Tissot indicatrices

Now you know something about spatial distortion, but it would be nice to know a little more—like which properties a particular projection distorts and how much.

A good technique for evaluating distortion is the Tissot Indicatrix.

The idea is that when a circle on a sphere is projected onto a flat surface, it is deformed into an ellipse. The size, shape, and orientation of this ellipse (which is called an indicatrix) describes the spatial distortion at that location. By placing indicatrices at regular intervals, you can see the distortion pattern across the surface.

The Miller Cylindrical projection is neither conformal nor equal area. There is no distortion of shape or area along the equator, but both properties are increasingly distorted toward the poles.

How big is an indicatrix?

Infinitely small. You can't directly compare a spherical figure (like a circle on the earth's surface) to a plane figure (like a projected ellipse). To compare them, you have to treat the circle as a plane figure, and you do this by making it infinitely small. An infinitely small circle doesn't work as a visual aid, so for practical purposes you have to draw it bigger. The size you choose is arbitrary—what matters is that the projected ellipse is drawn in correct proportion to it.

Here's another example:

In the Orthographic projection (polar aspect), there is a single point of no distortion at the north pole. (This point is not marked with an indicatrix.) Shape and area are increasingly distorted with distance from the center; in this case, area shrinks relative to its true size. Beyond a single hemisphere, severe distortion makes the projection unusable.

Tissot indicatrices aren't the only device for analyzing map distortion. Another technique is to use isolines to connect points of equal distortion (just as contour lines connect points of equal elevation). This course, however, relies on the good offices of Monsieur Tissot. In the next concept, we'll take a deeper look at his indicatrices.

Quantifying distortion

Tissot indicatrices can give you a good general sense of distortion. But you might want to know just how good “good” is. You can also use the indicatrix to quantify distortion at any point on a map.

Below is a more detailed drawing. The blue ellipse is the Tissot indicatrix. The green circle is a reference circle representing the original figure on the sphere.

The graphic elements of a Tissot indicatrix.

Area distortion
The indicatrix is larger than the reference circle and differently shaped. This tells you that both area and shape are distorted at this location. (If there were no distortion, the indicatrix and the circle would be identical.)

It's easy to see that area is distorted, but by how much? Would you say the ellipse is twice as big as the reference circle? Three times as big? Four?

Check the area distortion

At this location (60°N, 120°W) on a Miller Cylindrical projection, area is shown three times larger than its true size. Remember, the "area" we're talking about is that of an infinitely small ellipse, not the whole area covered by the graphic.

Distance distortion
The red line on the indicatrix represents scale (distance) along the meridian. The black line represents scale along the parallel. When either line goes past the green reference circle—or doesn't reach it—there is distance distortion along that line. Here, both scales are exaggerated, though by different amounts. (In fact, scale is exaggerated here in all directions, since any line drawn from the center of the circle to the ellipse would go beyond the reference circle.)

If you define the length of the reference circle's radius as 1 unit, then a scale factor of 1.0 along a given line means there is no scale distortion in that direction. (A scale factor of 2.0 along a given line means that the line's length is twice the radius of the reference circle.)

In the graphic above, what would you estimate the scale factor along the meridian to be? About 1.5? How about the scale factor along the parallel?

Check the scale factor

In this graphic, the scale factor along the meridian is, in fact, 1.5. The scale factor along the parallel is 2.0.

Shape distortion
Shape distortion is measured by angular deformation. Consider a pair of intersecting lines on a reference circle. When those lines are projected, the angle between them on the Tissot indicatrix will be different than it was on the reference circle (assuming the map is nonconformal and the lines don't happen to be the major and minor axes of the indicatrix).

Remember the following graphic from the Shape concept?

The angles formed by intersecting lines change when a reference circle is projected to an ellipse.

At each location, one pair of lines will undergo a greater change in angle than any other pair. This greatest angular change, measured in degrees, is called the maximum angular deviation (or deformation). If the location is conformal, this value is 0. (On a conformal map, it is 0 at all locations). If the location is nonconformal, this value varies from a few degrees (low distortion) to more than 90 degrees (severe distortion).

What about distortion of direction?

Distortion of this property can't be analyzed from a Tissot indicatrix.

Useful indicatrix measures
Location : The latitude-longitude coordinates of the location being analyzed.

Parallel scale: The scale factor along the parallel. A value of 1.0 means there is no distance distortion along the parallel. Values greater than one mean that the map overstates the true distance. Values less than one mean that the map understates the true distance.

More about parallel scale

Suppose you make a measurement along a given parallel. According to your map's scale bar, the distance you measured is 100 kilometers. That's true only if the parallel scale at that location is 1.0. If the parallel scale is 2.0, then the true distance is 50 kilometers. If the parallel scale is 0.5, then the true distance is 200 kilometers.

Meridian scale: The scale factor along the meridian. A value of 1.0 means there is no distance distortion along the meridian.

Maximum scale: The greatest scale factor in any direction (always the direction of the major axis of the indicatrix.). A value of 1.0 means that scale is true along the major axis and is not overstated in any direction.

Minimum scale: The smallest scale factor in any direction (always the direction of the minor axis of the indicatrix). A value of 1.0 means that scale is true along the minor axis and is not understated in any direction.

Area scale: The maximum scale multiplied by the minimum scale. A value of 1.0 means there is no area distortion. Values greater than one mean that the map overstates the true area. (For example, a value of 4.0 means that the map shows area as four times larger than it really is.) Values less than one mean that the map understates the true area.

Meridian-parallel angle: The projected angle between the meridian and the parallel. A value of 90 means that this angle is not distorted.

Maximum angular deviation: The greatest angular change in any pair of intersecting lines when a reference circle is projected. A value of 0 means that the location is conformal.