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It is well known that there is no map projection that can represent on the map plane exactly the distance between arbitrary two points on the globe. The technical term ‘equidistant projections’ means “map projections possessing the special property which preserves the Principal Scale either radially from the Point of Zero Distortion, as in the Azimuthal Equidistant Projection; or perpendicularly to the line of zero distortion, as in the Cylindrical Equidistant Projection and in the Conical Equidistant Projections（ ICA, 1973）.” Equidistant projections are usually the three projections mentioned above and the two point equidistant projection is often grouped into the projections.

But there are definitions of broader senses. One is to include projections preserving length along parallels and still broader is to include projections that have any line along which the length are kept correct. Many Japanese textbooks on map projections or cartography adopt these broader senses for the definition of equidistant projections except for the one by the author. Furthermore it is said that the equidistant properties are compatible with the equalarea property and conformal property of map projections.

These compatibilities are examined according to the definitions of the equidistant projections in narrow sense or broader senses. It is pointed out that equidistant properties are not compatible with equalarea nor conformal properties when the definition is a narrow sense. The author also argues that the narrow sense definition of the equidistant projection is appropriate considering the original meaning of the term. Therefore it should be paid attention to what definition is used for the term equidistant projection when discussing the compatibility of the map projection properties.

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