LINE GEOMETRY. In geometry, curves and other loci are frequently regarded as generated by a point, the position of which is defined by certain con ditions. A straight line however may be thought of as the generat ing element, such as a tangent to a curve, or as a generator of a cone. That branch of geometry in which the line is regarded as the generating element is called line geometry.
A point in space is determined by its distances from three planes which intersect in one point (i.e., by its co-ordinates), and a plane is defined by the totality of the points the co-ordinates of which satisfy a linear equation. The coefficients of the co-ordinates in the equation are called plane co ordinates. A point and a plane are in united position when their co-ordinates satisfy a certain bilinear equation (i.e., linear in the co-ordinates of the point, and also linear in the co-ordinates of the plane). The principle of duality (q.v.) asserts that for every theorem involving points, obtained from properties of point co ordinates, there is another involving planes, and conversely. These properties remain true if each point co-ordinate is replaced by any linear fractional homogeneous expression in four numbers xi, x2, x3, with common denominator and non-vanishing determinant. The numbers xi are called projective point co-ordinates. Simi larly, u3, u4 may be taken as projective plane co-ordinates, and the condition for united position may be expressed by the equation 2 uixi=o.
In the plane, the projective co-ordinates, may be defined either as point co-ordinates or as line co-ordinates, thus giving rise to point-line duality in the plane. In space a straight line is self dual ; it is uniquely determined by any two distinct points on it, or by any two distinct planes through it. The line itself may be regarded as an element; for many purposes in kinematics, in dynamics, and in optics this is more advantageous than to have it defined indirectly in terms of points or planes.
A line meets two distinct planes each in one point, finite or infinite, and a point in each is determined when its two co ordinates are fixed. Thus it follows that a line is determined by four independent co-ordinates, or there exist lines in space.
When the co-ordinates pile satisfy one equation, there are ac lines singled out, which constitute a complex; if they satisfy two such equations, the co' lines constitute a congruence; if they satisfy three, the co' lines belong to a ruled surface; if they satisfy four, there may be only a finite number of lines. The linear complex can, by a proper choice of a system of co-ordinates, be reduced to the form xy'—xiy-Fk(z—z') =o, in which x, y, z and y', z' are the cartesian co-ordinates of any two points on a line. If k=o, the complex is special; it now con sists of all the lines which meet the fixed line y=o, the axis. If k is not zero, the line x=o, y=o is still called the axis but not all the lines of the complex meet it. The complex can be visualized in various ways. It is identical with the null system of mechanics, i.e., the totality of those axes of rotation with re gard to which a system of forces in space have a moment of rota tion zero. Associated with every point (pole) there is (see POLE AND POLAR) a plane (polar) passing through it and conversely such that associated with every point is a pencil of lines of the complex through it, and lying in its polar plane.