Line (geometry)

A straight line or short line is an element of geometry. The shortest distance between two points is straight and is referred to as distance. A straight, infinitely long, infinitely thin and unlimited in both directions is called a straight line. However, modern axiomatic theories of geometry take it no relation ( Synthetic geometry). For them, a straight line is an object without internal properties, only the relations to other lines, points and levels are important. In analytical geometry, a straight line as a set of points is realized. More: In an affine space is a straight line a one-dimensional affine subspace.

Synthetic geometry

In his Elements, Euclid has given an explicit definition of a straight line that corresponds to the intuitive picture. For rates and their evidence, however, this definition does not matter. Therefore Modern axiom systems waive such definition.

A straight line in this case is a term to take the individual axioms reference. An example is the first axiom of Hilbert's axiom system:

The meaning of the term straight line results from the corpus of the axioms. An interpretation as an infinitely long, infinitely thin line is not mandatory, but only a suggestion of what you could vividly imagine it.

In the projective plane, the terms point and line even completely interchangeable ( duality ). Thus, it is possible here, a straight line is infinitely imagined as infinitely small point, and a long and infinitely thin as.

Analytic Geometry

In analytic geometry, the geometric space is represented as a - dimensional vector space over a field. A straight line is defined as one-dimensional affine subspace of this vector space, ie. Coset as a one-dimensional linear subspace

In three dimensions over the field of real numbers, the definition of lines of Analytical Geometry all conditions, which presupposes Hilbert in his system of axioms of geometry met. In this case is a straight line hence a straight line in the sense of Hilbert.

All you need is the position of two points to describe a straight line. One of the points is used as a " prop " the straight line on him " lies " it so to speak - hence this point is called the receptor point or base of the line. The second point to obtain the direction of the straight line. The direction is in this case given by the vector from the start point to the " direction point."

The line through the points and contains exactly the points whose position vector an illustration

Has, so

Here, the support vector, that is the position vector of the interpolation point and the direction vector.

The affine hull of two different vectors and

Is also a straight line.

Also, the solution space of a ( inhomogeneous ) linear system of equations with linearly independent equations is an affine subspace of dimension one and thus a straight line. In two dimensions, a straight line therefore by a linear equation

Be given, with or nonzero and must be. Is not equal to 0, then one speaks of a linear function on K.

Shortest path

In the real Euclidean space, the shortest path between two points is a straight line. Generalizing this property of straight lines on curved spaces ( manifolds ), we come to the concept of geodesic, short geodesic.

Determine the equation of a line in the plane

The equation of a line in the plane can be determined in three different ways:

Point towards equation:

  • May be a point and the tilt angle (slope angle).
  • Given a start point and the slope (the slope ).

Two -point equation:

  • Given are two points with.


Generating the straight line in space ℝ ⁿ

Point Directions equation

For each pair consisting of a position vector (ie, point ) and a direction vector, there is a straight line which includes and extends in the direction, namely

Two -point equation

Given two position vectors (ie, points). Then there exists a uniquely determined line that contains and, namely

Position of two straight lines to one another

Two straight lines can have the following situation relationships. You can:

  • Be the same: Both lines have all points in common.
  • An intersection own: Both lines have exactly one point in common (especially: perpendicular to each other ).
  • Mutually be genuine parallel: Both lines have no point in common and can be converted into each other by a shift.
  • Be skewed to one another: Both lines have no point in common, but can not alone by a shift into one another (from at least three dimensions).

In terms of the theory of relations is also called parallelism if both lines are identical, in particular, any straight line parallel to itself. To clarify then, a distinction between genuine parallel and identical.