Direction (geometry, geography)

Direction is the information vested in the relative position of a destination (or target) point with respect to a starting point, absent the distance information. Directions may be either relative to some indicated reference (the violins in a full orchestra are typically seated to the left of the conductor), or absolute according to some previously agreed upon frame of reference (New York City lies due west of Madrid). Direction is often indicated manually by an extended index finger or written as an arrow. On a vertically oriented sign representing a horizontal plane, such as a road sign, "forward" is usually indicated by an upward arrow. Mathematically, direction may be uniquely specified by a direction vector ( a unit vector used to represent spatial direction), or equivalently by the angles made by the most direct path with respect to a specified set of axes.

Examples of 2D and 3D directions represented numerically by direction vectors are shown. The same unit vector construct is used to specify spatial directions in 2D, 3D or higher dimensions. As illustrated, each unique 2D direction is equivalent numerically to a point on the unit circle, while each unique 3D direction may be represented numerically as a point on the unit sphere.

The primary advantage of direction vectors over angular representations comes in software computation, where the simplicity of software algorithms benefits from representations which present the fewest exceptions to be handled. For example, in software where 2D direction is represented by angle theta, there is a discontinuity in value that jumps from 2*pi --> 0 (assuming a preference for a 1:1 numerical representation). 2D direction vectors vary continuously while offering 1:1 representation, thus eliminating a singularity exception to be handled. Representational simplicity is even greater for direction vectors in 3D, as compared to polar angle-pair (phi, theta).

In Euclidean and affine geometry, the direction of a subspace is the vector space associated to this subspace. In the case of a subspace of dimension one (that is a line) the direction vectors are the nonzero vectors of this vector space.

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