GC5HDCB VINCENTY'S FORMULAE (Multi-cache) in North Carolina, United States created by Tandemguy

GC5HDCB ▼

A cache by Tandemguy Message this owner

Hidden : 12/5/2014

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Size: Size: regular (regular)

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Geocache Description:

This cache's alternate name is HOP, SKIP AND A LONG JUMP. It is a multi-cache with one offset.

Vincenty's formulae are two related iterative measures used in geodesy to calculate the distance between two points on the surface of a spheroid, developed by Thaddeus Vincenty (1975a) They are based on the assumption that the figure of the Earth is an oblate spheroid, and hence are more accurate than methods such as great circle distance which assume a spherical Earth.

Vincenty's formulae are the ones used by your gps device to solve all kinds of interesting math equations to accurately lead you to your smiley. For example, to find the final stage without today's electronic devices, you would have to do the following:

Given an initial point (φ₁, L₁) and initial azimuth, α₁, and a distance, s, along the geodesic the problem is to find the end point (φ₂, L₂) and azimuth, α₂.

Start by calculating the following:

$\tan U_1 = (1 - f)\tan \phi_1 \,$

$\sigma_1 = \arctan \left ( \frac{ \tan U_1}{ \cos \alpha_1} \right ) \,$ ^[2]

$\sin \alpha = \cos U_1 \sin \alpha_1; \,\,\,\, \cos^2 \alpha = (1 - \sin \alpha)(1 + \sin \alpha)$

$u^2 = \cos^2 \alpha \frac{a^2 - b^2}{b^2} \,$

$A = 1 + \frac{u^2}{16384} \left\{ 4096 + u^2 \left[ -768 +u^2 (320 - 175u^2) \right] \right\}$

$B = \frac{u^2}{1024} \left\{ 256 + u^2 \left[ -128 + u^2 (74-47 u^2) \right] \right\}$

Then, using an initial value $\sigma = \tfrac{s}{bA}$ , iterate the following equations until there is no significant change in σ:

$2 \sigma_m = 2 \sigma_1 + \sigma \,$

$\Delta \sigma = B \sin \sigma \Big\{ \cos(2 \sigma_m) + \tfrac{1}{4} B \big[ \cos \sigma \big(-1+2 \cos^2(2 \sigma_m) \big) - \tfrac{1}{6} B \cos(2 \sigma_m) (-3+4 \sin^2 \sigma) \big(-3+4 \cos^2 (2 \sigma_m)\big) \big] \Big\}$

$\sigma = \frac{s}{bA} + \Delta \sigma \,$

Once σ is obtained to sufficient accuracy evaluate:

$\phi_2 = \arctan \left( \frac{\sin U_1 \cos \sigma + \cos U_1 \sin \sigma \cos \alpha_1}{(1 - f) \sqrt{\sin^2 \alpha + (\sin U_1 \sin \sigma - \cos U_1 \cos \sigma \cos \alpha_1 )^2 } } \right) \,$ ^[2]

$\lambda = \arctan \left( \frac{\sin \sigma \sin \alpha_1}{\cos U_1 \cos \sigma - \sin U_1 \sin \sigma \cos \alpha_1} \right) \,$ ^[2]

$C = \frac{f}{16} \cos^2 \alpha \big[4 + f(4-3 \cos^2 \alpha) \big] \,$

$L = \lambda - (1-C) f \sin \alpha \left\{ \sigma + C \sin \sigma \left[\cos (2 \sigma_m) + C \cos \sigma (-1 + 2 \cos^2 (2 \sigma_m)) \right]\right\} \,$

$\alpha_2 = \arctan \left( \frac{\sin \alpha}{-\sin U_1 \sin \sigma + \cos U_1 \cos \sigma \cos \alpha_1} \right) \,$ ^[2]

Fortunately, you probably won't have to solve these equations to obtain the coordinates to the final.

Instead, you can do this:

Go to the posted coordinates for Stage 1.

There you will find a metal tag about 5 feet off the ground attached to a tree. The tag has two numbers on it. The first number is the number of meters that are to be added to 974 meters to give the distance in meters to the cache from Stage 1 coordinates. The second number is the number of degrees that are to be added to 22.76 degrees to give the bearing from true north to the cache from Stage 1 coordinates.

The cache container is a camouflaged ammo can. It is in base of a large broken off, hollow tree stump.

Among the uploaded images in this listing there is a picture of a survey monument which is located about 10 meters from the final ground zero. If you orient yourself with the picture, the stump and the cache will be to your right.

I like to stock my caches with decent swag that may be of interest to both young and old. It's a lot more fun if people trade even or up rather than just taking things and leaving behind junk and trash. Let's keep the fun going.

Please do not include spoilers in your logs. Logs with spoilers will be deleted immediately. If you believe there are any problems with the cache setup, please advise me privately by e-mail so that I may fix them if necessary.

Additional Hints (Decrypt)

Fgntr 1 ynory vf nggnpurq gb jung nccrnef gb or n Dhrephf nyon va gur zvqqyr bs n fgnaq bs gur fnzr fcrpvrf. Cerpvfvba cebwrpgvba va obgu qvfgnapr naq ornevat vf erpbzzraqrq.

Decryption Key

A|B|C|D|E|F|G|H|I|J|K|L|M
-------------------------
N|O|P|Q|R|S|T|U|V|W|X|Y|Z

(letter above equals below, and vice versa)

VINCENTY'S FORMULAE Multi-Cache

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