should be treated as rough rule of thumb approximations only. In astronomy, right ascension and declination are usually measured in angular units, expressed in terms of time, based on a 24-hour day. Angles between curves The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. , on both sides, κυρτός, convex) or cissoidal (Gr. κισσός, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίς, a tool for scraping), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave. Bisecting and trisecting angles The ancient Greek mathematicians knew how to bisect an angle (divide it into two angles of equal measure) using only a compass and straightedge, but could only trisect certain angles. In 1837, Pierre Wantzel showed that for most angles this construction cannot be performed. Dot product and generalisations In the Euclidean space, the angle θ between two Euclidean vectors u and v is related to their dot product and their lengths by the formula This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their normal vectors and between skew lines from their vector equations. Inner product To define angles in an abstract real inner product space, we replace the Euclidean dot product ( · ) by the inner product , i.e. In a complex inner product space, the expression for the cosine above may give non-real values, so it is replaced with or, more commonly, using the absolute value, with The latter definition ignores the direction of the vectors and thus describes the angle between one-dimensional subspaces and spanned by the vectors and correspondingly. Angles between subspaces The definition of the angle between one-dimensional subspaces and given by in a Hilbert space can be extended to subspaces of any finite dimensions. Given two subspaces , with , this leads to a definition of angles called canonical or principal angles between subspaces. Angles in Riemannian geometry In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where U and V are tangent vectors and gij are the components of the metric tensor G, Hyperbolic angle A hyperbolic angle is an argument of a hyperbolic function just as the circular angle is the argument