It is impossible to construct maps which represent an accurate image of the Earth's surface. In other words: There isn't any map on which all distances of the Earth are mapped with the property which preserves the principal scale.
This is due to the fact that on the one hand the sphere as well as the rotation ellipsoid (spheroid) and on the other hand the plane have different Gauss' curvatures.
Several curves, however, can be mapped length-preserving, e.g. the meridians onto the Plate Carrée Map.
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THEOREMA EGREGIUM by Gauss |
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standard parallels |