# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Navigational Algorithms - 3 COPs analytical solution**

**From:**Wolfgang K�berer

**Date:**2006 Oct 18, 11:35 -0700

Dear Andres Ruiz, by chance I have a copy of Navigation, Vol. 25, No. 4, with the article by Watkins and Janiczek. I' ll send you a scan off-list tomorrow. And there is a french book on "La navigation astronomique" by Philippe Bourbon published by the Institut oc�anographique (Paris 2000), which makes use of matrices for reduction of multiple sights. It costs about 40 � and may be ordered via Amazon or directly from the Institut oc�anographique. I haven't had the time to try to follow the math, though, so I cannot comment on its merits ot faults. By the way - it may interest other list members - I bought the copy of "Navigation" 26 years ago because it contained an article by D.W.Kerst on "Longitude by lunar observations and the pocket calculator". It is based on the Chauvenet (direct) method and I wrote a program at that time which solved the formulae on a HP 41. Never tried it in the real world, though. I just owned a Davis Mark III then and did'nt think that could be accurate enough for lunar observations. Regards, Wolfgang K�berer Andres Ruiz schrieb: > [NavList 1437] Re: Navigator's Newsletter, and other forums. > From: jrvz---net > Date: 17 Oct 2006 22:45 > > ... > (It does require inverting 3x3 matrices, so it's not well suited to > hand calculation.) > If someone remembers seeing something like this, I would appreciate a > reference. Otherwise I'll write it up in more detail. > > --------------------------------------------------------- > > > > > > Now I am writing some articles to be published on my web page under the name Navigational Algorithms. > > The draft of one about circles of position is: > > > > > 3 COPs analytical solution > > > > > Input data: 3 observations > > > > K > > Body > > Ho > > GHA > > Dec > > 1 > > Body1 > > Ho1 > > GHA1 > > Dec1 > > 2 > > Body2 > > Ho2 > > GHA2 > > Dec2 > > 3 > > Body3 > > Ho3 > > GHA3 > > Dec3 > > > > > > > > Fix: BF LF > > > > > > > > Manual Solution > > > > > > > > > > > > > > > M3 = x1*y2*z3+y1*z2*x3+z1*x2*y3-(z1*y2*x3+x1*z2*y3+y1*x2*z3) > > > > Mx = y2*z3+y1*z2+y3*z1-(y2*z1+y1*z3+y3*z2) > > My = z2*x3+z1*x2+z3*x1-(z2*x1+z1*x3+z3*x2) > > Mz = x2*y3+x1*y2+x3*y1-(x2*y1+x1*y3+x3*y2) > > > > > > > Example by hand > > > > > K > > Body > > Ho > > GHA > > Dec > > 1 > > Arcturus > > 22.2 > > 166.829 > > 19.335 > > 2 > > Vega > > 70.15 > > 115.277 > > 38.758 > > 3 > > Dubhe > > 20.35 > > 226.98 > > 61.909 > > > > > > | -2.432 0.569 0.876 | | 0.006| | 1 | > > | -0.354 0.750 0.666 | | 0.731| = | 1 | > > | -0.924 -0.990 2.537 | | 0.682| | 1 | > > > > B [0-90][+N/-S] > > L [0-360][+ W->E] > > > > Fix(42.982085093299339, 89.566484629127913) > > > > > > This method work only for one point of intersection. Otherwise a least squares method must be used to find the most probable celestial position. > > > > Help: > > I am looking for some information about sight reduction with matrices, I have find two articles: > > > > Dewit, C., "Some Remarks on Sight Reduction with Matrices," Navigation: J. Inst. Naviga., Vol. 26, No. 3, pp. 252-253 (1979). > > Watkins. R. and Janiczek. P. M., Sight Reduction with Matrices, NAVIGATION, Journal of The Institute of Navigation, Vol. 25, No. 4, Winter 1978-79, pp. 447-48. > > > > if someone could send me some copies or additional information by e-mail, I,ll be very thankful. > > > > Andr�s > > http://www.geocities.com/andresruizgonzalez > > > > > ------_=_NextPart_002_01C6F29C.75CAF27F > Content-Type: text/html; charset=iso-8859-1 > Content-Transfer-Encoding: quoted-printable > X-Google-AttachSize: 24915 > > > > > > > > > > > > > >> >> > > > > > ------_=_NextPart_002_01C6F29C.75CAF27F-- --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to NavList@fer3.com To unsubscribe, send email to NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---style='font-size:10.0pt;font-family:"Courier New"'>[NavList 1437] Re: > Navigator's Newsletter, and other forums.

> >

> From: jrvz---net

> Date: 17 Oct 2006 22:45`…`

`size=2 face="Courier New">(It does require inverting 3x3 matrices, so it's not well suited to`

`size=2 face="Courier New">hand calculation.)`

`size=2 face="Courier New">If someone remembers seeing something like this, I would appreciate a`

`size=2 face="Courier New">reference. Otherwise I'll write it up in more detail.`

> >---------------------------------------------------------

> >size=2 color=navy face=Arial>

> >size=2 color=navy face=Arial>

> >size=2 face=Arial>Now > I am writing some articles to be published on my web page under the name

> >.style='font-weight:bold;font-style:italic'>Navigational Algorithmssize=2 face=Arial>The > draft of one about circles of position is:

> >> >

> >> >## > >

3 COPs > analytical solution> >

Input data: 3 observations

> >> > style='border-collapse:collapse'> > > > > > > > > > > > > > > > > > > > > > > > > > > > > >

> >

> >

face=Arial>K> >

face=Arial>Body> >

face=Arial>Ho> >

face=Arial>GHA> >

face=Arial>Dec> 1

>> face=Arial>Body1

>> face=Arial>Ho1

>> face=Arial>GHA1

>> face=Arial>Dec1

>> 2

>> face=Arial>Body2

>> face=Arial>Ho2

>> face=Arial>GHA2

>> face=Arial>Dec2

>> 3

>> face=Arial>Body3

>> face=Arial>Ho3

>> face=Arial>GHA3

>> face=Arial>Dec3

>src="http://fer3.com/arc/feed.aspx?m2=1&cid=img/101441.image001-jpg-01C6F2AA-04F86E70">

> >> >

> >

Fix: BF LF

> >> >

> >> >## > >

## > >

Manual > Solution> > style='border-collapse:collapse'> > > > > >

> >

> src="http://fer3.com/arc/feed.aspx?m2=1&cid=img/101441.image010-gif-01C6F2A9-46607840">

>> src="http://fer3.com/arc/feed.aspx?m2=1&cid=img/101441.image012-gif-01C6F2A9-46607840">

>> >

style='font-size:12.0pt;font-family:"Courier New";color:navy'>

> >> >

M3 = > x1*y2*z3+y1*z2*x3+z1*x2*y3-(z1*y2*x3+x1*z2*y3+y1*x2*z3)

> >> >

Mx = y2*z3+y1*z2+y3*z1-(y2*z1+y1*z3+y3*z2)

> >style='font-size:12.0pt;font-family:"Courier New"'>My = > z2*x3+z1*x2+z3*x1-(z2*x1+z1*x3+z3*x2)

> >Mz = x2*y3+x1*y2+x3*y1-(x2*y1+x1*y3+x3*y2)

> >> >

clear=all style='page-break-before:always'> > > >## > >

style='font-size:16.0pt'>Example by hand> > style='border-collapse:collapse'> > > > > > > > > > > > > > > > > > > > > > > > > > > > > >

> >

> >

face=Arial>K> >

face=Arial>Body> >

face=Arial>Ho> >

face=Arial>GHA> >

face=Arial>Dec> 1

>> Arcturus

>> face=Arial>22.2

>> face=Arial>166.829

>> face=Arial>19.335

>> 2

>> Vega

>> face=Arial>70.15

>> face=Arial>115.277

>> face=Arial>38.758

>> 3

>> Dubhe

>> face=Arial>20.35

>> face=Arial>226.98

>> face=Arial>61.909

>> >

style='font-size:12.0pt;font-family:"Courier New"'>

> >style='font-size:12.0pt;font-family:"Courier New"'>| -2.432 0.569 > 0.876 | | 0.006| | 1 |

> >style='font-size:12.0pt;font-family:"Courier New"'>| -0.354 0.750 > 0.666 | | 0.731| = | 1 |

> >style='font-size:12.0pt;font-family:"Courier New"'>| -0.924 -0.990 2.537 > | | 0.682| | 1 |

> >> >

B [0-90][+N/-S]

> >L [0-360][+ W->E]

> >> >

Fix(42.982085093299339, 89.566484629127913)

> >> >

> >

This method work only for one point of intersection. > Otherwise a least squares method must be used to find the most probable > celestial position.

> >> >

Help:

> >I am looking for some information about sight reduction > with matrices, I have find two articles:

> >> >

Dewit, C., “Some Remarks on Sight Reduction > with Matrices,” Navigation: J. Inst. Naviga., Vol. 26, No. 3, pp. 252-253 > (1979).

> >Watkins. R. and Janiczek. P. M., Sight Reduction with > Matrices, NAVIGATION, Journal of The Institute of Navigation, Vol. 25, No. 4, > Winter 1978-79, pp. 447-48.

> >> >

if someone could send me some copies or additional > information by e-mail, I,ll be very thankful.

> >> >

Andr�s

> >http://www.geocities.com/andresruizgonzalez

> >> >