The Transverse cylindrical coordinate system projects the earth spheroid onto a cylinder wrapped around the earth along a line of longitude. This line is called the central meridian.
Let the z axis go along the earths axis and up through the north pole. Let the x axis go from the center of the earth along the equatorial plane to the central meridian. The y axis also lies in the equatorial plane and is perpendicular to and east of the x axis.
The parametric equations of a cylinder of elliptical crossection are :-
Note that in the same way that there are many different ways to define latitude, there are many different ways to perform projections. The manner chosen here is to project from the center of the earth to the point on the surface and onto the cylindrical mapping surface. An equally valid way is to project along lines normal to the surface of the spheroid.
If one multiplies the (x,y,z) point on the surface of the earth by a scalar, the
projection parameter v. This forms a parametric equation of a line going from the center
of the earth, through the point on the surface and projecting onto the cylinder. One finds
the projection point where this line crosses the cylinder...
Where v is the projection parameter. Now inserting the equation of the cylinder..
Note that the latitude of the projection point on the cylinder is not the same as the latitude of the point on the surface of the earth. We can solve for the latitude by dividing the projection equations and cancelling out v....
and then substitute back into a projection equation and simplyfy to obtain the projection parameter v...
If the cylinder were a circular cylinder touching the earth at the equator, then the
equations simplify to :-
Now one can use the parametric equations of the spheroid (4) to express the projection point latitude t' in terms of
the latitude and longitude.
Now the map coordinates are given as follows :-
Now from equation (55) we have...
Thus we can see that on the central meridian the Gauss conform projection matches the Transverse Cylindrical exactly.