Writing the Inverse Projection

Maps > Map Projections > Creating a New Projection > Writing the Inverse Projection

Before writing the inverse function for the Mercator projection, you should be familiar with the IlvProjection::inverse function.

The IlvProjection::inverse Function

The inverse function prepares the data for inversion and processes the data for appropriate offset.

This function:

suppresses the offset produced by the Cartesian coordinates and converts these coordinates to meters.

reverts the coordinates to their geographic values and applies them to a standard ellipsoid with a semi-major axis of value 1.

calls the function sInverse or eInverse depending on whether the ellipsoid is a sphere or not.

adds the value of the central meridian to the longitude and adjusts the longitude to the range [-PI;PI] if longitude reduction is used (the default value).

converts the latitude if the coordinates are geocentric.

Inverse Projection onto a Sphere

The inverse projection onto a sphere is performed via the sInverse function.

It is not necessary for you to implement the sInverse function. If you call the IlvProjection::inverse function for a projection that does not support the inverse function, the error code IlvProjection::UnsupportedFeatureError() is returned.

As we saw earlier with the sForward function, the projection does not modify the x value. Therefore, the inverse equation is applied only to the y value.

IlvMapsError

MercatorProjection::sInverse(IlvCoordinate& xy) const

{

xy.setY(IlvMaps::Pi()/2. - 2.*atan(exp(-xy.y())));

return IlvMaps::NoError();

}

Inverse Projection onto an Ellipsoid

The inverse projection onto an ellipsoid is performed via the eInverse function. This function assumes that the value of the semi-major axis of the ellipsoid is 1.

In the particular case of the Mercator projection, the implementation of this function is more complex for an ellipsoid than for a sphere. It requires iterations and might fail since there is no simple analytical inverse equation for the Mercator projection from a nonspherical ellipsoid.

IlvMapsError

Mercator::eInverse(IlvCoordinate& xy) const

{

IlvDouble ts = exp(-xy.y());

IlvDouble e = sqrt(getEllipsoid()->getES());

IlvDouble eccnth = 0.5 * e;

IlvDouble Phi = IlvMaps::Pi() / 2. -2. * atan(ts);

int i = 15;

IlvDoubledphi;

do {

IlvDouble con = e * sin(Phi);

dphi = IlvMaps::Pi()/2. -

2. * atan(ts * pow((1 - con)/(1 + con), eccnth)) - Phi;

Phi += dphi;

} while(fabs(dphi) > 1.e-10 && --i != 0);

if(i <= 0)

return ToleranceConditionError();

xy.setY(Phi);

return IlvMaps::NoError();

}

Version 6.0