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Tissot Indicatrices

What is a "Tissot indicatrix?"

A Tissot indicatrix ("Tiss-oh") displays properties of a map projection at a point.

A projection cannot preserve all the geometric properties of space at every point: length, area, angle.  The Tissot indicatrix is a figure that shows how a projection changes the geometry.
   

It does so in a simple manner: by showing what a small circle would look like on the map.

A truly small circle would not be visible on maps at most scales.  The indicatrix makes it much larger than scale so you can see it.
   

We add several elements to aid the interpretation.  A reference circle shows what the small circle would look like if the projection did not distort its geometry.  It will appear as a perfect circle.
   

The reference lines show what standard lengths look like along a parallel (line of latitude) and meridian (line of longitude) look like.
   

By displaying many indicatrices on a map, you can readily appreciate how a projection changes geometric properties.

This figure shows Tissot indicatrices for the Hammer-Aitoff projection.  The rapid changes in distortion near Australia are evident.

The mathematical explanation

All projections are differentiable at almost all points.  This means they can be well approximated locally by linear maps.  A linear map is an affine transformation  It can be expressed as a combination of a translation (which does not matter: all projections in some sense move a point on the earth onto a map), a rescaling, a rotation, and a skewing.

When one subjects a circle to these transformations, it always turns into an ellipse.  That is the Tissot indicatrix.

An affine transformation is completely described by its effect on two independent vectors.  It is natural to let those vectors be small displacements in the direction of increasing longitude (along a parallel) and increasing latitude (along a meridian).  We also show the images of these vectors.

Interpreting Tissot indicatrices

A conformal projection preserves local angles.  The indicatrix will be a circle.  It might be a different size than the reference circle.  Conversely, wherever the indicatrix is circular, the projection is conformal.

Tissot indicatrices for the Mercator projection, showing conformality.

The large distortion at extreme southern latitudes is evident in the discrepancies between the gray (reference) circles and the Tissot indicatrices (blue circles).

An area-preserving transformation preserves local areas.  The indicatrix may be an ellipse.  In that case, the expansion along the major axis will exactly compensate for the contraction along the minor axis in order to preserve area.  (The area of an ellipse is proportional to the product of its major and minor axis lengths.)

Tissot indicatrices for the Peters projection, showing its equal-area property.

The blue ellipses maintain constant areas, but the amount, degree, and direction of scale distortion varies dramatically from north to south, illustrating the large distortions present in this projection.

An equidistant transformation preserves distances along some path, often a parallel or meridian.  Along this path, the indicatrix and the reference circle will intersect.

Tissot indicatrices for an azimuthal equidistant projection from the south pole.

The blue ellipses meet the reference circles at their northern and southern tips, showing how scale is consistently preserved along meridians.  The indicatrices make the  increased east-west distortion to the north apparent.

Features and use

Tissot is an ArcView 3.x  tool you can apply to any point in a view.  Doing so brings up the Tissot dialog, which will remain until you dismiss it or select another tool.  You can readily move back and forth between ArcView and the Tissot dialog, which remains on top of the ArcView window.

lambda, phi: Natural ellipsoid coordinates, in degrees.

X, Y: Projected coordinates.

a, b: Scales along the indicatrix's major and minor axes, respectively.  This makes them the maximum and minimum linear scale distortion, respectively.  Values of 1.0 are distortion-free.

h, k: Scales along the meridian (north-south) and parallel (east-west).

s: Amount of distortion in area, equal to the product of a and b  A value of 1.0 is distortion-free; larger values indicate an apparent increase in area on the map relative to actual area.

omega: Maximum amount by which an angle will be distorted by the projection at the indicatrix's location.  A value of 0.0 is perfect (conformal).

theta: Projected angle between the meridian and the parallel.

g: How much to rotate map north so it points toward true north; negative values are clockwise, positive are counterclockwise.

(These letter names are fairly standard and agree with Snyder's notation.)

The dialog displays many projection properties.  A brief explanation of each appears when you press the Help button.

The projection's name appears in the title.  This will be automatic for a view whose projection is set.  (That is, the underlying data are represented in decimal degrees.)  When showing already-projected data, you will first be prompted for the projection to use.

The DMS/DD selection controls how degree measures are displayed: DMS for degrees-minutes seconds, DD for decimal degrees.

You can control the behavior of Tissot with this dialog:

Use Draw indicatrix to specify whether you want the indicatrix drawn in the view where you click.  The dialog will show you the projection properties, as above, regardless.
Use Erase previous to remove the previous indicatrices before drawing the next one.
Use Continuous mode to draw indicatrices beneath the path of the mouse as you sweep it across a view.

Continuous mode lets you quickly sketch out an array of indicatrices across a map to visually analyze its projection.

The size is roughly the diameter of the reference circle as a percentage of the view's width.  It controls the apparent sizes of the indicatrices drawn.  Changing it will change the sizes of all indicatrices the next time one is drawn.

The copy button places a text version of the dialog's contents on the clipboard for documentation or further analysis.

The erase all button removes all previous indicatrices from the active view.  Use it to conserve computing resources (and limit the size of ArcView's project file.)  (Of course you can select all graphics and delete them, but using this button is easier and eliminates the risk of erasing graphics, such as labels, you want to keep.)

The OK button dismisses the Tissot dialog.  Any indicatrices you have drawn will remain in the view.

Availability

"Tissot" is an ArcView 3.x extension that displays Tissot indicatrices on a map.  It is intended for educational purposes.  A few minutes spent with it can replace many hours of hard study.

Order Tissot on-line here for $4.95 (US).  A class license (for unlimited use by any number of people taking a class that uses ArcView) costs only $24.95.

References

Tissot uses formulas for elliptical spheroids published in John Snyder's Map Projections manual (pages 20-24).

 

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Last modified: Tuesday August 07, 2001.