Chapter 20, Problem 20.23P

To determine

The geodetic coordinates for a point A.

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The geodetic coordinate for point A is $\left(41°12\text{'}12.203\text{"}\text{N},78°26\text{'}30.3340"\text{W}\right)$ .

Given:

The state plane coordinates of points A and B are shown in the following table.

Point	E(m)	N(m)
A	$541983.399$	$115702.804$
B	$541457.526$	$115430.257$

Concept Used:

Write the expression for the actual $E$ coordinate of the $E$ grid axis.

$E\text{'}=E-E_o$...... (I)

Here, $E\text{'}$ is the actual coordinate, $E$ is the given coordinate of the $E$ grid axis and $E_0$ is the offset to the $N$ grid axis.

Write the expression for the actual $N$ coordinate of the $N$ grid axis.

$N\text{'}=R_b-\left(N-N_b\right)$ ...... (II)

Here, $N\text{'}$ is actual $N$ coordinate, $N$ is the given coordinate, $R_b$ and $N_b$ are the zone constants for the north zone of Pennsylvania.

Write the expression for the convergence angle.

$\gamma ={\tan }^{-1}\left(\frac{E\text{'}}{N\text{'}}\right)$......(III)

Here, $\gamma$ is the convergence angle.

Write the expression for the radius of the earth.

$R=\frac{N\text{'}}{\cos \gamma }$...... (IV)

Here, $R$ is the radius of the earth.

Write the expression for geodetic longitude.

$\lambda ={\lambda }_0-\frac{\gamma }{\sin {\varphi }_0}$ ....... (V)

Here, $\lambda$ is the geodetic longitude, ${\lambda }_0$ and $\text{sin}\varphi$ are zone constant of the north zone of Pennsylvania.

Write the expression for the number of arcs.

$\Delta \varphi "=\frac{\left(R_T-R\right)}{T.D}$…… (VI)

Here, $\Delta \varphi "$ is the number of arcs, $R$ is the computed radius of earth, $R_T$ is radius of earth obtained from the table for the Pennsylvania North Zone and $T.D$ is the tab difference.

Write the expression for geodetic latitude.

$\varphi =\text{Latitude}+\Delta \varphi "$ . ...... (VII)Calculations:

Calculate the value of $E\text{'}$ .

For Pennsylvania’s North Zone. $E_o$ is $\text{600000}\text{m}$ and $E$ is $\text{541983}\text{.399}\text{m}$

Substitute $\text{600000}\text{m}$ . for $E_o$ and $\text{541983}\text{.399}\text{m}$ for $E$ in Equation (I).

$\begin{array}{c}E\text{'}=\left(541983.399\text{m}-600000\text{m}\right)\\ \text{=}-\text{58016}\text{.601}\text{m}\end{array}$

Calculate the value of $N\text{'}$ .For Pennsylvania’s north zone $R_b$ is $7379348.3668\text{m}$ , $N$ is $115702.804\text{m}$ ,and $N_b$ is $0$ .

Substitute $7379348.3668\text{m}$ for $R_b$ , $115702.804\text{m}$ for $N$ and $0$ for $N_b$ in Equation (II).

$\begin{array}{c}N\text{'}=7379348.3668\text{m}-\left(115702.804\text{m}-\text{0}\text{.00}\text{m}\right)\\ =7263645.563\text{m}\end{array}$

Calculate the convergence angle.

Substitute $-\text{58016}\text{.601}\text{m}$ for $E\text{'}$ and $7263645.563\text{m}$ for $N\text{'}$ in Equation (III).

$\begin{array}{c}\gamma ={\tan }^{-1}\left(\frac{-58016.601\text{m}}{7263645.563\text{m}}\right)\\ ={\tan }^{-1}\left(-0.007987\right)\\ =-0°27\text{'}27.45"\end{array}$

Calculate the radius of Earth.

Substitute $7263645.563\text{m}$ for $N\text{'}$ and $\left(-0°27\text{'}27.45"\right)$ for $\gamma$ in Equation (IV).

$\begin{array}{c}R=\frac{7263645.563\text{m}}{\cos \left(-0°27\text{'}27.45"\right)}\\ =\frac{7263645.563\text{m}}{0.999}\\ =7263877.255\text{m}\end{array}$

Calculate the geodetic longitude of point $A$ .

For Pennsylvania’s north zone ${\lambda }_0$ is $77°45\text{'}$ and $\sin {\varphi }_0$ is $0.661539733812$ .

Substitute $77°45\text{'}$ for ${\lambda }_0$ , $0.661539733812$ for $\sin {\varphi }_0$ and $\left(-0.2°27\text{'}27.45"\right)$ for $\gamma$ in Equation (V).

$\begin{array}{c}\lambda =77°45\text{'}-\left(\frac{-0°27\text{'}27.45"}{0.661539733812}\right)\\ =77°45\text{'}-\left(-0°41\text{'}30.33"\right)\\ =78°26\text{'}30.3340"\text{W}\end{array}$

Calculate the geodetic latitude with the help of table showing various parameters for the North Zone of Pennsylvania.

Latitude   $\left(\varphi \right)$	Radius(m)	Tab. Difference	Scale factor   $\left(k\right)$ f
$41°12\text{'}$	7264593.050	30.84830	0.99996400
$41°13\text{'}$	7262742.152	30.84836	0.99996295

As per the calculations the computed radius, $R$ is between $41°12\text{'}$ and $41°13\text{'}$ . Hence add the number of arc seconds to $41°12\text{'}$ to determine the geodetic latitude.

Substitute $7264593.050\text{m}$ for $R_T$ , $7263877.128\text{m}$ for $R$ and $30.84830$ for $T.D$ in Equation (VI).

$\begin{array}{c}\Delta \varphi "=\frac{\left(7264593.050\text{m}-\text{7263877}\text{.255}\text{m}\right)}{30.84830}\\ =\frac{\left(715.795\text{m}\right)}{30.84830}\\ =23.203"\end{array}$

Calculate the geodetic latitude.

Substitute $41°12\text{'}$ for Latitude and $23.203"$ for $\Delta \varphi "$ in Equation (VI).

$\begin{array}{l}\varphi =41°12\text{'}+23.203"\\ \varphi =41°12\text{'}12.203"\text{N}\end{array}$

Conclusion:

The geodetic coordinate of point $A$ is $\left(41°12\text{'}12.203"\text{N},78°26\text{'}30.3340"\text{W}\right)$ .