I want to project a map to a particular shape, like a circle or ellipse. For example, let's imagine we want to project a map of the continental United states so that its outline is that of a circle. The basic strategy I have is illustrated below:

To project the outline is easy, the geographic center is at the center of the circle and the perimeter is divided into small sectors, perhaps 1-degree each. Then the points of the perimeter are equally divided within their sector and mapped to the corresponding sector on the perimeter of the circle.

To project the interior points of the map is the tricky part on which I need advice. In the image an example of such a point is shown as a blue dot. Then the procedure I have in mind for projecting it is as follows.

(1) Find the point on the geodesic perimeter of the United States which is closest to the point to be projected. This is indicated as a red dot in the illustrative image above.

(2) In the geodesic data (a sphere), draw an imaginary line to the red dot, the closest point.

(3) In the geodesic data (a sphere), draw two lines going outwards at 120 degrees in either direction relative to the red line. These two lines are shown in green. Find the points where these lines intersect the geodesic perimeter of the United States. These two points are shown as green dots.

(4) To project the point of interest, the blue dot, start at the red dot and move inwards along a line at the same angle as in the original geodesic data. So, for example, in illustrative image the red line is at on a bearing of of about 10 degrees relative to the red dot. That same angle is adopted in the projected image.

(5) To locate the blue dot, a ratio is determined between the sum of geodesic distances of the two green lines and the distance of the red line. Then, in the projected image, the blue dot is placed so that this ratio is the same.

My question is whether this is a reasonable strategy for doing this kind of projection, and how I would go about implementing it in Mathematica.

I want to project a map to a particular shape, like a circle or ellipse. For example, let's imagine we want to project a map of the continental United states so that its outline is that of a circle. The basic strategy I have is illustrated below:

To project the outline is easy, the geographic center is at the center of the circle and the perimeter is divided into small sectors, perhaps 1-degree each. Then the points of the perimeter are equally divided within their sector and mapped to the corresponding sector on the perimeter of the circle.

To project the interior points of the map is the tricky part on which I need advice. In the image an example of such a point is shown as a blue dot. Then the procedure I have in mind for projecting it is as follows.

(1) Find the point on the geodesic perimeter of the United States which is closest to the point to be projected. This is indicated as a red dot in the illustrative image above.

(2) In the geodesic data (a sphere), draw an imaginary line to the red dot, the closest point.

(3) In the geodesic data (a sphere), draw two lines going outwards at 120 degrees in either direction relative to the red line. These two lines are shown in green. Find the points where these lines intersect the geodesic perimeter of the United States. These two points are shown as green dots.

(4) To project the point of interest, the blue dot, start at the red dot and move inwards along a line at the same angle as in the original geodesic data. So, for example, in illustrative image the red line is at on a bearing of of about 10 degrees relative to the red dot. That same angle is adopted in the projected image.

(5) To locate the blue dot, a ratio is determined between the sum of geodesic distances of the two green lines and the distance of the red line. Then, in the projected image, the blue dot is placed so that this ratio is the same. Note that the green lines will not be at 120-degrees from the red line in the projected image.

My question is whether this is a reasonable strategy for doing this kind of projection, and how I would go about implementing it in Mathematica.