" at "ideal points". Unlike the projective case, ideal points form a boundary, not a submanifold. So, these lines do not "intersect" at an ideal point and such points, although well defined, do not belong to the hyperbolic space itself. The ideal points together form the Cayley absolute or boundary of a hyperbolic geometry.

neg: Benz plane mutually intersect at one point without being tangent (or no points when parallel). Adding to the point set the new point formula_1, defined to lie on every line results in every block being determined by exactly three points, as well as the intersection of any two blocks following a uniform pattern (intersecting at two points, tangent or non-intersecting). This homogeneous geometry is called classical inversive geometry or a Möbius plane. The inhomogeneity of the description (lines, circles, new point) can be seen to be non-substantive by using a 3-dimensional model. Using a stereographic projection, the classical Möbius plane may be

neg: Direct sum but is usually strictly smaller when the index set formula_23 is infinite, because direct products do not have the restriction that all but finitely many coordinates must be zero. For example, the "xy"-plane, a two-dimensional vector space, can be thought of as the direct sum of two one-dimensional vector spaces, namely the "x" and "y" axes. In this direct sum, the "x" and "y" axes intersect only at the origin (the zero vector). Addition is defined coordinate-wise, that is formula_24, which is the same as vector addition. Given two objects formula_4 and formula_5, their direct sum is written as formula_6.

neg: Projective plane projective plane is of the PG(2,"K") type, with "K" a division ring, a duality is called a reciprocity. These planes are always self-dual. By the fundamental theorem of projective geometry a reciprocity is the composition of an automorphic function of "K" and a homography. If the automorphism involved is the identity, then the reciprocity is called a projective correlation. A correlation of order two (an involution) is called a polarity. If a correlation φ is not a polarity then φ is a nontrivial collineation. It can be shown that a projective plane has the same number of lines as it

neg: Steiner conic mapping formula_6 of a pencil formula_7 onto a pencil formula_8 is a bijection (1-1 correspondence) such that corresponding lines intersect on a fixed line formula_12, which is called the "axis" of the perspectivity formula_6 (figure 2). A "projective" mapping is a finite product of perspective mappings. Examples of commonly used fields are the real numbers formula_14, the rational numbers formula_15 or the complex numbers formula_16. The construction also works over finite fields, providing examples in finite projective planes. "Remark:" The fundamental theorem for projective planes states, that a projective mapping in a projective plane over a field (pappian plane) is