Welcome to the NavList Message Boards.

NavList:

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

Compose Your Message

Message:αβγ
Message:abc
Add Images & Files
    or...
       
    Reply
    Re: distance with atan2
    From: Herbert Prinz
    Date: 2005 Nov 12, 13:59 -0500

    I hope Paul Hirose does not mind if I step in here to explain the
    essential concept behind his method and while I am at it, correct a tiny
    inaccuracy.
    
    Paul Hirose wrote:
    
    > One way to calculate distance (separation angle) on a sphere is to use
    > vectors and the atan2 function or rectangular to polar conversion. Set x
    > to the dot product of the two vectors, and y to the magnitude of their
    > cross product. The separation angle is atan2(y, x), or you can get the
    > angle by converting x and y to polar form.
    
    [...]
    
    > An advantage of the vector method is its accuracy at all angles. A large
    > portion of Meeus' chapter on separation angle discusses ways to avoid
    > the inaccuracy of the traditional spherical trig formula near 0 or 180
    > degrees.
    >
    
    The numerical advantage of the method is not due to it being a vector
    method, but rather  lies in the avoidance of the arccos function. The
    goal is to replace  arccos by arctan.
    
    If we didn't mind the arccos, the vector method would be rather simple.
    The  angle between two unit vectors is DEFINED (!) as the arccos of
    their dot product. Therefore, if you want the angular separation of two
    such vectors representing positions on the celestial sphere or earth,
    compute their dot product (which is a cheap operation once you have the
    cartesian representation), take the arccos and you are done! Check this
    with the numbers in Paul's example.
    
    (Side note. As I have said before on this list, this simple method is
    good to at least 1 arc second on all programmable pocket calculators and
    any modern computer under a suitable programming language. In fact, on
    my PC, using 80 bit floating numbers and the C# math library, I can
    produce a maximum error of 3 milliarcseconds. The more refined methods
    in Meeus are for astronomers, not for boat navigators!)
    
    But let's go ahead and avoid the arccos at all cost. Since tan x = sin x
    / cos x, we can work with the arctan if we obtain the sine of the given
    angle in addition to the cosine, which we already have. That's where the
    X product comes in. Never mind its exact definition, the essential
    feature that we need here is that it is a vector the magnitude of which
    is equal to the area subtended by the two given vectors. Since these are
    unit vectors, it's easy to see that said area is equal to the sine of
    the angle. You can again verify this with the numbers in Paul's example.
    
    Now that we have the sine of the required angle (from the magnitude of
    the x product) and the cosine of the required angle (from the dot
    product), we may as well build the quotient (i.e. its tangent) and
    obtain the angle from it.
    
    The x product is, of course not a cheap operation on most calculators,
    and this is where the rub is.
    
    I hope that this comment is a useful starting point for those who want
    to explore the inner workings of the given recipe.
    
    Herbert Prinz
    
    
    

       
    Reply
    Browse Files

    Drop Files

    NavList

    What is NavList?

    Join NavList

    Name:
    (please, no nicknames or handles)
    Email:
    Do you want to receive all group messages by email?
    Yes No

    You can also join by posting. Your first on-topic post automatically makes you a member.

    Posting Code

    Enter the email address associated with your NavList messages. Your posting code will be emailed to you immediately.
    Email:

    Email Settings

    Posting Code:

    Custom Index

    Subject:
    Author:
    Start date: (yyyymm dd)
    End date: (yyyymm dd)

    Visit this site
    Visit this site
    Visit this site
    Visit this site
    Visit this site
    Visit this site