Numerical Examples for Van der Grinten Projection #
SPHERE #
Forward Equations #
Given
| Radius of sphere: | $R=\;\;$ unit |
| Central meridian: | $\lambda_0=\;$° |
| Point: | $\phi=\;$° |
| $\lambda=\;$° | |
From equations (29-6), (29-3), (29-4), (29-5), and (29-6a) in order,
$$
\eqalign{
\theta &= \arcsin |2\times(-50^\circ)/180^\circ| \cr
&= 33.7489886^\circ
}
$$
$$
\eqalign{
A &= ½|180^\circ/(-160^\circ-(-85^\circ))-(-160^\circ-(-85^\circ))/180^\circ| \cr
&= ½|(-2.4000000) - (-0.4166667)| \cr
&= 0.9916667
}
$$
$$
\eqalign{
G &= \cos33.7489886^\circ/(\sin33.7489886^\circ+\cos33.7489886^\circ-1) \cr
&= 2.1483315
}
$$
$$
\eqalign{
P &= 2.1483315/(2/\sin33.7489886^\circ - 1) \cr
&= 5.5856618
}
$$
$$
\eqalign{
Q &= 0.9916667^2+2.1483315 \cr
&= 3.1317343
}
$$
From equation (29-1),
$$
\eqalign{
x =& -\pi\times1.0\{ 0.9916667\times(2.1483315-5.5856618^2) \cr
& +[0.9916667^2\times(2.1483315 - 5.5856618^2)\cr
& -(5.5856618^2 + 0.9916667^2)\times(2.1483315^2 - 5.5856618^2)]^{1/2}\}/ \cr
& (5.5856618^2+0.9916667^2)\cr
=& -1.1954154\text{ units}
}
$$
taking the initial “—” sign because $(\lambda-\lambda_0)$ is negative.
Note that $\pi$ is not converted to $180^\circ$ here, since there is no angle in degrees to offset it. From equation (29-2),$$
\eqalign{
y =& -\pi\times1.0\{ 5.5856618\times3.1317343 - 0.9916667\cr
& \times[(0.9916667^2+1)\times(7.0000000+0.9916667^2) \cr
& -3.1317343^2]^{1/2} \}/(5.5856618^2 + 0.9916667^2) \cr
=& -0.9960733\text{ units}
}
$$
taking the initial “—” sign because $\phi$ is negative.
Inverse Equations #
Inversing forward example:
Given: $R, \lambda_0$, for forward example
| $x=\;$ units |
| $y=\;$ units |
Using equations (29-9) through (29-19) in order,
$$
\eqalign{
X &= -1.1954154/(\pi\times1.0) \cr
&= -0.3805125
}
$$
$$
\eqalign{
Y &= -0.9960733/(\pi\times1.0) \cr
&= -0.3170600
}
$$
$$
\eqalign{
c_1 &= -|-0.3170600|\times(1+(-0.3805125)^2+(-0.3170600)^2)\cr
&= -0.3948401
}
$$
$$
\eqalign{
c_2 &= -0.3948401-2\times(-0.3170600)^2+(-0.3805125)^2 \cr
&= -0.4511044
}
$$
$$
\eqalign{
c_1 &= -|-0.3170600|\times[1+(-0.3805125)^2+(-0.3170600)^2]\cr
&= -0.3948401
}
$$
$$
\eqalign{
d =& (-0.3170600)^2/2.0509147+[2\times(-0.4511044)^3/2.0509147^3 \cr
& - 9\times(-0.3948401)\times(-0.4511044)/2.0509147^2]/27 \cr
=& 0.0341124
}
$$
$$
\eqalign{
a_1 =& [(-0.3948401)-(-0.4511044)^2/(3\times2.0509147)]/2.0509147\cr
=& -0.2086455
}
$$
$$
\eqalign{
m_1 &= 2\times[-(-0.2086455)/3]^{1/2} \cr
&= 0.5274409
}
$$
$$
\eqalign{
\theta_1 &= ⅓\arccos(3\times0.0341124/((-0.2086455)\times0.5274409)) \cr
&= ⅓\arccos(-0.9299322) \cr
&= 52.8080829^\circ
}
$$
$$
\eqalign{
\phi =& -180^\circ\times[(-0.5274409)\cos(52.8080829^\circ+60^\circ) \cr
& (-0.4511044)/(3\times0.5274409)] \cr
=& -49.9999985^\circ
}
$$
taking the initial “—” sign because $y$ is negative.
$$
$$