Talk:Shape

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Contents 1 Definition of shape 2 Procrustes/Shape 2.1 Dilations and scaling 3 Definition 4 Describing shapes 4.1 Descriptive names for shapes 4.2 Global and local properties of a shape 4.3 Shape invariants 4.4 Merge

[edit] Definition of shape

Hi Oleg,

I adjusted the definition on the shapes page.

I changed the definition, because I am interested in comparing shapes. If two shapes are exactly the same, then there is no difference between the definitions. That there is no difference between the shapes after Euclidean transformations are filtered out, is equivalent to there being no difference between all properties invariant to Euclidean transformations.

The difference between the two definitions becomes apparent when comparing two objects which do not have exactly the same shape. It may not be possible to filter out the effects of location, scale and rotation. Methods, such as Procrustes analysis, are based on heuristic algorithms to filter out these transformations. It may be more precise to write that the difference between two shapes is described by the differences between corresponding invariant properties.

I will not pursue the issue. I do not agree, however, with your description of the extra words as 'clumsy', and do not see how such a description helps anybody.

Best regards. --anon

(The relevant diff Oleg Alexandrov (talk) 16:50, 24 January 2006 (UTC)) I see your point. I still believe you were trying to insert a rather confusing definition of shape (and now I see it was motivated by complicated heuristical algorithms) in an otherwise general purpose article called shape. Maybe if it was better written or better motivated, it would have made sense. The way it was when I removed it I still believe it was not helpful. Oleg Alexandrov (talk) 16:17, 19 January 2006 (UTC)

Perhaps we should just reword the sentence into anon's version, to remove ambiguity? In either case, the definition discusses scale, location and rotation. What about reflection? Shouldn't that be mentioned as well? -- Meni Rosenfeld (talk)

On second thought, less chat, more hat. I will now do the above changes, revert them if you disagree. -- Meni Rosenfeld (talk) 16:52, 19 January 2006 (UTC)

Actually this is a question of whether mirror images are considered different shapes? They aren't as far as I know. Am I mistaken? -- Meni Rosenfeld (talk) 16:58, 19 January 2006 (UTC)

"Maybe if it was better written or better motivated, it would have made sense. The way it was when I removed it I still believe it was not helpful."

Dear Oleg, there is no need to be rude. If you are going to make such comments on the clarity of my writing, please remember that people in glass houses ought not throw stones.

My argument was not motivated by complicated heuristic algorithms. I merely wanted the word 'filtered' taken out. The word 'filtered' suggests a, potentially complicated, heuristic algorithm.

Dear Meni, you ask whether mirror images are considered the same shapes. It depends on whether different parts of the shape have labels. A mirror image of a cube has the same shape. If the corners of the cube each have a unique label, however, they will not be considered to have the same shape.

Best regards, anon.

So I guess Meni found the right solution. No, what you did, duplicating the definition and taking one word out was not helpful. Oleg Alexandrov (talk) 17:49, 19 January 2006 (UTC)

Statistical shape analysis by Dryden and Mardia, 1998, uses your original definition. An invariant approach to the statistical analysis of shapes by Lele and Richtsmeier, 2001, uses the newer definition. I feel that the latter definition implies the former, more than the other way around. I did not duplicate the definition. I gave the two, competing, definitions. I also gave the reason that there were two definitions.

"No, what you did, duplicating the definition and taking one word out was not helpful. " Stating your opinion as if it were fact does not make it fact. I would assume a fellow mathematician would know that.

About anon's edit: I apologize for speaking in Oleg's name, but I'm sure he did not mean to be rude, and that he just wanted to simply state his opinion against your edit, without any cumbersome silk gloves. And I will have to agree with him that the way you have put it was too much of a duplication, and that the article is about a mathematical concept, not a computer science one. So whether there is an algorithm that filters out those properties is irrelevant to this article - in math, you can do things without algorithms (isn't math great?)

About mirror image:anon's answer missed the point of my question - there is no argument whether the cube you have mentioned cannot be transformed to its mirror image with translation and rotation. But are these two really considered different shapes? Oleg, what do you think? -- Meni Rosenfeld (talk) 18:07, 19 January 2006 (UTC)

I guess the matter is easier to decide in 2D. But even there, I don't really know if two shapes which are mirror image of each other would be considered the same shape. My best guess that that they won't. Oleg Alexandrov (talk) 21:20, 19 January 2006 (UTC)

Hi again guys,

"And I will have to agree with him that the way you have put it was too much of a duplication, and that the article is about a mathematical concept, not a computer science one. So whether there is an algorithm that filters out those properties is irrelevant to this article - in math, you can do things without algorithms (isn't math great?)" Nobody mentioned computer science. The Procrustes algorithm dates to Gower, 1975. It has only recently been used as a mathematical tool in computer science.

Just in case you think algorithms are confined to computer science, I got this from wikipedia: "Al-Khwarizmi, the 9th century Persian astronomer of the Caliph of Baghdad, wrote several important books, on the Hindu-Arabid numerals and on methods for solving equations. The word algorithm is derived from his name, and the word algebra from the title of one of his works, Al-Jabr wa-al-Muqabilah."

With respect to the word 'duplication', is this not an encyclopaedia? There are two definitions in the mathematical literature. Is it not more comprehensive to include both? That my original alteration lacked sufficient clarity is unfortunate, but not necessarily a reason to discount it entirely.

I used a cube because it is easy to visualise. I should have made that more clear. As in Oleg's original definition, two objects have the same shape if one can be transformed to exactly match the other, using only Euclidean transformations. These Euclidean transformations are also called rigid motions, or Euclidean motions, in geometry. The question of equivalence of reflections comes up quite a lot. The definition of shape is sometimes extended to include all orthogonal transformations, plus scaling, i.e., transformations that preserve relative distances and angles (google mathworld, orthogonal transformations for a good definition). Orthogonal transformations include reflection. This implies that the shape of an object is equivalent to the shape of its reflection.

"But are these two really considered different shapes?" This question is not clear to me, I am afraid. I would appreciate it if you would expand a little. Are you asking if the object and its reflection have the same shape? Or, are you asking if the object and its reflection are the same objects? Euclidean transformations involve scaling, rotating and translating. The reflection question is a good question, and there is no perfect answer to it.

Finally, Oleg, if you knew what you were talking about, you would know that shape is clearly defined in $n$-dimensional space by the likes of Kendall and Le, and Goodall. And reflection is discussed.

I admire your enthusiasm in extending Wikipedia. My intention was merely to do the same, by adding the extra knowledge I had about the definition of shape. Your reaction was not appreciated.

Okay, before you two start throwing punches, I'll say that what I understood from Oleg's reply is not that defining shape is technically harder in higher dimensions, which we all know is not the case, but rather that in lower dimensions it is easier to visualize the problem and have an intuitive feel for what the definition should look like. So I believe your comment "if you knew what you were talking about" was not appropriate, and just potentially lead to a more hostile argument. Regarding my original question, I agree that if "identical shape" is taken to mean you can transform one object to the other using a rigid transformation and scaling, then mirror images do not have the same shape.

Regarding algorithms, I agree that algorithms are used not just in computers. But note that "computer science" is a somewhat misleading concept. Most of what is studied in it has no direct relation to modern computers - It's just that it makes much more sense when using such machines. So, it is possible to discuss algorithms in pure mathematics, but to most questions they are irrelevant. For example, you know an equation such as x⁵ + x⁴ + x³ + x² + x = 2 can be solved (has a solution), regardless of whether you have an algorithm to find the solution or a data structure to represent it. Similarly, "properties that remain after...is filtered" and "properties that are invariant to..." are identical in the context of mathematics - it doesn't matter if you have an actual algorithm to filter these things out. In CS (which has, again, a loose relation to modern computers) they are different, and the latter is probably a more successful one. ---- Meni Rosenfeld (talk) 13:53, 20 January 2006 (UTC)

"According to one common definition, the shape of an object is all the geometrical information that remains after location, scale and rotational effects are filtered out. This definition implicitly assumes that it is possible to filter these effects out. Another definition is that the shape of an object is all of the geometrical information that is invariant to location, scale and rotation."

The definition used in the first sentence originates, according to Dryden & Mardia, 1998, in a 1977 paper by Kendall, "The Diffusion of Shape". Kendall's 1984 paper started off the whole Procrustes analysis methodology, linked to at the bottom of the shapes page.

I agree that I may be incorrect in concluding that the first definition "implicitly assumes" a filtration method is possible. It would be better, in my updated opinion, if I had said "suggests". I feel that this definition unnecessarily motivates Procrustes analysis, a filtration method, to mathematically analyse differences between shapes.

So, yes, the definitions are mathematically equivalent. They are not, however, equivalent. Both definitions are in common use in modern mathematics. As such, it seems reasonable to me that both definitions should be included in an article in an Encyclopaedia, and the reason that there are two definitions should also be included. This is why I had three sentences; two definitions and a reason.

My alterations may have needed improvement. I do not, however, think it is good practice to just try to wipe them out, and call the words I used clumsy without justification. My last comment may have been somewhat antagonistic, but I am not yet ready to retract it. --anon

Anon, I would still be weary of putting all that compicated stuff i the second paragraph in the introduction. But I have an idea. Since you are the expert, could you please write a good new section in the article on shape analysis, discussing both definitions, differences between them, uses, etc, with references? I am weary of having all that complicated stuff in the introduction, but a standalone section somewhere down the article would be much appreciated. Oleg Alexandrov (talk) 16:23, 20 January 2006 (UTC)

Hi again Oleg,

may I have the reference for your opening paragraph? The bit about size being dilation, specifically. Regards, anon

By the way, how about logging in? You will get a true name and signature. Also a watchlist.

About shapes. This is not my opening paragraph, and I did not write this article. I don't have references either. All started with me removing some rather complicated/almost duplicated definition.

You are the expert, please work on this article if you wish. All I care about is to keep at least the first part of this article accessible to the general public. As far as anything else is concerened, please feel free to do any work you feel like it, actually I would ask you to do so, you may add some useful things. Hope that helps. :) Oleg Alexandrov (talk) 16:50, 24 January 2006 (UTC)

[edit] Procrustes/Shape

Sorry for getting into the discussion late. For what its worth I was one of Mardia's postdocs. Some of this might want to be discussed under Talk:Procrustes analysis rather than here. Also it would be good if anon could register.

A few points.

Actually this is a question of whether mirror images are considered different shapes?

Generally considered to be different shapes. Especially important when considering things like chemical compounds when left handed and right handed compounds can have different properties.

Another extension is to consider shape and scale or shape and size when the size of an object is important.

Dialation not commonly used in the statistics field, normally talk about scaling instead. (Is dilation a more common American term?)

The common usage of the term is much less precise than a mathematical definition. Consider What shape is it? its a rectangle! (even though rectangles have infinitly many different shapes acording to the above definiton). There is a case for including wider definitions of shape with a more topological feel to them.

As to the definition I'd be happier with just the newer definition

The shape of an object is all of the geometrical information that is invariant to location, scale and rotation.

I don't think anyone in the field would object to that definition or indeed say that it was a different definition, just a better way of expressing the same concept in English. You would probably find a slight different wording in each paper on the subject.

The pedants among us might object to the concept of the shape of an object instead talk about whether two objects have the same shape. Indeed Procrustes is technically closer to the latter.

To be really pedantic the definition of shape should be expressed in mathematics and not English. The Shape of on object is an equivalence class of objects which can be transformed onto each other by rotation, translation and scaling.

Finally its important to distinguish Statistical shape analysis (Procrustes, Kendal, Goodall, Mardia, Lele etc) from the general concept of shape which has been around since Euclid. Procrustes is one method of discussing shape which has particular statistical underpinning. --Salix alba (talk) 19:29, 25 January 2006 (UTC)

I never heard that "rectangle" is one shape.--Patrick 02:03, 26 January 2006 (UTC)

I've been pondering a a bit about quite what an appropriate definition of shape would be.

First we need to consider the history of the word. Its been around for a long time and it is only reciently that a particular definition of shape has been used in statistics.

In Wiktionary shape is defined as

shape (plural: shapes) The status or condition of something She checked to see what kind of shape the animal was in. The appearance of something, especially it's outline: He cut a square shape out of the cake. A figure with unspecified appearance; especially a geometric figure What shape shall we use for the cookies? Stars, circles, or diamonds? (medicine): Condition of personal health, especially muscular health.

There are lots of occurences of the word in mathematical work with a less strict definition. From a text book aimed at 16 year olds. We have

A triangle is any three sided shape

Shapes which are exactly the same size and shape are called congruent shapes.

This leads to both a strict and loose definition of shape. --Salix alba (talk) 19:54, 26 January 2006 (UTC)

[edit] Dilations and scaling

Using dilation reather than scaling is confusing. The only property required for a dilation is that its distance preserving, hence both reflection and translation are dilations. Dilations also include reflection so are not strictly shape preserving. Hence dilation has to go. --Salix alba (talk) 19:54, 26 January 2006 (UTC)

I like the way this article is evolving, but I do not think it is finished yet. I have some points.

"For an object of greater than 2 dimensions, one can always reduce the dimensions of the shape by considering the shape of a cross-section or a projection." What does this mean? One is not reducing the dimensions of the shape. One is instead looking at a piece of the shape.

There are two uses of the words shape. There is 'a shape', such as a triangle, square, whatever, where shape is an object. Then, there is 'the shape of an object', where 'shape' is the property of an object. The article seems to switch between the two definitions haphazardly.

For example: "The shape does not depend on changes in orientation/direction. However, a mirror image could be called a different shape." Would it be more appropriate to say that the mirror image of an object may have the same shape, if it is symmetric?

Next, the scientific community is divided on whether the shape of an object is considered to be invariant to its size. This is primarily because multiple definitions of size, or scale, exist. For a closed, convex surface, is scale the surface area or the enclosed volume?

For example, consider the sphere. The surface area of the sphere is 4\pi r^2, whereas the enclosed volume is 4/3 \pi r^3. We could consider scaling the Euclidean space in which the sphere lies, but what about when the centroid size is used as the measure of scale of landmark configurations?

To quote Lele and Richtsmeier, The promise of geometric morphometrics, 2002, "Focusing specifically on landmark data, we discuss the concepts of size and shape, and reiterate that since no unique definition of size exists, shape can only be recognized with reference to a chosen surrogate for size." If this is true, then the shape of an object is invariant to scale, and scale has multiple definitions, then a single object can have multiple shapes. This does not seem to me to be a desirable property of a mathematical definition.

Anon -- —The preceding unsigned comment was added by 193.1.100.8 (talk • contribs).

[edit] Definition

"In other words, the shape of a set is all the geometrical information that is invariant to location, scale and rotation." You have got to be kidding. First of all, this isn't much of a definition, in that it tells what shape is not, rather than what it is. Second, it's a bit technical in the introduction. Please fix. 71.102.186.234 01:30, 30 November 2006 (UTC)

[edit] Describing shapes

This article is still very week here's a few thoughts on a new section

[edit] Descriptive names for shapes

A variety of descriptives names have been used to describe shapes, these are often in comparison with well know objects. Frequently there are many different objects which fit each class

Geometrical shapes

• triangular -
• rectangular -
• square -
• Cylindrical
• Conical
• Spherical
• Circular
• Spiral (or helical)

Letter like shapes

• L-shaped, S-shaped (sigmoid), T-shaped, U-shaped, V-shaped, (crossed shaped) X-shaped

Comparison with everyday objects

• Hourglass shapes
• Figure-8 shaped
• Teardrop
• Bell shaped
• Star shaped
• Doughnut (with a hole i.e. toroidal)

[edit] Global and local properties of a shape

It is also common to describe one aspect of a shape as a whole. The sign of the Gaussian curvature can be used to describe an aspect of the local shape

• Convex - positive Gaussian curvature
• Flat - zero Gaussian curvature
• Saddle-shaped - negative Gaussian curvature
• Concave

• Symmetrical, generally refers to mirror symmetry
• polygonal
• polyhedral

[edit] Shape invariants

Certain points and curves on an object will be invariant under rotation, scaling and translation

• inflections of plane curves (when the curvature is zero)
• vertices of plane curves (where the curvature has a local minimum or maximum)
• parabolic lines of a surface (when Gaussian curvature is zero)
• ridges (local extremum of principal curvature)

Topological properties are also invariant under a wider class of transformations

--Salix alba (talk) 13:53, 1 February 2007 (UTC)

That's fine with me, as long as the extra details appear in sections towards the bottom. And a hint: use a browser with spell-checker, like recent versions of Firefox. :) Oleg Alexandrov (talk) 16:01, 1 February 2007 (UTC)

Point taken about the spelling, it needs a lot of fleshing out, basicially just a half hour brainstorm. GHope to work on this a bit more in the next couple of days. --Salix alba (talk) 00:14, 2 February 2007 (UTC)

[edit] Merge

Salix Alba is essentially proposing this merger above; and it was supported at Wikipedia:Articles for deletion/Glossary of shapes with metaphorical names; so was deleting that list. The declared purpose of the list (having some place for V-shaped to redirect to) can as well be served by this article; and it may produce lesss of an indiscriminate collection of information. Septentrionalis PMAnderson 05:32, 9 February 2007 (UTC)

I disagree with the merger. That article is much bigger than this one and all that listcruft would overwhelm this article. I would agree with moving here a minimal list of most important shapes rather than the whole list. Or otherwise I think that article better stay separate or get deleted. Oleg Alexandrov (talk) 15:59, 9 February 2007 (UTC)

I proposed the deletion of the "metaphorical" article; I'd be just as happy with a consensus to make it a redirect, and quarry its history here. Septentrionalis PMAnderson 03:40, 10 February 2007 (UTC)