A sphere cannot be flattened without stretching it, so every map projection destroys something. The families are named for what each one refuses to give up.
The shading is the distortion field: how wrong the map is at every point. Switch between area and angle and watch what happens. A conformal projection's angle field goes completely dark. An equal-area projection's area field goes completely dark. No projection can black out both.
Drag anywhere to move the probe: an 800 km circle drawn on the actual globe. Watch it swell, shrink, and shear.
a and b are the principal
scale factors: the semi-axes of Tissot's indicatrix, the ellipse a tiny circle
turns into.
a×b, relative to the
map's nominal scale. Angular deformation is
ω = 2·asin((a−b)/(a+b)), the largest angle
error at that point.
a = b everywhere, so ω is exactly 0.
Equal-area means a×b = 1 everywhere. Gauss's
Theorema Egregium says no projection of a sphere can do both.
Numbers are computed live in your browser, not looked up. Distance mode compares straight-line map distance from the probe against true great-circle distance. Equidistant projections hold distance only from one point or along one set of lines, so unlike the other two families they have no edge of their own on the trade-off chart.