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Memorandum N 2
24.06.997
Arrangement on storing coordinate values and operations on them

Vector values 3D. In memorandums are marked by bold letters. Example: x=(x1;x2;x3). Stored in the form of the one-dimensional array of three numbers. Basic coordinate system - ICRS (international celestial reference system), more exactly ICRF (international celestial reference frame) as a system of three coordinate axes with high accuracy (~10–3 angular seconds) binded to motionless extragalactic sources of the electromagnetic radiation. This system is close to the orthonormal trihedron, in which the first vector is directed to the mean 0 Aries at the epoch J2000, and third - to the mean celestial pole at the same epoch. No need to to convert vector values to calculate their components in other reference systems, since here the vector calculation system of all necessary Cartesian and angular values has adopted. Instead of vector components the orthonormal trihedrons of coordinate systems are subjected to transformations.

Orthonormal trihedrons(reference frames). Orthonormal trihedron is a system of three unit vectors, which components are given in ICRS. Their length is equal to 1, and they perpendicular to each other. To specify a reference frame it is necessary to fix two perpendicular unit vectors with their ordinal numbers. To complete the reference frame it is necessary to determine the third vector so that the resulting orthonormal trihedron will be right-handed. So to calculate the orientation of some coordinate system it is sufficiently to calculate 2 elements of trihedron.

Scalar (or inner) product. Symbol and example: x y=x1y1+x2y2+x3y3.

Vector product. Symbol and example: x y=(x2y3–x3y2)1+(x3y1–x1y3)2+(x1y2–x2y1)3=(x2y3–x3y2; x3y1–x1y3; x1y2–x2y1).

Cartesian coordinate of vector.Calculated by scalar product of this vector with the corresponding element of reference frame. Example: x1=x 1

Admissible transformations. Only the ortogonal transformation are admissible as they save the length of vector and the angle between two vectors.

Length of vector. Root of scalar square of vector. Invariant.

Normalized vector. Vector, which components are divided by the length of this vector.Symbol and example: {x}=(x1/lx;x2/lx;x3/lx)

Angle between 2 vectors. Angle is calculated as an arccos of scalar product of this two vectors after their normalization. a=arccos({x} {y}).

Angular polar coordinate of vector (q). Calculated by taking an scalar product of the third element of reference frame with the required vector after its normalization.
q=arccos({x} 3)

Angular azimuth coordinate of vector (j). At first two scalar products of vector {{x}({x} 3)3} with the first and second elements of reference frame are calculated. First innerproduct is equal to a1=cosj, second — a2=sinj. According these two values using the function arctan azimuth angle is calculated within the range of [0, 2p]: j=atan2(a2;a1).

Rotation of vector by angle e around the axis z (in positive direction). Symbol and example:
R3(e)x=(x1cose–x2sine;x1sine+x2cose;x3)

Rotation of vector by angle e around the axis y (in positive direction). Symbol and example:
R3(e)x=(x3sine+x1cose;x2;x3cose–x1sine)

Rotation of vector by angle e around the axis x (in positive direction). Symbol and example:
R3(e)x=(x1;x2cose–x3sine;x3cose+x2sine)