Tissot's indicatrix

It is the geometry that results from projecting a circle of infinitesimal radius from a curved geometric model, such as a globe, onto a map.

Tissot proved that the resulting diagram is an ellipse whose axes indicate the two principal directions along which scale is maximal and minimal at that point on the map.

A common scheme places them at each intersection of displayed meridians and parallels.

These schematics are important in the study of map projections, both to illustrate distortion and to provide the basis for the calculations that represent the magnitude of distortion precisely at each point.

Because the infinitesimal circles represented by the ellipses on the map all have the same area on the underlying curved geometric model, the distortion imposed by the map projection is evident.

There is a one-to-one correspondence between the Tissot indicatrix and the metric tensor of the map projection coordinate conversion.

[1] Tissot's theory was developed in the context of cartographic analysis.

Generally the geometric model represents the Earth, and comes in the form of a sphere or ellipsoid.

Tissot's indicatrices illustrate linear, angular, and areal distortions of maps: In conformal maps, where each point preserves angles projected from the geometric model, the Tissot's indicatrices are all circles of size varying by location, possibly also with varying orientation (given the four circle quadrants split by meridians and parallels).

In arbitrary projections, both area and shape vary across the map.

Even though the radii of the original circle and its distortion ellipse will all be infinitesimal, by employing differential calculus the ratios between them can still be meaningfully calculated.

along the parallel and along the meridian may undergo a change of length and a rotation during projection.

For a given point, it is common in the literature to represent the scale along the meridian as

In general, which angle that is and how it is oriented do not figure prominently into distortion analysis; it is the magnitude of the change that is significant.

are the maximum and minimum scale factors, analogous to the semimajor and semiminor axes in the diagram;

, such that at each point the ellipse degenerates into a circle, with the radius being equal to the scale factor.

For equal-area such as the sinusoidal projection, the semi-major axis of the ellipse is the reciprocal of the semi-minor axis, such that every ellipse has equal area even as their eccentricities vary.

For arbitrary projections, the shape and the area of the ellipses at each point are largely independent from one another.

[3] Another way to understand and derive Tissot's indicatrix is through the differential geometry of surfaces.

[4] This approach lends itself well to modern numerical methods, as the parameters of Tissot's indicatrix can be computed using singular value decomposition (SVD) and central difference approximation.

: Substituting these values into the first fundamental form gives the formula for elemental distance on the ellipsoid: This result relates the measure of distance on the ellipsoid surface as a function of the spherical coordinate system.

Recall that the purpose of Tissot's indicatrix is to relate how distances on the sphere change when mapped to a planar surface.

that relates differential distance along the bases of the spherical coordinate system to differential distance along the bases of the Cartesian coordinate system on the planar map.

can be performed directly from the equation above, yielding: For the purposes of this computation, it is useful to express this relationship as a matrix operation: Now, in order to relate the distances on the ellipsoid surface to those on the plane, we need to relate the coordinate systems.

Expressed in this form, SVD can be used to parcel out the important components of the local transformation.

In order to extract the desired distortion information, at any given location in the spherical coordinate system, the values of

Once these values are computed, SVD can be applied to each transformation matrix to extract the local distortion information.

Remember that, because distortion is local, every location on the map will have its own transformation.

The next operation, represented by the diagonal singular value matrix, scales the circle along its axes, deforming it to an ellipse.

Thus, the singular values represent the scale factors along axes of the ellipse.