represented as −45° is effectively equivalent to an orientation represented as 360° − 45° or 315°. Although the final position is the same, a physical rotation (movement) of −45° is not the same as a rotation of 315° (for example, the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor). In three-dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined in terms of an orientation, which is typically defined by a normal vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie. In navigation, bearings or azimuth are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so a north-west orientation corresponds to a bearing of 315°. Alternative ways of measuring an angle For an angular unit, it is definitional that the angle addition postulate holds. Some angle measurements where the angle addition postulate does not hold include: The slope or gradient is equal to the tangent of the angle; a gradient is often expressed as a percentage. For very small values (less than 5%), the slope of a line is approximately the measure in radians of its angle with the horizontal direction. The spread between two lines is defined in rational geometry as the square of the sine of the angle between the lines. As the sine of an angle and the sine of its supplementary angle are the same, any angle of rotation that maps one of the lines into the other leads to the same value for the spread between the lines. Although done rarely, one can report the direct results of trigonometric functions, such as the sine of the angle. Astronomical approximations Astronomers measure apparent sizes of and distances between objects in degrees from their point of observation. 0.5° is the approximate diameter of the Sun and of the Moon as viewed from Earth. 1° is the approximate width of the little finger at arm's length. 10° is the approximate width of a closed fist at arm's length. 20° is the approximate width of a handspan at arm's length. These measurements clearly depend on the individual subject, and the above