Line coordinates
Encyclopedia
In geometry
, line coordinates are used to specify the position of a line
just as point coordinates (or simply coordinates
) are used to specify the position of a point.
and b is the x-intercept. This system specifies coordinates for all lines that are not vertical. However, it is more common and simpler algebraically to use coordinates where the equation of the line is lx + my + 1 = 0. This system specifies coordinates for all lines except those that pass through the origin. The geometrical interpretations of l and m are the negative reciprocals of the x and y-intercept
respectively.
The exclusion of lines passing through the origin can be resolved by using a system of three coordinates to specify the line in which the equation, lx + my + n = 0. Here l and m may not both be 0. In this equation, only the ratios between l, m and n are significant, in other words if the coordinates are multiplied by a non-zero scalar then line represented remains the same. So is a system of homogeneous coordinates
for the line.
If points in the plane are represented by homogeneous coordinates , the equation of the line is lx + my + nz = 0. In this context, l, m and n may not all be 0. In particular, represents the line z = 0, which is the line at infinity
in the projective plane
. The coordinates and represent the x and y-axes respectively.
as a subset of the points in the plane, the equation φ(l, m) = 0 represents a subset of the lines on the plain. The set of lines on the plane may, in an abstract sense, be thought of as the set of points in a projective plane, the dual
of the original plane. The equation φ(l, m) = 0 then represents a curve in the dual plane.
For a curve f(x, y) = 0 in the plane, the tangent
s to the curve form a curve in the dual space called the dual curve
. If φ(l, m) = 0 is the equation of the dual curve, then it is called the tangential equation, for the original curve. A given equation φ(l, m) = 0 represents a curve in the original plane determined as the envelope
of the lines that satisfy this equation. Similarly, if φ(l, m, n) is a homogeneous function
then φ(l, m, n) = 0 represents a curve in the dual space given in homogeneous coordinates, and may be called the homogeneous tangential equation of the enveloped curve.
Tangential equations are useful in the study of curves defined as envelopes, just as Cartesian equations are useful in the study of curves defined as loci.
By Cramer's rule
, the solution is
The lines (l1, m1), (l2, m2), and (l3, m3) are concurrent
when the determinant
For homogeneous coordinates, the intersection of the lines (l1, m1, n1) and (l2, m2, n2) is
The lines (l1, m1, n1), (l2, m2, n2) and (l3, m3, n3) are concurrent
when the determinant
Dually, the coordinates of the line containing (x1, y1, z1) and (x2, y2, z2) are
determine the line containing them. Similarly, for two points in three-dimensional space (x1, y1, z1, w1) and (x2, y2, z2, w2), the line containing them is determined by the six determinants
This is the basis for a system of homogeneous line coordinates in three-dimensional space called Plücker coordinates. Six numbers in a set of coordinates only represent a line when they satisfy an additional equation. This system maps the space of lines in three-dimensional space to a projective space of dimension five, but with the additional requirement the space of lines is a manifold
of dimension four.
More generally, the lines in n-dimensional projective space are determined by a system of n(n − 1)/2 homogeneous coordinates that satisfy a set of (n − 2)(n − 3)/2 conditions, resulting in a manifold of dimension 2(n − 1).
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, line coordinates are used to specify the position of a line
Line (geometry)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
just as point coordinates (or simply coordinates
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...
) are used to specify the position of a point.
Lines in the plane
There several possible ways to specify the position of a line in the plane. A simple way is by the pair where the equation of the line is y =mx + b. Here m is the slopeSlope
In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....
and b is the x-intercept. This system specifies coordinates for all lines that are not vertical. However, it is more common and simpler algebraically to use coordinates where the equation of the line is lx + my + 1 = 0. This system specifies coordinates for all lines except those that pass through the origin. The geometrical interpretations of l and m are the negative reciprocals of the x and y-intercept
Y-intercept
In coordinate geometry, using the common convention that the horizontal axis represents a variable x and the vertical axis represents a variable y, a y-intercept is a point where the graph of a function or relation intersects with the y-axis of the coordinate system...
respectively.
The exclusion of lines passing through the origin can be resolved by using a system of three coordinates to specify the line in which the equation, lx + my + n = 0. Here l and m may not both be 0. In this equation, only the ratios between l, m and n are significant, in other words if the coordinates are multiplied by a non-zero scalar then line represented remains the same. So is a system of homogeneous coordinates
Homogeneous coordinates
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
for the line.
If points in the plane are represented by homogeneous coordinates , the equation of the line is lx + my + nz = 0. In this context, l, m and n may not all be 0. In particular, represents the line z = 0, which is the line at infinity
Line at infinity
In geometry and topology, the line at infinity is a line that is added to the real 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.-Geometric formulation:In...
in the 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 that do not intersect...
. The coordinates and represent the x and y-axes respectively.
Tangential equations
Just as f(x, y) = 0 can represent a curveCurve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
as a subset of the points in the plane, the equation φ(l, m) = 0 represents a subset of the lines on the plain. The set of lines on the plane may, in an abstract sense, be thought of as the set of points in a projective plane, the dual
Duality (projective geometry)
A striking feature of projective planes is the "symmetry" of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this metamathematical concept. There are two approaches to the subject of duality, one through language and the other a more...
of the original plane. The equation φ(l, m) = 0 then represents a curve in the dual plane.
For a curve f(x, y) = 0 in the plane, the tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...
s to the curve form a curve in the dual space called the dual curve
Dual curve
In projective geometry, a dual curve of a given plane curve C is a curve in the dual projective plane consisting of the set of lines tangent to C. There is a map from a curve to its dual, sending each point to the point dual to its tangent line. If C is algebraic then so is its dual and the degree...
. If φ(l, m) = 0 is the equation of the dual curve, then it is called the tangential equation, for the original curve. A given equation φ(l, m) = 0 represents a curve in the original plane determined as the envelope
Envelope (mathematics)
In geometry, an envelope of a family of curves in the plane is a curve that is tangent to each member of the family at some point. Classically, a point on the envelope can be thought of as the intersection of two "adjacent" curves, meaning the limit of intersections of nearby curves...
of the lines that satisfy this equation. Similarly, if φ(l, m, n) is a homogeneous function
Homogeneous function
In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...
then φ(l, m, n) = 0 represents a curve in the dual space given in homogeneous coordinates, and may be called the homogeneous tangential equation of the enveloped curve.
Tangential equations are useful in the study of curves defined as envelopes, just as Cartesian equations are useful in the study of curves defined as loci.
Tangential equation of a point
A linear equation in line coordinates has the form al + bm + c = 0, where a, b and c are constants. Suppose (l, m) is a line that satisfies this equation. If c is not 0 then lx + my + 1 = 0, where x = a/c and y = b/c, so every line satisfying the original equation passes though the point (x, y). Conversely, any line through (x, y) satisfies the original equation, so al + bm + c = 0 is the equation of set of lines through (x, y). For a given point (x, y), the equation of the set of lines though it is lx + my + 1 = 0, so this may be defined as the tangential equation of the point. Similarly, for a point (x, y, z) given in homogeneous coordinates, then the equation of the point in homogeneous tangential coordinates is (lx, my, nz) = 0.Formulas
The intersection of the lines (l1, m1) and (l2, m2) is the solution to the linear equationsBy Cramer's rule
Cramer's rule
In linear algebra, Cramer's rule is a theorem, which gives an expression for the solution of a system of linear equations with as many equations as unknowns, valid in those cases where there is a unique solution...
, the solution is
The lines (l1, m1), (l2, m2), and (l3, m3) are concurrent
Concurrent lines
In geometry, two or more lines are said to be concurrent if they intersect at a single point.In a triangle, four basic types of sets of concurrent lines are altitudes, angle bisectors, medians, and perpendicular bisectors:...
when the determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
For homogeneous coordinates, the intersection of the lines (l1, m1, n1) and (l2, m2, n2) is
The lines (l1, m1, n1), (l2, m2, n2) and (l3, m3, n3) are concurrent
Concurrent lines
In geometry, two or more lines are said to be concurrent if they intersect at a single point.In a triangle, four basic types of sets of concurrent lines are altitudes, angle bisectors, medians, and perpendicular bisectors:...
when the determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
Dually, the coordinates of the line containing (x1, y1, z1) and (x2, y2, z2) are
Lines in three-dimensional space
For two given points in the plane, (x1, y1, z1) and (x2, y2, z2), the three determinantsdetermine the line containing them. Similarly, for two points in three-dimensional space (x1, y1, z1, w1) and (x2, y2, z2, w2), the line containing them is determined by the six determinants
This is the basis for a system of homogeneous line coordinates in three-dimensional space called Plücker coordinates. Six numbers in a set of coordinates only represent a line when they satisfy an additional equation. This system maps the space of lines in three-dimensional space to a projective space of dimension five, but with the additional requirement the space of lines is a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
of dimension four.
More generally, the lines in n-dimensional projective space are determined by a system of n(n − 1)/2 homogeneous coordinates that satisfy a set of (n − 2)(n − 3)/2 conditions, resulting in a manifold of dimension 2(n − 1).