First, what is the Mercator projection? Conformally map the unit sphere with latitude λ and longitude φ onto the unit cylinder, (λ,φ) → (z,φ), and unroll the cylinder. The arc of length cosλ dφ on the sphere maps to an arc of length dφ on the cylinder, so the scale factor is secλ. The arc of length dλ on the sphere maps to an arc of length dz on the cylinder. For the map to be conformal, dz/dλ = secλ. The clever trick is to take the integral as z = asinh tanλ. Second what is the equation of a great circle? Let the normal to the plane of the great circle lie on the international date line, longitude φ = π, at latitude μ. This normal has coordinates n = (-cosμ, 0, sinμ). The general point on the sphere has coordinates r = (cosλ cosφ, cosλ sinφ, sinλ). The points on the great circle satisfy n.r = 0, which gives the relation tanλ = cotμ cosλ. Putting these together, the map of a great circle is z = asinh (cotμ cosλ). The map is approximately sinusoidal when cotμ is small, i.e. when the great circle is near the equator. -- Gene
________________________________ From: Henry Baker <hbaker1@pipeline.com> To: math-fun@mailman.xmission.com Sent: Friday, June 6, 2014 9:17 AM Subject: [math-fun] Stupid Mercator map question
Yes, I know this question is 400 years old, but what is the shape of the image of a great circle on a Mercator projection map?
I've seen it described as "sinusoid" shaped, but that can't be right, can it? (Yes, we've all seen those satellite location maps on TV, and they look pretty sinusoidal, but are they really?)
I'm not trying to measure lengths or angles, but am simply interested in the mathematical shapes of these great circle on the flat Mercator map.
I'm also not interested in the images of non-great circles.