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gnomonic.jl

Astrometry.SOFA.tpors — Function
tpors(ξ::Float64, η::Float64, a::Float64, b::Float64)

In the tangent plane projection, given the rectangular coordinates of a star and its spherical coordinates, determine the spherical coordinates of the tangent point.

Input

  • ξ, η – rectangular coordinates of star image (Note 2)
  • a, b – star's spherical coordinates (Note 3)

Output

  • a01, b01 – tangent point's spherical coordinates, Soln. 1
  • a02, b02 – tangent point's spherical coordinates, Soln. 2

Note

  1. The tangent plane projection is also called the "gnomonic projection" and the "central projection".

  2. The eta axis points due north in the adopted coordinate system. If the spherical coordinates are observed (RA,Dec), the tangent plane coordinates (ξ,η) are conventionally called the "standard coordinates". If the spherical coordinates are with respect to a right-handed triad, (ξ,η) are also right-handed. The units of (ξ,η) are, effectively, radians at the tangent point.

  3. All angular arguments are in radians.

  4. The angles a01 and a02 are returned in the range 0-2π. The angles b01 and b02 are returned in the range +/-π, but in the usual, non-pole-crossing, case, the range is +/-π/2.

  5. Cases where there is no solution can arise only near the poles. For example, it is clearly impossible for a star at the pole itself to have a non-zero ξ value, and hence it is meaningless to ask where the tangent point would have to be to bring about this combination of ξ and dec.

  6. Also near the poles, cases can arise where there are two useful solutions. The return value indicates whether the second of the two solutions returned is useful; 1 indicates only one useful solution, the usual case.

  7. The basis of the algorithm is to solve the spherical triangle PSC, where P is the north celestial pole, S is the star and C is the tangent point. The spherical coordinates of the tangent point are [a0,b0]; writing ρ^2 = (ξ^2+η^2) and r^2 = (1+ρ^2), side c is then (π/2-b), side p is sqrt(ξ^2+η^2) and side s (to be found) is (π/2-b0). Angle C is given by sin(C) = ξ/ρ and cos(C) = η/ρ. Angle P (to be found) is the longitude difference between star and tangent point (a-a0).

  8. This function is a member of the following set:

    spherical      vector         solve for

eraTpxes      eraTpxev          ξ,η
eraTpsts      eraTpstv          star

    eraTpors < eraTporv origin

References

Calabretta M.R. & Greisen, E.W., 2002, "Representations of celestial coordinates in FITS", Astron.Astrophys. 395, 1077

Green, R.M., "Spherical Astronomy", Cambridge University Press, 1987, Chapter 13.

source
Astrometry.SOFA.tporv — Function
tporv(ξ::Float64, η::Float64, v::Vector{Float64})

In the tangent plane projection, given the rectangular coordinates of a star and its direction cosines, determine the direction cosines of the tangent point.

Input

  • ξ, η – rectangular coordinates of star image (Note 2)
  • v – star's direction cosines (Note 3)

Output

  • v01 – tangent point's direction cosines, Solution 1
  • v02 – tangent point's direction cosines, Solution 2

Note

  1. The tangent plane projection is also called the "gnomonic projection" and the "central projection".

  2. The eta axis points due north in the adopted coordinate system. If the direction cosines represent observed (RA,Dec), the tangent plane coordinates (ξ,η) are conventionally called the "standard coordinates". If the direction cosines are with respect to a right-handed triad, (ξ,η) are also right-handed. The units of (ξ,η) are, effectively, radians at the tangent point.

  3. The vector v must be of unit length or the result will be wrong.

  4. Cases where there is no solution can arise only near the poles. For example, it is clearly impossible for a star at the pole itself to have a non-zero xi value, and hence it is meaningless to ask where the tangent point would have to be.

  5. Also near the poles, cases can arise where there are two useful solutions. The return value indicates whether the second of the two solutions returned is useful; 1 indicates only one useful solution, the usual case.

  6. The basis of the algorithm is to solve the spherical triangle PSC, where P is the north celestial pole, S is the star and C is the tangent point. Calling the celestial spherical coordinates of the star and tangent point (a,b) and (a0,b0) respectively, and writing ρ^2 = (ξ^2+η^2) and r^2 = (1+ρ^2), and transforming the vector v into (a,b) in the normal way, side c is then (π/2-b), side p is sqrt(ξ^2+η^2) and side s (to be found) is (π/2-b0), while angle C is given by sin(C) = ξ/ρ and cos(C) = η/ρ; angle P (to be found) is (a-a0). After solving the spherical triangle, the result (a0,b0) can be expressed in vector form as v0.

  7. This function is a member of the following set:

    spherical      vector         solve for

eraTpxes      eraTpxev         xi,eta
eraTpsts      eraTpstv          star
eraTpors    > eraTporv <       origin

References

Calabretta M.R. & Greisen, E.W., 2002, "Representations of celestial coordinates in FITS", Astron.Astrophys. 395, 1077

Green, R.M., "Spherical Astronomy", Cambridge University Press, 1987, Chapter 13.

source
Astrometry.SOFA.tpsts — Function
tpsts(ξ::Float64, η::Float64, a0::Float64, b0::Float64)

In the tangent plane projection, given the star's rectangular coordinates and the spherical coordinates of the tangent point, solve for the spherical coordinates of the star.

Input

  • ξ, η – rectangular coordinates of star image (Note 2)
  • a0, b0 – tangent point's spherical coordinates

Output

  • a, b – star's spherical coordinates

Note

  1. The tangent plane projection is also called the "gnomonic projection" and the "central projection".

  2. The eta axis points due north in the adopted coordinate system. If the spherical coordinates are observed (RA,Dec), the tangent plane coordinates (ξ,η) are conventionally called the "standard coordinates". If the spherical coordinates are with respect to a right-handed triad, (ξ,η) are also right-handed. The units of (ξ,η) are, effectively, radians at the tangent point.

  3. All angular arguments are in radians.

  4. This function is a member of the following set:

    spherical      vector         solve for

eraTpxes      eraTpxev         xi,eta

    eraTpsts < eraTpstv star

    eraTpors      eraTporv         origin

References

Calabretta M.R. & Greisen, E.W., 2002, "Representations of celestial coordinates in FITS", Astron.Astrophys. 395, 1077

Green, R.M., "Spherical Astronomy", Cambridge University Press, 1987, Chapter 13.

source
Astrometry.SOFA.tpstv — Function
tpstv(ξ::Float64, η::Float64, v0::Vector{Float64})

In the tangent plane projection, given the star's rectangular coordinates and the direction cosines of the tangent point, solve for the direction cosines of the star.

Input

  • ξ, η – rectangular coordinates of star image (Note 2)
  • v0 – tangent point's direction cosines

Output

  • v – star's direction cosines

Note

  1. The tangent plane projection is also called the "gnomonic projection" and the "central projection".

  2. The η axis points due north in the adopted coordinate system. If the direction cosines represent observed (RA,Dec), the tangent plane coordinates (ξ,η) are conventionally called the "standard coordinates". If the direction cosines are with respect to a right-handed triad, (ξ,η) are also right-handed. The units of (ξ,η) are, effectively, radians at the tangent point.

  3. The method used is to complete the star vector in the (xi,eta) based triad and normalize it, then rotate the triad to put the tangent point at the pole with the x-axis aligned to zero longitude. Writing (a0,b0) for the celestial spherical coordinates of the tangent point, the sequence of rotations is (b-π/2) around the x-axis followed by (-a-π/2) around the z-axis.

  4. If vector v0 is not of unit length, the returned vector v will be wrong.

  5. If vector v0 points at a pole, the returned vector v will be based on the arbitrary assumption that the longitude coordinate of the tangent point is zero.

  6. This function is a member of the following set:

    spherical      vector         solve for

eraTpxes      eraTpxev           ξ,η
eraTpsts    > eraTpstv <        star
eraTpors      eraTporv         origin

References

Calabretta M.R. & Greisen, E.W., 2002, "Representations of celestial coordinates in FITS", Astron.Astrophys. 395, 1077

Green, R.M., "Spherical Astronomy", Cambridge University Press, 1987, Chapter 13.

source
Astrometry.SOFA.tpxes — Function
tpxes(a::Float64, b::Float64, a0::Float64, b0::Float64)

In the tangent plane projection, given celestial spherical coordinates for a star and the tangent point, solve for the star's rectangular coordinates in the tangent plane.

Input

  • a, b – star's spherical coordinates
  • a0, b0 – tangent point's spherical coordinates

Output

  • ξ, η – rectangular coordinates of star image (Note 2)

Note

  1. The tangent plane projection is also called the "gnomonic projection" and the "central projection".

  2. The η axis points due north in the adopted coordinate system. If the spherical coordinates are observed (RA,Dec), the tangent plane coordinates (ξ,η) are conventionally called the "standard coordinates". For right-handed spherical coordinates, (ξ,η) are also right-handed. The units of (ξ,η) are, effectively, radians at the tangent point.

  3. All angular arguments are in radians.

  4. This function is a member of the following set:

    spherical      vector         solve for

    eraTpxes < eraTpxev ξ,η

    eraTpsts      eraTpstv          star
eraTpors      eraTporv         origin

References

Calabretta M.R. & Greisen, E.W., 2002, "Representations of celestial coordinates in FITS", Astron.Astrophys. 395, 1077

Green, R.M., "Spherical Astronomy", Cambridge University Press, 1987, Chapter 13.

source
Astrometry.SOFA.tpxev — Function
tpxev(v::Vector{Float64}, v0::Vector{Float64})

In the tangent plane projection, given celestial direction cosines for a star and the tangent point, solve for the star's rectangular coordinates in the tangent plane.

Input

  • v – direction cosines of star (Note 4)
  • v0 – direction cosines of tangent point (Note 4)

Output

  • ξ, η – tangent plane coordinates of star

Note

  1. The tangent plane projection is also called the "gnomonic projection" and the "central projection".

  2. The eta axis points due north in the adopted coordinate system. If the direction cosines represent observed (RA,Dec), the tangent plane coordinates (ξ,η) are conventionally called the "standard coordinates". If the direction cosines are with respect to a right-handed triad, (ξ,η) are also right-handed. The units of (ξ,η) are, effectively, radians at the tangent point.

  3. The method used is to extend the star vector to the tangent plane and then rotate the triad so that (x,y) becomes (ξ,η). Writing (a,b) for the celestial spherical coordinates of the star, the sequence of rotations is (a+π/2) around the z-axis followed by (π/2-b) around the x-axis.

  4. If vector v0 is not of unit length, or if vector v is of zero length, the results will be wrong.

  5. If v0 points at a pole, the returned (ξ,η) will be based on the arbitrary assumption that the longitude coordinate of the tangent point is zero.

  6. This function is a member of the following set:

    spherical      vector         solve for

eraTpxes    > eraTpxev <         ξ,η
eraTpsts      eraTpstv          star
eraTpors      eraTporv         origin

References

Calabretta M.R. & Greisen, E.W., 2002, "Representations of celestial coordinates in FITS", Astron.Astrophys. 395, 1077

Green, R.M., "Spherical Astronomy", Cambridge University Press, 1987, Chapter 13.

source