The loxodrome and two projections of the sphere 
Mercator projection - construction
In Mercator's projection, the points in the sphere are mapped onto points on a cylinder tangent to the sphere at the Equatorial line.
Imagine the sphere filling like a balloon inside a cylinder tangent to the sphere associated with the Equatorial line.
When a point on the balloons surface touches the cylinder, it stays fixed. This allows us to obtain the projection of the corresponding points on the sphere. In a similar fashion, as the balloon expands, the meridians and parallels approximate the cylinder until they are fixed on its surface.
This process continues and, in particular, the poles will never touch the cylinder. This means that as the points approximate the poles, the absolute value of the z components of their projections grows at a non-constant rate and goes to infinity. Therefore, only the points with latitude between a certain predefined minimum and maximum are projected.
Once this procedure is done, one just needs to unfold the cylinder to obtain Mercator's map. In this map, parallels are projected onto horizontal parallel line segments with increasingly spaced intervals. The meridians are projected onto vertical parallel line segment, equally spaced.
When a point on the balloons surface touches the cylinder, it stays fixed. This allows us to obtain the projection of the corresponding points on the sphere. In a similar fashion, as the balloon expands, the meridians and parallels approximate the cylinder until they are fixed on its surface.
This process continues and, in particular, the poles will never touch the cylinder. This means that as the points approximate the poles, the absolute value of the z components of their projections grows at a non-constant rate and goes to infinity. Therefore, only the points with latitude between a certain predefined minimum and maximum are projected.
Once this procedure is done, one just needs to unfold the cylinder to obtain Mercator's map. In this map, parallels are projected onto horizontal parallel line segments with increasingly spaced intervals. The meridians are projected onto vertical parallel line segment, equally spaced.
The following applet illustrates this process.
- Initially, a sphere is displayed, tangent to a cylinder at the Equatorial line. If you move the cursor Build the Projection, you can fill the sphere like a balloon until it touches the cylinder and then you can see the cylinder opening. The surface of the cylinder corresponds to the map of Mercator's projection.
- In red, a movable point is displayed and the parallel and meridian passing through that point are also shown. Initially, the point is located at the surface of the sphere and, after blowing up the sphere like a balloon, that point will be projected onto a point in the surface of the cylinder.
- In Maximum Latitude, you can choose the maximum absolute value of the latitude of the points that will be projected in such a way that, if you select 80º, the points projected will have latitude between -80º and 80º.
- Selecting World Map you can see a part of the world's map being projected onto Mercator's map.
