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    Re: Hc formula difference why?
    From: Bruce Cutting
    Date: 2017 Nov 21, 12:40 -0700

    The "subtraction" rule is from numerical analysis.  Always avoid
    subtraction and division when possible.  It is best to aggregate the
    numbers before doing either subtraction or division.  Te reason is that
    subtraction and division produce larger errors because the precision of
    the components are added for thee operations.
    > *Herman you wrote:
    > The formula for solving the calculated height, in the celestial triangle,
    > is written in different order and ways.*
    > *You see Hc=arcSin((Cos L • Cos d • Cos t) + (Sin L • Sin d))
    >               Hc=arcSin((Cos L • Cos d • Cos t) +/- (Sin L • Sin d))
    >               Hc=arcSin((Cos L • Cos d • Cos t) ~ (Sin L • Sin d))
    >               Hc=arcSin((Sin L • Sin d)+(Cos L • Cos d • Cos t) )*
    > *And people ad it with different rules. For instance I remember ~ says
    > subtract the smallest value from the biggest??*
    > *Why al these differences, is it a kind of interpretation?*
    > *What is the "original" formula?*
    > *Regards,
    > HermanD*
    > Speaking as a chap who can barely remember the rules for plane triangles
    > these days and who never could remember anything to do with spherical
    > triangles for more than half an hour, this is my interpretation.
    > What is being attempted here is to find the third side of a spherical
    > triangle from two sides and an included angle.  However, Hc, latitude,
    > declination, and LHA are not necessarily the sides and angle in the
    > triangle.  E.g. in this case the unknown side is co-alt (90-Hc), not Hc,
    > and t the known angle might be LHA or 360-LHA depending upon where the
    > body is in relation to the observer.  Moreover, the body might be in the
    > same hemisphere as the observer or the opposite, and P in the PZX triangle
    > might be the North or the South Pole, so it all gets a bit complicated.
    > E.g. if P and Z are in one hemisphere and X is in the other, the
    > declination side is 90+dec not co-dec. Finally, the known sides or the
    > angle might be more that 90, so the trig functions will only be positive
    > according to the All, Sin, Tan, Cos rule, and these latter two examples
    > are where I would suggest the + or – comes in.   If you want an original
    > formula, I think you probably have to go back to a spherical triangle with
    > angles A, B, and C, and sides a, b, and c, and adapt that to co-alt,
    > co-lat, co-dec, and LHA or 360-LHA as suits your particular case.
    > So from the Cosine rule:
    > cosa=cosb.cosc + sinb.sinc.cosA
    > or:
    > cosco-alt=cosco-lat.cosco-dec+sinco-lat.sinco-dec.cost
    > but: sinx=90-cosx and cosx=90-sinx
    > so:
    > sinalt=sinlat.sindec+coslat.cosdec.cost
    > or:
    > Hc = arcsin(coslat.cosdec.cost + sinlat.sindec)
    > DaveP
    > [plain text auto-generated]
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    > http://fer3.com/arc/m2.aspx/Hc-formula-difference-why-DavidPike-nov-2017-g40658

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