Numerical Examples - Bonne Projection

# Numerical Examples for Bonne Projection #

## SPHERE #

### Forward Equations #

Given

 Radius of sphere: $R=\;\;$ units Standard parallel: $\phi_1=\;$ ° Central meridian: $\lambda_0=\;$ ° Point: $\phi=\;$ ° $\lambda=\;$ °

Find $x, y$

Using equations (19-1) through (19-4) in order,

\eqalign{ \rho &= 1.0\times[\cot 40^\circ +(40^\circ - 30^\circ)\times \pi/180^\circ] \cr &= 1.3662865\;\text{units} }
\eqalign{ E &= 1.0\times[-85^\circ-(-75^\circ)]\cos30^\circ/1.3662865 \cr &= -6.3385344^\circ }
\eqalign{ x &= 1.3662865\sin (-6.3385344^\circ) \cr &= -0.1508418\;\text{units} }
\eqalign{ y &= 1.0\cot 40^\circ - 1.3662865\cos(-6.3385344^\circ) \cr &= -0.1661807\;\text{units} }

### Inverse Equations #

Inversing forward example: Given $R, \phi_1, \lambda_0$ for forward example

 Point: $x=\;$ units $y=\;$ units

Find $\phi, \lambda$

Using equations (19-5) through (19-7) in order

\eqalign{ \rho &= [(-0.1508418)^2 + (1\cot40^\circ-(-0.1661807))^2]^{1/2} \cr &= 1.3662865\;\text{units} }
\eqalign{ \phi &= (\cot40^\circ)\times 180^\circ/\pi + 40^\circ -(1.3662865/1.0)\times 180^\circ/\pi \cr &= 30.0000012^\circ }
\eqalign{ \lambda =& -75^\circ + 1.3662865\times\{\arctan[-0.1508418/(1.0\cot40^\circ) \cr & - (-0.1661807)]\}/(1.0\cos30.0000012^\circ) \cr =& -84.9999985^\circ }

## ELLIPSOID #

### Forward Equations #

Given:

 Clarke 1866WGS-84 ellipsoid $a=$ 6378206.4 m $e^2=$ 0.00676866 Standard parallel: $\phi_1=$ ° Central meridian: $\lambda_0=$ ° Point: $\phi=$ ° $\lambda=$ °
Find $x, y$.

Using equations (14-15) and (3-21),

\eqalign{ m &= \cos 30^\circ/(1-0.0067687\sin^230^\circ) \cr &= 0.8667591 }
\eqalign{ M =&6378206.4[(1-0.0067687/4-3\times 0.0067687^2/64 - 5\times 0.0067687^3/256)\times 30^\circ\times\pi/180^\circ \cr &-(3\times 0.0067687/8+3\times 0.0067687^2/32 +45\times0.0067687^3/1024)\sin(2\times30^\circ) \cr &+(15\times 0.0067687^2/256 +45\times 0.0067687^3/1024)\sin(4\times 30^\circ) \cr &-(35\times 0.0067687^3/3072)\sin(6\times30^\circ)] \cr =& 3319933.29\;\text{m} }
Using the same equations, but with $\phi_1=40^\circ$ in place of $30^\circ$ ,
$$m_1 = 0.7671179$$
$$M_1 = 4429318.90\;\text{m}$$
Using equations (19-8) through (19-11) in order,
\eqalign{ \rho &= 6378206.4\times0.7671179/\sin40^\circ+4429318.90-3319933.29 \cr &= 8721287.35\;\text{m} }
\eqalign{ E &= 6378206.40\times0.8667591\times[-75^\circ - (-85^\circ)]/8721287.35 \cr &= -6.3389360^\circ }
\eqalign{ x &= 8721287.35\sin(-6.3389360^\circ) \cr &= -962915.09\;\text{m} }
\eqalign{ y &= 6378206.40\times0.7671179\sin40^\circ-8721287.35\cos(-6.3389360^\circ) \cr &= -1056065.01\;\text{m} }

### Inverse Equations #

Inversing forward example:

Given

 $x=\;$m $y=\;$m
Find: $\phi, \lambda$

Using equations (14-15) and (3-21), $m_1$, and $M_1$ are calculated as in the forward example:

$$m_1 = 0.7671179$$
$$M_1 = 4429318.90\;\text{m}$$
Using equations (19-12), (19-13), (7-19), (3-24), and (3-26) in order,
\eqalign{ \rho &= [(-962915.09)^2+(6378206.4\times0.7671179/\sin40^\circ - (-1056065.01))^2]^{1/2} \cr &= 8721287.36\;\text{m} }
\eqalign{ M &= 6378206.4\times0.7671179/\sin40^\circ + 4429318.90 - 8721287.36 \cr &= 3319933.29\;\text{m} }
\eqalign{ \mu =& \{3319933.29/[6378206.4\times(1-0.0067687/4 \cr & -3\times0.0067687^2/64 - 5\times0.0067687^3/256)]\} \times 180^\circ/\pi \cr =& 29.8737593^\circ }
\eqalign{ e_1 &= [1-(1-0.0067687)^{1/2}]/[1+(1-0.0067687)^{1/2}] \cr &= 0.001697916 }
\eqalign{ \phi =& 29.8737593^\circ +[(3\times0.0016979/2 - 27\times 0.0016979^3/32)\sin(2\times29.8737593^\circ) \cr & + (21\times 0.0016979^2/16-55\times0.0016979^4/32)\sin(4\times29.8737593^\circ) \cr & + (151\times0.0016979^3/96)\sin(6\times29.8737593^\circ) \cr & + (1097\times0.0016979^4/512)\sin(8\times29.8737593^\circ)]\times180^\circ/\pi \cr =& 30.0000000^\circ }
Using equation (14-15),
\eqalign{ m &= \cos30^\circ/(1-0.0067687\sin^2 30^\circ)^{1/2} \cr &= 0.8667591 }
Using equation (19-14),
\eqalign{ \lambda =& -75^\circ + \{ 8721287.356278036\times \arctan[-962915.09/ \cr & (6378206.4\times0.7671179/\sin40^\circ-(-1056065.01))]/ \cr & (6378206.4\times 0.8667591)\}\times 180^\circ/\pi \cr =& -85.0000000^\circ }