Talk:Tissot's indicatrix/Archives/ 1

Wrong?

I think -- and I'll admit I could be wrong about this -- that the following sentence taken from the last paragraph in this article is wrong. The sentence reads:

Because MOA<MOA’, there is an angular distortion.

I think that should read:

Because M'OA<MOA, there is an angular distortion.

• You are right, that was a mistake. It's fixed now - Alvesgaspar (talk) 21:05, 26 December 2007 (UTC)

Plural

The plural of "indicatrix" would be "indicatrices"... AnonMoos (talk) 03:36, 22 June 2008 (UTC)

Technical

This article is so mathematical and jargony. I changed the first paragraph a tiny bit to try to make it more accessible. I removed "When the Tissot’s indicatrix reduces to a circle it means that, at that particular point, the scale is independent of direction." because I didn't understand what it means (or more precisely what it's trying to convey - i.e. what is the implication of scale being independent of direction, i.e. why would I care? Direction of what?) Axlrosen (talk) 12:13, 17 April 2009 (UTC)

Infinitesimal radius?

From the lead of the article:

It is the diagram that results from projecting a circle of infinitesimal radius from a curved geometric model onto the projection plane.

Wait, what? Infinitesimal radius? Either this is some new use of the word "infinitesimal" with which I'm unfamiliar, or it's flat-out wrong. Actually it seems to me that the radius of the circle used is arbitrary. Before "infinitesimal", the lead prescribed "unit radius" (and later in the article, in the Mathematics section, a "circle of unit area" is still called for) -- but in practical applications, "unit radius" is completely dependent on the units of scale, surely? I've changed "infinitesimal" to "arbitrary", but if someone can explain why "infinitesimal" was correct, please do educate me here and change it back there! -- Perey (talk)

Wait, wait, never mind, I figured it out. The circle is infinitesimal in theory, representing distortions at a single point. However, its radius is arbitrary in practice, because drawing infinitesimal circles on a map really isn't helpful to anybody. (And the "unit radius" part is just used to make the maths easier. "Unit area" is used for the same reasons, though I was confused about that at first, since "unit area" and "unit radius" are mutually exclusive unless π = 1.) -- Perey (talk) 18:04, 22 February 2011 (UTC)

Sorry; the new edits are confused, and for some reason you have deleted an entire paragraph that seems important. I’ll be restoring the original text; we can work out how to address your point which, it seems to me, amounts to reconciling the three different units used in Tissot ellipses: the infinitesimal nature of the mathematics, the unit radius used to describe what happens in the infinitesimals, and the arbitrary size chosen to display them. It’s a reasonable point, but your edits broke too many things. Strebe (talk) 19:25, 22 February 2011 (UTC)

Okay, sorry for the confusion. I think I was tired. (On the other hand, I'm positive I didn't delete any paragraphs -- I merely merged two and added one.) I've got the whole thing much clearer in my mind now. Let's take my issues (and edits) one at a time.

Principle vs. Principal curvature

Simple error, uncontroversial. I've restored this already.

Thank you. That was an embarrassment. Strebe (talk) 19:52, 2 March 2011 (UTC)

Second paragraph of "Description" put first

This is the one starting "Tissot's theory was developed...". I felt it fit better as the first paragraph, since it tells us what the indicatrix is for.

Appearance on conformal and equal-area maps

Agreed. In fact, I think the content of the first paragraph in that section should go last. Strebe (talk) 19:52, 2 March 2011 (UTC)

This I moved to be part of that first paragraph, with heavy rephrasing to help put everything in context for those unfamiliar with the topic.

New paragraph about the radius of the indicatrix

This is the crux of the issue, the whole thing that confused my poor tired brain in the first place. You put it very well: there are three different scales used.

Now, I think the second and third (reordering and contextualising) can probably be considered separate points to the mathematical phrasing, though there's no point restoring them until we've agreed on how to phrase the explanation of the different radii. -- Perey (talk) 17:51, 2 March 2011 (UTC)

An article’s lede is generally a tricky balance between concise description and omissions that some people find confusing or too imprecise. I’m not sure the lede should get into the matter of the different radii, even though it seems your confusion started there. I do think the body of the article should elaborate in order to eliminate that confusion, however.

Part of the reason no distinction is made between “infinitesimal” and “unit” radius is that, to mathematicians, they are the same thing: the result of taking the derivative of a function is in the units of the function space, yet the derivative describes behavior at the level of infinitesimals. Hence the circle is infinitesimal but the analysis is at the scale of units of function space. It’s going to take some finesse to convey this clearly. Strebe (talk) 19:52, 2 March 2011 (UTC)

I'd begun to wonder whether the indicatrices are projections of small circles of equal radius onto the map or representations of the scale and distortion at a set of discrete points. The graphic showing small circles on the surface of a globe tempts one to think the former ... but I see now that it's indeed the latter. On this basis, I've come up with this explanation:

The Tissot's indicatrix, for a given point on the globe $P$ and constant $r$, is the limit as $\epsilon \rightarrow 0$ of a small circle centred on $P$ of radius $\epsilon$, projected onto the map plane and then enlarged about the projection of $P$ by a factor ${\frac {r}{\epsilon }}$.

But still, this might be too technical for some people. What do you guys think? — Smjg (talk) 15:53, 21 June 2014 (UTC)

That’s a fair description of how the diagram is constructed. This situation illustrates a fundamental tension I keep noticing in Wikipedia: We’re obliged to cite sources. Yet in many more-or-less obscure domains, some things just go unsaid because they’re obvious to the expert but too difficult for general readership. I think your description would help bridge the gap for the range of readers who are versed in beginning to intermediate calculus. I don’t know what fraction of Wikipedia’s readership that amounts to. Unfortunately the description is also WP:OR. Strebe (talk) 19:49, 21 June 2014 (UTC)

That's a grey area. How does one decide whether an interpretation of known statements constitutes OR or not?

But I realise now that what I've said is basically equivalent to most of the second paragraph of the Mathematics section. But there are things in that section that don't make sense - I'll start a new thread about this. — Smjg (talk) 00:12, 22 June 2014 (UTC)

Some statements in the Mathematics section don't make sense

"Linear scale is not conserved along these two directions, since OA’ is not equal to OA and OB’ is not equal to OB." What is meant by "linear scale" here? Most maps of the world are several orders of magnitude smaller than the world itself, so of course OA′ won't equal OA. I wouldn't have thought that for a map to be considered to conserve linear scale along a given line it would have to show that line at actual size. If a map is scaled up or down, it's still the same projection, and doesn't become any more or less to scale.

"The area of circle ABCD is, by definition, equal to 1. Because the area of ellipse A’B’ is less than 1, a distortion of area has occurred." Similarly here. Scaling area down doesn't constitute distorting it. Surely what matters is whether the area of A′B′ is the same at all points on the map. — Smjg (talk) 00:31, 22 June 2014 (UTC)

I think I see what you’re getting at here. Right, in an absolute sense, scaling the map does not matter at all, since all you have done is redeclare the nominal scale, which (to some extent) is arbitrary. It’s the change in area across the map that means something.

However, the same cannot be said of angular deformation; it is an absolute metric. Because areal deformation is absolute and areal (in/de)flation is relative, they are not directly comparable without some amount of arbitrary calibration. What the diagram in the article shows is what the areal (in/de)flation is given a specific nominal scale. Strebe (talk) 01:26, 22 June 2014 (UTC)

Clearing up the math

Maybe it would be worthwhile to explain a' and b', and perhaps to add an illustration which aligns with the variable names used in the deformation equations? Marcman411 (talk) 16:40, 9 April 2019 (UTC)

Tissot software demonstration video

This isn't suitable material. It's demonstration of software that's used for computing Tissot indicatrices, but doesn't add anything to the understanding of the article or topic. I have reverted it. Strebe (talk) 19:37, 10 May 2016 (UTC)

Maybe a wrong picture?

Among the five images on the right-hand side of the page the last labelled "The Azimuthal projection with Tissot's indicatrices" is the same as the first labelled "View on a sphere: all are identical circles". Is this really intended? I'm not an expert on projections but this seams odd to me. - Eltom13 (talk) 12:51, 11 January 2017 (UTC)

External links modified (January 2018)

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In popular culture

This page may get more hits than usual because of the mention in XKCD at https://xkcd.com/2255/ Eastmain (talk • contribs) 04:17, 16 January 2020 (UTC)

Added numerical computation

After spending weeks trying to understand Tissot's indicatrix from the equations in Snyder's book, I came across Laskowski's article, which really clarified things. As my purposes were for software, being able to compute the indicatrix numerically is a useful result. I've added a section to the page walking through the derivation as explained by Laskowski with some of my own consolidation and simplification. Marcman411 (talk) 18:57, 28 June 2020 (UTC)

AveRaComp

I just removed the section on AveRaComp, which really has little to do with Tissot's indicatrix other than being an alternative distortion metric. As these already have a place here, it seemed out of place. If anyone really feels that AveRaComp belongs on the Tissot's indicatrix article, feel free to engage here, or just revert my edit. If you do revert it, can you please explain its relevance? Marcman411 (talk) 19:28, 28 June 2020 (UTC)

Tissot indicatrix at a singularity

Currently the "Guyou hemisphere-in-a-square projection" article claims that "Tissot's indicatrix of deformation... at the singular points... the indicatrix would be infinite in size." However, the articles Peirce quincuncial projection and Adams hemisphere-in-a-square projection both show indicatrix at the corresponding singular points that is obviously not infinite in size, even though all three are aspects of the same map projection.

1. How can we fix the inconsistency in those articles?

2. Should this "Tissot indicatrix" article say a few words about such singularities? Perhaps explicitly saying something like:

"The Tissot indicatrix is typically a small ellipse. However, some map projections have a few locations (singularities) where the Tissot indicatrix is some other shape. For example, the Tissot indicatrix is a triangle at the poles of the sinusoidal projection, 1/2 of circle at the North pole of the Lee conformal world in a tetrahedron, 3/4 of a circle at the corners of many cube-based polyhedral map projections, 5/6 of a circle at the corners of many icosahedron-based polyhedral map projections, etc."

--DavidCary (talk) 22:34, 22 August 2022 (UTC)

I’m sure Guyou (and the other rearrangements) go to infinity at their singular points. I only looked at Guyou (which, as you note, is just a rearrangement of the same projection as Peirce &c.) to check your assertions. I used arbitrary precision in evaluating the major and minor axes of the Tissot ellipse for Guyou. The closer to the singularity I got, the greater the lengths of the axes, without bound. I did not try to derive the analytical result, but there is no reason an analytical result would differ. The fact that somebody’s Tissot rendering yields a finite result should be viewed skeptically. Certainly article text needs to get fixed where inconsistent, but I don’t see that here. Similarly, I’m sure Lee (Cox) also goes to infinity. As far as sinusoidal goes, I think I understand where you are coming from, but I’m skeptical that “indicatrix” means anything useful applied to the poles given that the shape of the indicatrix abruptly snaps from an ellipse to a triangle (in your scheme) there with no gradual transition. Similarly, in the other conformal cases you mention, the radius of the circle (or portion of a circle, in your scheme) changes smoothly all the way to the singularity, but the shape of the indicatrix abruptly snaps from a circle to a fraction of a circle. The literature doesn’t define the Tissot ellipse at such locations, as far as I have found, and so anything we say about such things would be original research. Strebe (talk) 23:16, 26 August 2022 (UTC)