Numerical Examples for Sinusoidal Projection #

$$ \eqalign{ x &= 1.0\times[-75^\circ-(-90^\circ)]\cos(-50^\circ) \cr &= 0.1682814\text{ units} } $$

$$ \eqalign{ y &= 1.0\times(-50^\circ)\times\pi/180^\circ \cr &= -0.8726646\text{ units} } $$

$$ \eqalign{ h &= \{1+ [-75^\circ-(-90^\circ)]^2\times(\pi/180^\circ)^2\times\sin^2(-50^\circ) \}^{1/2} \cr &= 1.0199119 } $$

$$ \eqalign{ k &= 1.0 } $$

$$ \eqalign{ \theta' &= \arcsin(1/1.0199119) \cr &= 78.6597719^\circ } $$

$$ \eqalign{ \omega &= 2\arctan[(1/2)[-75^\circ-(-90^\circ)]\times(\pi/180^\circ)\times\sin(-50^\circ)] \cr &= 11.4523842^\circ } $$

$$ \eqalign{ \phi &= (-0.8726646/1.0)\times180^\circ/\pi \cr &= -49.9999985^\circ } $$

$$ \eqalign{ \lambda &= -85^\circ+0.1682814/[1.0\cos(-49.9999985^\circ)]\times180^\circ/\pi \cr &= -70.0000007^\circ } $$

$$ \eqalign{ x =& 6378206.4\times[-75^\circ-(-90^\circ)]\times(\pi/180^\circ)\cos(-50^\circ)/ \cr & [1-0.0067687\sin^2(-50^\circ)]^{1/2} \cr =& 1075471.54\text{ m} } $$

$$ \eqalign{ M =& 6378206.4\times[(1-0.0067687/4-3\times0.0067687^2/64 \cr & -5\times0.0067687^3/256)\times (-50^\circ)\times \pi/180^\circ \cr & -(3\times0.0067687/8+3\times0.0067687^2/32 \cr & +45\times0.0067687^3/1024)\times\sin(2\times(-50^\circ)) \cr & +(15\times0.0067687^2/256+45\times0.0067687^3/1024) \cr & \times \sin(4\times(-50^\circ)) -(35\times0.0067687^3/3072) \cr & \times\sin(6\times(-50^\circ))] \cr =& -5540628.03\;\text{m} } $$

$$ y = -5540628.03\text{ m} $$

$$ M = -5540628.03 $$

$$ \eqalign{ \mu =& -5540628.03 /[6378206.4\times(1-0.0067687/4 \cr & -3\times0.0067687^2/64 - 5\times0.0067687^3/256)] \cr =& -0.8701555 \;\text{radians} = -49.8562392^\circ } $$

$$ \eqalign{ e_1 &= [1-(1-0.0067687)^{1/2}]/[1+(1-0.0067687)^{1/2}] \cr &= 0.001697916 } $$

$$ \eqalign{ \phi =& -49.8562392^\circ+[(3\times0.0016979/2-27\times0.0016979^3/32)\sin(2\times(-49.8562392^\circ)) \cr &+(21\times0.0016979^2/16-55\times0.0016979^4/32)\sin(4\times(-49.8562392^\circ)) \cr &+(151\times0.0016979^3/96)\sin(6\times(-49.8562392^\circ)) \cr &+(1097\times0.0016979^4/512)\sin(8\times(-49.8562392^\circ))]\times180^\circ/\pi \cr =&-50.0000001^\circ } $$

$$ \eqalign{ \lambda =& -90^\circ+\{1075471.54\times[1-0.0067687\sin^2(-50.0000001^\circ)]^{1/2}/ \cr & [6378206.4\cos(-50.0000001^\circ)]\}\times(180^\circ/\pi) \cr =& -75^\circ } $$