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Die tissotsche Indikatrix

Orthographic Tissot

Tissot’s Indicatrix is a method to visualize the distortions of a map projection. It was introduced in 1859 by the French mathematician Nicolas Auguste Tissot.

Imagine someone painting a lot of circles of identical size, at regular intervals, onto the Earth’s surface. When he’s done with it, the Earth viewed from space will look like shown in the image.

When you project the painted Earth’s surface to a flat map, the circles will be distorted. But depending on the map projection you choose, they will be distorted differently. Let’s have a look at the Mercator projection with Tissot’s Indikatix:

Mercator Tissot

The circles kept their circular shape but grow bigger and bigger as you approach the poles.
And that’s exactly what the Mercator projection does: It keeps the shapes (locally) but it causes different regions on the map to have disproportionate sizes with respect to each other.
(By the way, on YouTube you can find a nice video called Visualisierung der Mercatorprojektionvisualization of the Mercator projection – which points out very nicely how the distortions arise.)

In contrast, look at the result of an equa-area projection, namely Wagner IV:

Wagner 4 Tissot

Now all the red dots kept their size – measured in square centimeters, they all cover the same area, but alas! Most of them have turned to ovals now!
The only dots that are close to a circle are to be found at the junction of the central meridian and 40° North/South – for the northern hemisphere, that’s the dot that covers France and the north-east of Spain. (At 42°59´ North/South it’d be perfectly circular.)

And that’s what an equal-area projection does: It doesn’t distort areal relationships, but it distorts the shapes. So Tissot’s indicatrix is a tool to help you finding out which kind of distortion is to be found at which regions of a given map projetion at a glance.

 

Sometimes, though, it’s not that easy to decipher the information of the Tissot indicatrix at a glane. For example on the Hammer projection:

Hammer Tissot Klein

The dot is perfectly circular at the junction of equator and central meridian. But what about the one covering Alaska at 60° North, 150° West? Well, obviously, it’s quite distorted – but does it have the same size as the one mentioned before? I can’t tell by just looking at them, can you?

Hammer Tissot Equator Hammer Tissot Alaska

As long as I know that the Hammer projection in equal-area, I don’t have to tell by looking. Because I’m aware that there are no distortions of areal relationships, so all that I’m interested in is the distortion of shapes – and then, Tissot’s indicatrix is helpful. But as long as I’m not aware that the Hammer projection is equal-area, the Tissot indicatrix is helpful only to a limited extent.

There are other ways to visualize the distortions of a map projection which is better in that regard. Have a look at the distortion images at mapthematics.com:
The lighter the color, the less distortion.
The redder, the more angular distortion.
The greener, the more areal inflation or deflation.


Verzerrungen der Mercator-Projektion
 

Verzerrungen des Wagner VII
 

Verzerrungen der Hammer-Projektion

Regrettably I can’t generate images of that kind for all projections that are included on this website. So I had to put up with the Tissot indicatrix in the compare section.

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Comments

14 comments

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Timmy

Nice website
( ͡° ͜ʖ ͡°)
Tue Aug 27, 2019 8:35 pm CEST
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Jeffery L. Hensal

It is very interesting how the distortion is in relation to your perception
Mon Aug 31, 2020 7:58 pm CEST
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Isabella

The circles from Tissot's concept are helpful. I may need a little more explanation for the Hammer Projection though.
Wed June 02, 2021 4:23 am CEST   –    One Reply

Tobias Jung

The Hammer projection is merely an example, showing that in using the Tissot indicatrix it sometimes can be hard to tell whether a projection is really equal-area, or if it’s just very close. Does this answer your question or was it about something different?
However: Reading that section again I guess I be rephrased a bit; so thanks for pointing me there!
Wed June 02, 2021 1:12 pm CEST
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Grace

I thought it was really interesting to see how Tissot's Indicatrix actually uses a visual aid with the dots to show map distortion versus just hearing about it and trying to grasp it verbally.
Sun June 06, 2021 7:47 pm CEST
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Kaetlyn Hight

The connection between personal perception and distortion in relation to maps is very interesting and I am intrigued by the concept. I hope to learn more about this. The visual aid was very helpful as well.
Wed Aug 25, 2021 8:33 pm CEST
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Joslyn

It is incredible to see just how disoriented flat maps can be and that its never 100% accurate to the actual sphere that Earth is.
Thu Aug 26, 2021 6:50 pm CEST
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McKenzie Moreland

I learned that the Tissot's Indicator shows the distortion of map projections and how you can tell is that the circles KEEP their shape but grow bigger and bigger when they approach the poles. However Equal-Area projection they don't grow they actually change their shape. So it doesn't distort areal relationships, but distorts the shape. Most turn to ovals.
Fri Aug 27, 2021 7:12 pm CEST
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Madi Dietz

this was very interesting! I have never heard of Tissot's Indicatrix before!
Fri Aug 27, 2021 11:36 pm CEST
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Lesley Tapia

The Tossot's Indicatrix technique is amazing. I had never heard of it before, but it is fascinating how it helps us find the distortions of map projections.
Sun Aug 29, 2021 7:22 am CEST
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Omar Baasiri

t's unbelievable to see just how chaotic a flat map can be, but it does make sense on how this all works.
Mon Aug 30, 2021 3:31 am CEST
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Maurice

Very interesting to see the distortions. We learn that the earth isn't exactly spherical but it's interesting to see where the distortions lie.
Mon Aug 30, 2021 10:11 pm CEST
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Shaquille Parkin

That is an interesting concept tp see the distortion.
Tue Aug 31, 2021 2:14 am CEST
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Renee Sparks

The Tissot Indicatrix is a pretty cool concept cause it relates to your perception
Tue Aug 31, 2021 3:15 am CEST
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Krystal

Simple distortion, almost like placing same size objects ahead of one another. The one on top is always going to look bigger??
Wed Sep 08, 2021 6:31 am CEST   –    One Reply

Tobias Jung

Well, in almost all projections, distortions increase towards the edges of the map. And unless the projection is equal-area, the poles will be inflated most.
Wed Sep 08, 2021 12:16 pm CEST
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