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Formulae

Equatorial Coordinate

R = tan(π/2 - δ/2)

θ = α

X = R cos(θ)

Y = R sin(θ)

where (α, δ) = (Right ascension, Declination)

Horizontal Coordinate

R = tan(π/2 - alt/2)

θ = az

X = R cos(θ)

Y = R sin(θ)

where (az, alt) = (Azimuth, Altitude)

Wikipedia

Stereographic projection

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