Map Projection Distortion, Explained
After reading this you will be able to look at any flat world map, name what it lies about (shape, area, or both), and quantify that lie using Tissot's indicatrix.
What a projection is and why it must distort
The Earth is (very nearly) a sphere. A map is a flat sheet. To draw the globe on paper you need a rule that turns every point (latitude \phi, longitude \lambda) into a pair of flat coordinates (x, y). That rule is a projection.
Here is the catch. A sphere and a plane have different intrinsic curvature. The sphere has positive Gaussian curvature 1/R^2; the plane has zero. Gauss's Theorema Egregium says curvature is preserved by any distance-preserving map. So no projection can preserve all distances. Something always stretches. The only freedom you have is choosing what to sacrifice.
The famous example: on a Mercator map, Greenland looks about the same size as Africa. In reality Africa is roughly 14 times larger (30.4 million km² versus 2.16 million km²). Mercator inflates high-latitude land enormously. This tool makes that inflation visible by drawing small circles on the globe and showing what each projection does to them.
Tissot's indicatrix in one idea
Take a tiny circle on the globe, small enough that the projection looks like a linear map across it. A linear map sends circles to ellipses. That output ellipse is Tissot's indicatrix. Its two axes tell you everything about local distortion.
- Semi-major and semi-minor axes (a, b)
- The maximum and minimum scale factors at that point. If the original circle had radius 1, the ellipse has axes a and b.
- Area scale
- The factor a \cdot b. If it equals 1 everywhere, the projection is equal-area.
- Shape (angular) distortion
- Governed by the ratio a/b. If a = b everywhere, the ellipse stays a circle and the projection is conformal (angle-preserving).
A round indicatrix means shape is safe. A flat, cigar-shaped indicatrix means angles are wrong. A large indicatrix means area is inflated. You read distortion straight off the drawn ellipse.
The scale factors, from the map equations
Work in a local frame aligned with the meridians and parallels. Two scale factors matter. The meridian scale h measures north-south stretch, the parallel scale k measures east-west stretch. For a projection written as x = x(\lambda), y = y(\phi) on a sphere of radius R:
Here h is the map's vertical rate of change per unit latitude, divided by R. The k factor carries a \cos\phi because parallels shrink toward the poles: one degree of longitude covers less ground near the pole than at the equator. When the meridians and parallels stay perpendicular on the map (true for all five projections here along the standard aspect), a and b are just h and k sorted by size.
Two tests summarize a projection. Conformal means h = k at every point (ellipse stays round). Equal-area means h \cdot k = 1 at every point (ellipse keeps unit area). Gauss forbids both holding at once except on a flat surface.
Worked example: Mercator at four latitudes
The tool's defaults draw the standard sphere with a grid of indicatrices. Mercator uses x = R\lambda and y = R \ln \tan\left(\frac{\pi}{4} + \frac{\phi}{2}\right). Differentiate and you get a clean result:
Computing the scale factors
- Parallel scale: k = \frac{1}{R\cos\phi} \cdot R = \frac{1}{\cos\phi} = \sec\phi.
- Meridian scale: differentiating the log-tangent gives h = \frac{1}{\cos\phi} = \sec\phi as well.
- Because h = k everywhere, Mercator is conformal. The indicatrix is always a circle. Good for angles.
- Area scale is h \cdot k = \sec^2\phi. This is the number that explodes.
Now plug in latitudes. At the equator \phi = 0, area scale is \sec^2 0 = 1: no inflation. At \phi = 45^\circ, \sec^2 45^\circ = (\sqrt 2)^2 = 2: areas double. At \phi = 60^\circ, \sec^2 60^\circ = 2^2 = 4. At \phi = 75^\circ, \sec^2 75^\circ \approx 14.9. That factor of 15 near Greenland is exactly why Greenland matches Africa on the wall map.
How the five projections compare
Each projection makes a different trade. The table shows their behavior in the standard aspect, with the equirectangular, Mercator, and Mollweide values computed at latitude 60 degrees for a concrete comparison.
| Projection | Conformal? | Equal-area? | Area scale at 60° | Signature flaw |
|---|---|---|---|---|
| Equirectangular | No | No | 2.0 | Stretches east-west near poles |
| Mercator | Yes | No | 4.0 | Polar area explosion |
| Mollweide | No | Yes | 1.0 | Sheared shapes at map edges |
| Orthographic | No | No | varies | Compresses the visible rim to zero |
| Stereographic | Yes | No | varies | Inflates the far hemisphere hugely |
Equirectangular is the plainest: x = R\lambda, y = R\phi. So h = 1 always but k = \sec\phi, giving indicatrices that stay one unit tall and stretch wider toward the poles. Mollweide fixes area at 1 and pays with sheared ellipses that lean over near the left and right edges. Orthographic is the view of the globe from infinitely far away: it looks like a photograph, and the indicatrices flatten to slivers at the visible edge because you are seeing that terrain edge-on. Stereographic is conformal like Mercator, so its circles stay round, but the scale factor is \sec^2(\theta/2) in terms of angular distance \theta from the projection point, which blows up as you approach the opposite pole.
Reading and interpreting the indicatrices
Three quick readings cover most cases. Look at roundness first: if the ellipse is a circle, angles and shapes are locally faithful even if the size is wrong. Look at area second: a big ellipse means that region is exaggerated relative to the equator. Look at tilt third: a leaning ellipse means the map shears the terrain, so a north arrow drawn there would not point straight up.
Put numbers on it. The maximum angular distortion (how far a right angle can be bent) is given by:
Here a and b are the ellipse's semi-axes. When a = b the fraction is 0 so \omega = 0: no angular error, which is the conformal case. For equirectangular at 60 degrees, a = 2 and b = 1, so \omega = 2\arcsin(1/3) \approx 38.9^\circ. A right angle on the ground can appear as anything from about 51 to 129 degrees on that part of the map.
Common mistakes when reading maps
Do not read area off a Mercator map. It is conformal, not equal-area, so it is built to lie about size. Comparing country sizes on Mercator is the single most common error, and it consistently makes northern countries look bigger than they are.
A second mistake is assuming a round indicatrix means the map is undistorted. Roundness only guarantees that local angles are right. Stereographic keeps every indicatrix perfectly round yet inflates a distant continent by a factor of thousands. Shape is preserved point by point; the global picture is still wildly out of scale.
A third mistake is trusting straight lines. On Mercator a straight line is a rhumb line (constant compass bearing), not the shortest path. The shortest route from London to Tokyo curves far north on a Mercator map even though the flight really is shorter. Do not confuse "straight on this map" with "shortest on the globe."
Finally, remember these are toy globes: a perfect sphere of radius R. The real Earth is an oblate spheroid, flattened by about 0.3 percent at the poles. That difference matters for surveying and GPS but changes none of the qualitative distortion patterns you see here.
Related tools
Projections are one kind of map from a curved space to a flat one. If you want to see space that genuinely cannot be flattened without overlap or gap, the Hyperbolic Tiling Explorer tiles the Poincaré disk with polygons that would never fit in the plane. To watch how a 2×2 matrix stretches and shears a flat grid (the same linear algebra behind a local indicatrix), try the Linear Transformation Playground. For the four-dimensional analogue of projecting a higher shape into a lower space, the Tesseract Projection casts a hypercube's shadow into 3D. And to check Euler's formula on curved-looking solids, spin the Platonic Solids Explorer.
Frequently asked questions
Why can't we just make a map with no distortion?
Because a sphere and a plane have different Gaussian curvature, and Gauss's Theorema Egregium proves that curvature is preserved under any distance-preserving map. Flattening the sphere must therefore change some distances. Every projection is a choice about which errors to accept.
Which projection is the "most accurate"?
There is no single answer because "accurate" depends on the task. For navigation with a compass, Mercator is best because it preserves bearings. For comparing country sizes, choose an equal-area map like Mollweide. For a view that looks like a photo of the globe, orthographic. Match the projection to the question.
What does a leaning Tissot ellipse mean?
It means the map shears the ground at that point: the meridian and parallel directions are not drawn at right angles. A leaning ellipse always signals angular distortion, so that projection is not conformal there. Mollweide's edges show this clearly.
Why does Mercator inflate the poles so much?
Its area scale is \sec^2\phi, which is 1 at the equator, 4 at 60 degrees, and about 15 at 75 degrees. The secant grows without bound toward the pole, so the pole itself sits at infinity and can never be drawn. That is the polar explosion.
Is stereographic projection conformal like Mercator?
Yes. Both keep every Tissot indicatrix perfectly round, so both preserve local angles. They differ in how the scale grows: Mercator blows up toward the poles along the standard aspect, while stereographic blows up toward the single point opposite its projection center.
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