[Proj] Arc length of meridian, parametric latitude, elliptic functions

[Charles Karney]

A note on the arc length of the meridian, parametric latitude, elliptic
functions.

The arc length of the meridian can be written in terms of the geographic
or parametric (= reduced) latitudes.  The conversion between the two
forms is simple enough.  However when expressed in terms of elliptic
functions the relation becomes simpler still.

Consider an ellipse (x/a)^2 + (z/b)^2 = 1.  Let

  phi = geographic latitude
  beta = parametric latitude
  y = arc length from equator
  k' = b/a
  k = sqrt(1 - k'^2)
  a*E = quarter meridian (E = 2nd complete elliptic integral)

The modulus for the elliptic integrals and functions here is k.

Define beta', phi', and y' as the complementary quantities:

  beta + beta' = phi + phi' = pi/2
  y + y' = a*E

Note that beta and phi and their complements have a dual relationship,
namely

  tan(beta) = k' * tan(phi)
  tan(phi') = k' * tan(beta')

Now, the parametric equations for the meridian are

  x = a * sin(beta') =   a   * cos(phi) / sqrt(1 - k^2 * sin(phi)^2)
  z = b * cos(beta') = b^2/a * sin(phi) / sqrt(1 - k^2 * sin(phi)^2)

Computing y' and y as integrals over beta' and phi

  y' =   a   * integrate( (1 - k^2 * sin(beta')^2)^( 1/2), beta' )
  y  = b^2/a * integrate( (1 - k^2 * sin( phi )^2)^(-3/2),  phi  )

Introduce u such that (am = Jacobi amplitude function)

  phi = am(u)

Define its complement by (K = 1st complete elliptic integral)

  u + u' = K

Then it is easily verified that

  beta' = am(u')

The equations for x, z above become

  x = a * sn(u') =   a   * cd(u)
  z = b * cn(u') = b^2/a * sd(u)

Similarly y' and y as integrals over u' and u

  y' =   a   * integrate( dn(u')^2, u' ) = a *      Eps(u')
  y  = b^2/a * integrate( nd(u )^2, u  ) = a * (E - Eps(u'))

where Eps is the Jacobi's Epsilon function http://dlmf.nist.gov/22.16.ii
(Beware: the upper limit of Eq. 22.16.14 should be sn(x,k).)  Note that

  Eps(u') = E(beta')

(Here, E = 2nd incomplete elliptic integral.)

The interrelation implied by phi = am(u) and beta' = am(u') can be
expressed as

  F(phi) + F(beta') = K

(F = 1st incomplete elliptic integral).  http://dlmf.nist.gov/19.11.E9
gives the relation explicitly

  tan(phi) = 1/(k' * tan(beta'))
  tan(beta) = k' * tan(phi)

i.e., the standard definition of the parametric latitude.

Connection with the transverse Mercator projection:

  y = Gauss-Krueger northing on the central meridian
  u = Thompson northing on the central meridian

  --Charles

-- 
Charles Karney <ckarney at sarnoff.com>
Sarnoff Corporation, Princeton, NJ 08543-5300

Tel: +1 609 734 2312
Fax: +1 609 734 2662