Tangents and secants

The plane of a map projection is just another name for the surface of a map projection and you will see it used throughout map projection documentation. There are three developable surfaces, or two-dimensional geometric planes: the cone, the cylinder, and the plane.

Imagine the earth wrapped up in one of these developable surfaces. The line of contact between the earth and this surface is called a tangent. If there are two such lines, they are called secants. The contact point (or points) between the spheroidal earth's surface and the plane of the map projection is the only location where the properties of the projection are true. It's important, therefore, to know exactly how and where the plane of a map projection contacts the earth's surface since these tangents or secants are lines of zero distortion on the map.

Now examine a globe or imagine the earth as an object before you. You could look at it from different points of view: straight on, sideways, from above, or from some odd angle. In a map projection, these points of view are called aspects.


Some map projections contact the earth's surface along a point or line, called a tangent, while other map projections contact the earth's surface along two lines, called secants. Both tangents and secants represent locations on the map projection where there is no distortion. That is, there is no distortion of shape, area, distance, direction, or scale. Features at these locations on the projection are represented as they really are on the earth's surface.


Tangent and secant conic map projections


These examples show tangent and secant conic map projections. As you move farther away from the tangent or secant lines, distortion increases.


As you move farther away from the tangent or the secants in either a northerly or southerly direction, distortion increases. For a tangent map projection, this places the center of the map on a line that does not exhibit any distortion. However, on a map projection that uses secants, this causes the center of the map projection to be located in a distortion zone. Although in a secant map projection the center exhibits distortion, the distortion between the two secants is not the same as that found in areas north and south of the secants and the overall map projection's distortion is distributed more evenly for the area you are mapping. In the next concept you will examine this in more detail.