<html><body name="Mail Message Editor"><div><br></div><div>Charles & the interested:</div><div><br></div><div>Daylight casts unflattering light on blemishes. This line is not cogent:</div><div><br></div><div><div>>X⁻¹ inverts X, but acts on (ψ, 0.0), ψ being the complex "latitude"</div><div><br></div><div>Amending just that line in the complete description:</div><div><br></div><div><div>Define</div><div> Project to complex plane, ψ = X + iΥ</div><div> Wallis I TM, complex ζ = π/2 - X⁻¹(ψ)</div><div> Wallis II TM, complex ξ = η⁻¹(ζ)</div><div> Gauß-Krüger, complex F(ξ)</div><div><br></div><div>where</div><div> X(φ, λ), Υ(φ, λ) is any conformal mapping to the plane; e.g.,</div><div> X = r cos λ </div><div> Υ = r sin λ</div><div> r = [(1-sin φ)/cos φ][(1+e sin φ)/(1-e sin φ)]^(e/2).</div><div> X⁻¹ finds μ such that ψ = X(μ, 0), μ being a complex "latitude"</div><div> η⁻¹ is inverse parametric latitude, acting on a complex "latitude"</div><div> η⁻¹(p) = tan⁻¹ [tan(p)/√(1-e²)]</div><div> F is the elliptic integral of the second kind with parameter e.</div><div> e is eccentricity</div><div><br></div></div></div><div><br></div><div>Regards,</div><div>— daan Strebe</div><div><br></div><br>On Feb 3, 2009, at 2:54:30 AM, strebe <strebe@aol.com> wrote:<br><blockquote style="padding-left: 5px; margin-left: 5px; border-left-width: 2px; border-left-style: solid; border-left-color: blue; color: blue; "><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0; "><div style="width: 100%; "><div id="felix-mail-header-block" style="color: black; background-color: white; border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: silver; padding-bottom: 1em; margin-bottom: 1em; width: 100%; ">Wallis happens to project to the ellipsoidal stereographic to arrive onto the plane from the ellipsoid. The specific projection to the plane is not important, other than the simpler the better for practical purposes. The point is to arrive at Wallis I, where parallels are evenly spaced on the complex plane.<br></div><div id="felix-mail-content-block" style="color: black; background-color: white; width: 100%; "><div><div><div><div><br></div><div><div>Define</div><div> Project to complex plane, ψ = X + iΥ</div><div> Wallis I TM, complex ζ = π/2 - X⁻¹ (ψ)<br></div><div> Wallis II TM, complex ξ = η⁻¹ (ζ)</div><div> Gauß-Krüger, complex F (ξ)</div><div><br></div></div><div>where</div><div> X (φ, λ), Υ (φ, λ) is any conformal mapping to the plane; e.g.,<br></div><div> X = r cos λ </div><div> Υ = r sin λ</div><div> r = [(1-sin φ)/cos φ][(1+e sin φ)/(1-e sin φ)]^(e/2).</div><div> X⁻¹ inverts X, but acts on (ψ, 0.0), ψ being the complex "latitude"</div><div> η⁻¹ is inverse parametric latitude, acting on a complex "latitude"</div><div> η⁻¹ (p) = tan⁻¹ [tan (p)/√(1-e²)]</div><div> F is the elliptic integral of the second kind with parameter e.</div><div><div> e is eccentricity</div><div><br></div></div></div><div>As usual, the devil is in the details.</div><div><br></div></div><div>Regards,</div><div>— daan Strebe</div><div><br></div></div></div></div></span></blockquote><div class="aol_ad_footer" id="uF2BD0336352F4ED59492178A42511840"></div></body></html>