# Tissot's indicatrix

Last updated

In cartography, a Tissot's indicatrix (Tissot indicatrix, Tissot's ellipse, Tissot ellipse, ellipse of distortion) (plural: "Tissot's indicatrices") is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 and 1871 in order to characterize local distortions due to map projection. 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.

## Contents

A single indicatrix describes the distortion at a single point. Because distortion varies across a map, generally Tissot's indicatrices are placed across a map to illustrate the spatial change in distortion. 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 all ellipses on the map occupy the same area, 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. 

## Description

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:

• A map distorts distances (linear distortion) wherever the quotient between the lengths of an infinitesimally short line as projected onto the projection surface, and as it originally is on the Earth model, deviates from 1. The quotient is called the scale factor. Unless the projection is conformal at the point being considered, the scale factor varies by direction around the point.
• A map distorts angles wherever the angles measured on the model of the Earth are not conserved in the projection. This is expressed by an ellipse of distortion which is not a circle.
• A map distorts areas wherever areas measured in the model of the Earth are not conserved in the projection. This is expressed by ellipses of distortion whose areas vary across the map.

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 equal-area projections, where area proportions between objects are conserved, the Tissot's indicatrices all have the same area, though their shapes and orientations vary with location. In arbitrary projections, both area and shape vary across the map.

## Mathematics

In the diagram below, the circle $ABCD$ has unit area as defined on the surface of a sphere. The circle ${A'B'C'D'}$ is the Tissot's indicatrix that results from some projection of $ABCD$ onto a plane. Linear scale has not been preserved in this projection, as ${OA'\ncong OA}$ and $OB'\ncong OB$ . Because ${\angle M'OA'\ncong \angle MOA}$ , we know that there is an angular distortion. Because $\operatorname {Area} (A'B'C'D')\neq \operatorname {Area} (ABCD)$ , we know there is an areal distortion.

The original circle in the above example had a radius of 1, but when dealing with a Tissot indicatrix, one deals with ellipses of infinitesimal radius. 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. For example, if the ratio between the radius of the input circle and a projected circle is equal to 1, then the indicatrix is drawn with as a circle with an area of 1. The size that the indicatrix gets drawn on the map is arbitrary: they are all scaled by the same factor so that their sizes are proportional to one another. Like $M$ in the diagram, the axes from $O$ 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 $h$ and the scale along the parallel as $k$ . Unless the projection is conformal, all angles except the one subtended by the semi-major axis and semi-minor axis of the ellipse may have changed as well. A particular angle will have changed the most, and the value of that maximum change is known as the angular deformation, denoted as $\theta$ . 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. The values of $h$ , $k$ , and $\theta$ can be computed as follows:  :24

{\begin{aligned}h&={\frac {1}{R}}{\sqrt {{{\left({\frac {\partial x}{\partial \varphi }}\right)}^{2}}+{{\left({\frac {\partial y}{\partial \varphi }}\right)}^{2}}}}\\k&={\frac {1}{R\cos \varphi }}{\sqrt {{{\left({\frac {\partial x}{\partial \lambda }}\right)}^{2}}+{{\left({\frac {\partial y}{\partial \lambda }}\right)}^{2}}}}\\\sin \theta '&={\frac {1}{R^{2}hk\cos \varphi }}\left({{\frac {\partial y}{\partial \varphi }}{\frac {\partial x}{\partial \lambda }}-{\frac {\partial x}{\partial \varphi }}{\frac {\partial y}{\partial \lambda }}}\right)\\a'&={\sqrt {{h^{2}}+{k^{2}}+2hk\sin \theta '}}\\b'&={\sqrt {{h^{2}}+{k^{2}}-2hk\sin \theta '}}\\a&={\frac {a'+b'}{2}}\\b&={\frac {a'-b'}{2}}\\s&=hk\sin \theta '\\\omega &=2\arcsin \left({\frac {b'}{a'}}\right)\end{aligned}} where $\varphi$ and $\lambda$ are the latitude and longitude coordinates of a point, $R$ is the radius of the globe, and $x$ and $y$ are the point's resulting coordinates after projection.

In the result for any given point, $a$ and $b$ are the maximum and minimum scale factors, analogous to the semimajor and semiminor axes in the diagram; $s$ represents the amount of inflation or deflation in area, and $\omega$ represents the maximum angular distortion.

For conformal projections such as the Mercator projection, $h=k$ and $\theta ={\pi \over 2}$ , 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. 

## An alternative derivation for numerical computation

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

### Differential distance on the ellipsoid

Let a 3D point, ${\hat {X}}$ , on an ellipsoid be parameterized as:

${\hat {X}}(\lambda ,\phi )=\left[{\begin{matrix}N\cos {\lambda }\cos {\phi }\\-N(1-e^{2})\sin {\phi }\\N\sin {\lambda }\cos {\phi }\end{matrix}}\right]$ where $(\lambda ,\phi )$ are longitude and latitude, respectively, and $N$ is a function of the equatorial radius, $R$ , and eccentricity, $e$ :

$N={\frac {R}{\sqrt {1-e^{2}\sin ^{2}(\phi )}}}$ The element of distance on the sphere, $ds$ is defined by the first fundamental form:

$ds^{2}={\begin{bmatrix}d\lambda &d\phi \end{bmatrix}}{\begin{bmatrix}E&F\\F&G\end{bmatrix}}{\begin{bmatrix}d\lambda \\d\phi \end{bmatrix}}$ whose coefficients are defined as:

{\begin{aligned}&E={\frac {\partial {\hat {X}}}{\partial \lambda }}{\boldsymbol {\cdot }}{\frac {\partial {\hat {X}}}{\partial \lambda }}\\&F={\frac {\partial {\hat {X}}}{\partial \lambda }}{\boldsymbol {\cdot }}{\frac {\partial {\hat {X}}}{\partial \phi }}\\&G={\frac {\partial {\hat {X}}}{\partial \phi }}{\boldsymbol {\cdot }}{\frac {\partial {\hat {X}}}{\partial \phi }}\\\end{aligned}} Computing the necessary derivatives gives:

${\frac {\partial {\hat {X}}}{\partial \lambda }}=\left[{\begin{matrix}-N\sin {\lambda }\cos {\phi }\\0\\N\cos {\lambda }\cos {\phi }\end{matrix}}\right]\qquad \qquad {\frac {\partial {\hat {X}}}{\partial \phi }}=\left[{\begin{matrix}-M\cos {\lambda }\sin {\phi }\\-M\cos {\phi }\\M\sin {\lambda }\sin {\phi }\end{matrix}}\right]$ where $M$ is a function of the equatorial radius, $R$ , and the ellipsoid eccentricity, $e$ :

$M={\frac {R(1-e^{2})}{(1-e^{2}\sin ^{2}(\phi ))^{\frac {3}{2}}}}$ Substituting these values into the first fundamental form gives the formula for elemental distance on the ellipsoid:

$ds^{2}=\left(N\cos {\phi }\right)^{2}d\lambda ^{2}+M^{2}d\phi ^{2}$ This result relates the measure of distance on the ellipsoid surface as a function of the spherical coordinate system.

### Transforming the element of distance

Recall that the purpose of Tissot's indicatrix is to relate how distances on the sphere change when mapped to a planar surface. Specifically, the desired relation is the transform ${\mathcal {T}}$ 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. This can be expressed by the relation:

${\begin{bmatrix}dx\\dy\end{bmatrix}}={\mathcal {T}}{\begin{bmatrix}ds(\lambda ,0)\\ds(0,\phi )\end{bmatrix}}$ where $ds(\lambda ,0)$ and $ds(0,\phi )$ represent the computation of $ds$ along the longitudinal and latitudinal axes, respectively. Computation of $ds(\lambda ,0)$ and $ds(0,\phi )$ can be performed directly from the equation above, yielding:

{\begin{aligned}&ds(\lambda ,0)=N\cos(\phi )d\lambda \\&ds(0,\phi )=Md\phi \end{aligned}} For the purposes of this computation, it is useful to express this relationship as a matrix operation:

${\begin{bmatrix}d\lambda \\d\phi \end{bmatrix}}=K{\begin{bmatrix}ds(\lambda ,0)\\ds(0,\phi )\end{bmatrix}},\qquad K={\begin{bmatrix}{\frac {1}{N\cos {\phi }}}&0\\0&{\frac {1}{M}}\end{bmatrix}}$ Now, in order to relate the distances on the ellipsoid surface to those on the plane, we need to relate the coordinate systems. From the chain rule, we can write:

${\begin{bmatrix}dx\\dy\end{bmatrix}}=J{\begin{bmatrix}d\lambda \\d\phi \end{bmatrix}}$ where J is the Jacobian matrix:

$J={\begin{bmatrix}{\frac {\partial x}{\partial \lambda }}&{\frac {\partial x}{\partial \phi }}\\{\frac {\partial y}{\partial \lambda }}&{\frac {\partial y}{\partial \phi }}\end{bmatrix}}$ Plugging in the matrix expression for $d\lambda$ and $d\phi$ yields the definition of the transform ${\mathcal {T}}$ represented by the indicatrix:

${\begin{bmatrix}dx\\dy\end{bmatrix}}=JK{\begin{bmatrix}ds(\lambda ,0)\\ds(0,\phi )\end{bmatrix}}$ ${\mathcal {T}}=JK$ This transform ${\mathcal {T}}$ encapsulates the mapping from the ellipsoid surface to the plane. Expressed in this form, SVD can be used to parcel out the important components of the local transformation.

### Numerical computation and SVD

In order to extract the desired distortion information, at any given location in the spherical coordinate system, the values of $K$ can be computed directly. The Jacobian, $J$ , can be computed analytically from the mapping function itself, but it is often simpler to numerically approximate the values at any location on the map using central differences. 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.

Recall the definition of SVD:

$\mathrm {SVD} ({\mathcal {T}})=U\Lambda V^{T}$ It is the decomposition of the transformation, ${\mathcal {T}}$ , into a rotation in the source domain (i.e. the ellipsoid surface), $V^{T}$ , a scaling along the basis, $\Lambda$ , and a subsequent second rotation, $U$ . For understanding distortion, the first rotation is irrelevant, as it rotates the axes of the circle but has no bearing on the final orientation of the ellipse. 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. The first singular value provides the semi-major axis, $a$ , and the second provides the semi-minor axis, $b$ , which are the directional scaling factors of distortion. Scale distortion can be computed as the area of the ellipse, $ab$ , or equivalently by the determinant of ${\mathcal {T}}$ . Finally, the orientation of the ellipse, $\theta$ , can be extracted from the first column of $U$ as:

$\theta =\arctan \left({\frac {u_{1,0}}{u_{0,0}}}\right)$ ## Related Research Articles The Mercator projection is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because it is unique in representing north as up and south as down everywhere while preserving local directions and shapes. The map is thereby conformal. As a side effect, the Mercator projection inflates the size of objects away from the equator. This inflation is very small near the equator but accelerates with increasing latitude to become infinite at the poles. As a result, landmasses such as Greenland, Antarctica, Canada and Russia appear far larger than they actually are relative to landmasses near the equator, such as Central Africa. An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. In navigation, a rhumb line, rhumb, or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true north.

In geodesy, conversion among different geographic coordinate systems is made necessary by the different geographic coordinate systems in use across the world and over time. Coordinate conversion is composed of a number of different types of conversion: format change of geographic coordinates, conversion of coordinate systems, or transformation to different geodetic datums. Geographic coordinate conversion has applications in cartography, surveying, navigation and geographic information systems.

In probability theory, the Borel–Kolmogorov paradox is a paradox relating to conditional probability with respect to an event of probability zero. It is named after Émile Borel and Andrey Kolmogorov. The Mollweide projection is an equal-area, pseudocylindrical map projection generally used for maps of the world or celestial sphere. It is also known as the Babinet projection, homalographic projection, homolographic projection, and elliptical projection. The projection trades accuracy of angle and shape for accuracy of proportions in area, and as such is used where that property is needed, such as maps depicting global distributions. The azimuthal equidistant projection is an azimuthal map projection. It has the useful properties that all points on the map are at proportionally correct distances from the center point, and that all points on the map are at the correct azimuth (direction) from the center point. A useful application for this type of projection is a polar projection which shows all meridians as straight, with distances from the pole represented correctly. The flag of the United Nations contains an example of a polar azimuthal equidistant projection. The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways.

The Schwarzschild solution describes spacetime under the influence of a massive, non-rotating, spherically symmetric object. It is considered by some to be one of the simplest and most useful solutions to the Einstein field equations. A (pseudo-)Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation.

Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system that generalizes the two-dimensional elliptic coordinate system. Unlike most three-dimensional orthogonal coordinate systems that feature quadratic coordinate surfaces, the ellipsoidal coordinate system is based on confocal quadrics. The van der Grinten projection is a compromise map projection, which means that it is neither equal-area nor conformal. Unlike perspective projections, the van der Grinten projection is an arbitrary geometric construction on the plane. Van der Grinten projects the entire Earth into a circle. It largely preserves the familiar shapes of the Mercator projection while modestly reducing Mercator's distortion. Polar regions are subject to extreme distortion. Lines of longitude converge to points at the poles. The Winkel tripel projection, a modified azimuthal map projection of the world, is one of three projections proposed by German cartographer Oswald Winkel in 1921. The projection is the arithmetic mean of the equirectangular projection and the Aitoff projection: The name tripel refers to Winkel's goal of minimizing three kinds of distortion: area, direction, and distance. The Cassini projection is a map projection described by César-François Cassini de Thury in 1745. It is the transverse aspect of the equirectangular projection, in that the globe is first rotated so the central meridian becomes the "equator", and then the normal equirectangular projection is applied. Considering the earth as a sphere, the projection is composed of the operations: The General Perspective projection is a map projection. When the Earth is photographed from space, the camera records the view as a perspective projection. When the camera is aimed toward the center of the Earth, the resulting projection is called Vertical Perspective. When aimed in other directions, the resulting projection is called a Tilted Perspective. In general relativity, a point mass deflects a light ray with impact parameter by an angle approximately equal to Geographical distance or geodetic distance is the distance measured along the surface of the earth. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude. This distance is an element in solving the second (inverse) geodetic problem. The Eckert IV projection is an equal-area pseudocylindrical map projection. The length of the polar lines is half that of the equator, and lines of longitude are semiellipses, or portions of ellipses. It was first described by Max Eckert in 1906 as one of a series of three pairs of pseudocylindrical projections. Within each pair, meridians are the same whereas parallels differ. Odd-numbered projections have parallels spaced equally, whereas even-numbered projections have parallels spaced to preserve area. Eckert IV is paired with Eckert III. The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an oblate ellipsoid, a slightly flattened sphere. A geodesic is the shortest path between two points on a curved surface, analogous to a straight line on a plane surface. The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry. In cartography, an equivalent, authalic, or equal-area projection is a map projection that preserves relative area measure between any and all map regions. Equivalent projections are widely used for thematic maps showing scenario distribution such as population, farmland distribution, forested areas, and so forth, because an equal-area map does not change apparent density of the phenomenon being mapped.

1. Goldberg, David M.; Gott III, J. Richard (2007). "Flexion and Skewness in Map Projections of the Earth" (PDF). Cartographica. 42 (4): 297–318. arXiv:. doi:10.3138/carto.42.4.297. S2CID   11359702 . Retrieved 2011-11-14.
2. Snyder, John P. (1987). Map projections—A working manual. Professional Paper 1395. Denver: USGS. p. 383. ISBN   978-1782662228 . Retrieved 2015-11-26.
3. More general example of Tissot's indicatrix: the Winkel tripel projection.
4. Laskowski, Piotr (1989). "The Traditional and Modern Look at Tissot's Indicatrix". The American Cartographer. 16 (2): 123–133. doi:10.1559/152304089783875497.