Villanova University, Department of Mathematical Sciences

Cartographiometry (MAT 1210/ GEO 1700)

(©  copyright 1996-1999, Timothy G. Feeman and Elaine F. Bosowski.)

The Mollweide map: Supplemental questions

(mollmapq.htm)

The following questions concern the mathematical construction of the base graticule for a Mollweide map centered on the equator and the prime meridian. The questions assume that the graticule is designed using a model globe of radius 1 unit (so R=1).

1. On  a globe of radius 1 unit, what is the surface area of the portion of the globe that lies between the meridians at 20° West and 20° East? between 120° West and 120° East?
2. For each area you computed in the previous question, find the corresponding value for   "a" so that the ellipse with equation (x²/a²) + (y²/2) = 1 has the same area.
3. We determined that the parallel at latitude u° North should be shown on the map as a segment of the horizontal line with equation y = sqrt(2)*sin(t°), where the value of  t  is determined by the equation numbered (27) on page 73 of your notes. Use a calculator or computational software to find the values for   t  and y  that correspond to the latitudes u = 40 and u = 60.
4. In the text (page 73 again) it is stated that the parallel at 30° North corresponds to the horizontal line y = .571304 . What is the equation for the horizontal line corresponding to the parallel at   30° South ?
5. The Mollweide map shows all areas in their correct proportions, but it distorts the shapes of regions. In what parts of your finished map does the distortion of shapes appear to be the most severe? the least severe? Describe the distortions that you see. (Use your finished map as a visual aid.)