# Talk:Orientation (geometry)

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## orientation

I wonder how standard this "orientation" term is, I confuse it with the other notion of orientation.

It is unfortunate that the word has two meanings, especially because they are sometimes used both in the same context. What else would you call it?--Patrick 10:34, 3 November 2005 (UTC)

Right. However, my question was still whether "orientation" used in this context is standard. Oleg Alexandrov (talk) 15:15, 3 November 2005 (UTC)
Yes, I think so.--Patrick 17:00, 3 November 2005 (UTC)
Yes, the orientation of an object in space is standard informal meaning the direction in which the object is pointed. Don't confuse it with the orientation of vector space basis, which is only an order on the basis vectors.--MarSch 14:25, 6 November 2005 (UTC)

## Question

I don't understand what this article wants to say. The first sentence says that orientation changes by rotation. But what is orientation to start with? This is not clear from the article. Oleg Alexandrov (talk) 03:42, 4 November 2005 (UTC)

The term is so basic that it is hard to find even more basic terms to explain it in. The article explains the relation with another basic term, rotation, and points out that orientation has also another meaning.--Patrick 11:34, 4 November 2005 (UTC)
Wikipedia is not a dictionary, but orientation can be described by the directions of axes fixed in the body.--Patrick 11:48, 4 November 2005 (UTC)
The term "rotation" is also basic, but it has a rigurous math definition eventually, it is an orthogonal matrix with positive determinant. The "orientation" concept must be defined in some rigurous mathematical way too. That something is basic is no excuse not to define it. Pretending for a moment that I don't know what orientation is (and actually, I am not sure I have a good grasp of that concept), this article does not help in explaining it. Oleg Alexandrov (talk) 16:47, 4 November 2005 (UTC)
I added a formal part.--Patrick 23:20, 4 November 2005 (UTC)
Thanks. Oleg Alexandrov (talk) 01:51, 5 November 2005 (UTC)

If you know what a torsor is: the group of rotations is a torsor on the orientations. Basically this means they are isomorphic but NOT canonically. You need to choose a ``starting`` orientation to identify with the unit rotation. Now each rotation corresponds bijectively with an orientation. A rotation describes how to get from one orientation to another.--MarSch 14:43, 6 November 2005 (UTC)

## Summary style

This article is not meant to cover representations in detail, as there are already articles for that. In particular there are articles on the different representations such as Euler angles, yaw, pitch and roll (Tait-Bryan angles redirects there), rotation matrix etc., as well as an article Rotation representation (mathematics) on the mathematics of all these.

This article is meant to be in summary style. It should not include lengthy sections on any one representation as that would make it unbalanced, requiring all sections to be expanded in a similar way. But that would make it far to long and complex given that there are seven such sections. It especially does not need content from other articles just copied here out of order: there is already a section on yaw, pitch and roll/Tait-Bryan angles with a link to the main article. The number of images was also excessive (as it is excessive at yaw, pitch and roll), and making them smaller to better fit made them unreadable. So I've reverted the changes. --JohnBlackburnewordsdeeds 13:04, 3 October 2010 (UTC)

I do not agree with your removal. The conventions of axes are something that affects all the orientation representations, not just Tait-Bryan angles. For sure it should be here instead of there. The section in the yaw, pitch and roll article is the one that could be removed leaving a link to this place.--Juansempere (talk) 22:42, 3 October 2010 (UTC)
That's incorrect. Most mathematical formulations are independent of the axis conventions. You need axes to write them down numerically but they can be derived and worked on in an entirely coordinate-free way. Even matrices, though dependent on the basis, care little for the conventions: you can reorder the axes and simply reorder the rows and columns. It's only Euler angles where you have to be careful as the order the rotations are done matters (which is one reason they are less often used in mathematics and geometry today). I.e. it's something at needs to be covered at yaw, pitch and roll but not in a more general article: unless the detail of all the articles was merged here, but that make this too big and defeat the purpose of a summary article.--JohnBlackburnewordsdeeds 22:56, 3 October 2010 (UTC)
You didn't understand me. I will try to reword my sentence. Though most mathematical formulations are independent of the axis conventions, all of them will require an axis definition at some point, as you said. Therefore the place to explain those possibilities and standards is an article that speaks about all of them (this one). The other possibility would be to copy that text inside the articles of Euler angles, matrices and quaternions (yes, the quaternion representation depends of the axis definition) --Juansempere (talk) 05:43, 4 October 2010 (UTC)

No, they don't require any of that. All they require is the three dimensions: x, y and z. These can be established by considering vectors, which once you have this them other methods can be defined from them: the quaternion to rotate between two vectors is given by (xy, xy + yx - x × y) for example. The 3d fundamentals are covered in Cartesian coordinate system linked at the top of this article. Even Euler angles do not usually go into such details as you added; that's why we have two articles, Euler angles for the more abstract mathematical treatment and yaw, pitch and roll for their real world applications.--JohnBlackburnewordsdeeds 09:46, 4 October 2010 (UTC)

It is clear that I am not expressing myself clearly. I will put an example. You are a satellite engineer and you receive an orientation matrix to reorient your satellite. The person that sends the matrix to you says "This is referred to the norm DIN-9300". Of course, then you would go to wikipedia to find what the hell is that, and you would write "attitude", which is redirecting here. You will find nothing because that information is the article about Tait-Bryan angles. A convention which you are not using and which probably you never will.
The same can happen with quaternions or Euler angles. They can be defined without a specific axis convention, but at some point the axis will have an important role, and that should be documented in an article that is common for all the orientation conventions.--Juansempere (talk) 18:29, 4 October 2010 (UTC)
I suspect a satellite engineer will have the specialist knowledge to understand such instructions: I am not one so would not know. From my own experience, imagine you are a computer programmer asked to write a dynamics system, or a camera system, or an animation system. You use quaternions and matrices, the former being more compact and quicker to do calculations with, the latter better for applying transformations. Not once do you need any of the theory you are referring to.
More usefully, as anecdotes are not a useful guide to what to what should be in an article, look at any reference on quaternions. The one here is pretty good and pretty typical, and manages to define and explain quaternions without any reference to DIN-3000. The same is true of matrices and axis-angle. Even the theory of Euler angles doesn't depend on knowing all the different conventions used by different modes of transport, their names etc. – hence the two articles, Euler angles and yaw, pitch and roll.--JohnBlackburnewordsdeeds 20:05, 4 October 2010 (UTC)
Of course the anecdote was only to make you understand my point of view, not as a guide to include anything. One thing that you have stated (and I cannot agree with) is that we should not cover some information here because the people that would look for it will already know about it. What is wikipedia for then? Other thing that you said is that all the attitude description conventions can be defined independently of the axis convention. That is true, but they are related, and these axis conventions should be covered in an article independent of the tools to describe orientation. Would you agree to create a new article called Attitude description conventions or something like that? --Juansempere (talk) 20:37, 4 October 2010 (UTC)

We should, and already do, cover it at yaw, pitch and roll, which is an good place for it as it's to do with that way of describing rotations, and anyone can find it there who needs to. It's not needed at all for other methods. We have multiple articles on rotations because there is far too much for one article. They are all related to some extent, and there is some unnecessary overlap, but mostly they keep on topic. This article covers them all but largely in summary style, so should not include lengthy content from any other.

It is easy and common to study and work with orientation in 3D, including Euler angles, without ever encountering the various conventions: I for one never came across them until recently on Wikipedia. I.e. it's something that's peripheral to most modern geometry and the maths behind it. Whether it needs to be split from Yaw, pitch and roll is another question, but I don't think so: the text and diagrams are largely concerned with yaw, pitch and roll, so much so that it's abbreviated. And in a sense the split has already been done, as Euler angles is concerned with the theory and Yaw, pitch and roll with the different applications and conventions.--JohnBlackburnewordsdeeds 21:12, 4 October 2010 (UTC)

I'm with John on this - to try to describe the million and one different axis conventions used in the world is far too much detail for an article on such a general topic as "Orientation (geometry)". It's probably even too much detail for the Yaw, pitch and roll article, so putting it in a separate article might be the best thing to do. Djr32 (talk) 21:40, 4 October 2010 (UTC)
I would say that what John wants is NOT to put it in other article nor here. He wants to leave that information inside "Yaw, pitch and roll". While I really can admit most of his reasoning about the problem, I cannot agree with his proposed solution. To leave inside "Yaw, pitch and roll" is possibly the worst solution of them all. I would want to know your oppinion about this. Do you support to make a new article?--Juansempere (talk) 09:25, 5 October 2010 (UTC)
I think that it would probably be sensible to create a new article. The axis naming conventions used in different areas of engineering are separate from concept of Euler angles, and keeping the explanation separate seems to have the best chance of avoiding leaving everyone more confused than they started. (Though this seems to be an area which always causes no end of difficulties whatever you do...) It's definitely an issue for engineering rather than mathematics, i.e. applied rather than theory! Djr32 (talk) 21:51, 5 October 2010 (UTC)
Well, then I propose to make an article called axes conventions or Attitude (engineering) and to move there the removed information. Who agrees? Who dissagrees?--Juansempere (talk) 17:21, 6 October 2010 (UTC)
I have just done it. Please take a look.--Juansempere (talk) 22:26, 8 October 2010 (UTC)

## Caption of figure in lead

The present caption states “Changing orientation is the same as moving the coordinate axes.” That caption is wrong in some interpretations of it. For example, the orientation of a rigid body is changed if the coordinate axes fixed to the body are changed, but it is not changed if the coordinate axes used for description of the body are changed. Brews ohare (talk) 15:56, 5 October 2010 (UTC)

I agree. I think it is enough to say something like “For a rigid body, changing orientation is the same as moving some coordinate axes fixed on it”. Would this be OK?--Juansempere (talk) 17:07, 5 October 2010 (UTC)
How about something like: “For a rigid body, changing its orientation is the same as moving a set of coordinate axes affixed to it”. Brews ohare (talk) 17:23, 5 October 2010 (UTC)
I like yours more than mine.--Juansempere (talk) 17:28, 6 October 2010 (UTC)

## Definition

The definition of orientation should not contain the word orientation (as in "rotation from a base or initial orientation").

It may be defined as the rotation that, together with a traslation, would ideally move the body from an arbitrary reference position to its current position (I use here the word position to mean both linear and angular position).

Can you find some simpler (but equally valid) way to define it?

Paolo.dL (talk) 17:28, 20 November 2010 (UTC)

As of the current revision, the lead requires cleanup. It is too long, contains too much specific detail, and lacks any coherent organizational structure. The lead is just supposed to introduce the topic and summarize the article. It should be organized into no more than three paragraphs. Currently, I count seven "paragraphs" that are basically just loosely connected ideas. The lead in the older revision was more in line with the manual of style. I think the old revision should probably be restored, unless there is some reason not to. Sławomir Biały (talk) 14:05, 9 January 2011 (UTC)

The reason why the older revision is not there anymore is quite evident. The older revision only focused on rigid bodies. It was shorter just because it was incomplete. But you are right: the intro is too long now.
To solve this problem, I moved detailed explanations about math representations and degrees of freedom (in 3-D and 2-D) to the first section (overview about math representations), where they actually belong in.
Paolo.dL (talk) 21:51, 9 January 2011 (UTC)
The older revision actually did mention lines and planes (which btw are also "rigid bodies"). It did nor treat them in full detail, but nor should it. Sławomir Biały (talk) 12:15, 10 January 2011 (UTC)
But, it looks better now. The older version of the article lacked a good top level organization that the current "Mathematical representations" now fulfills. Another nitpick is that the section on orientation in geometric algebra is a special case of the notion in linear algebra, which is not related to the subject of this article. I will go ahead and move it to the more appropriate article. Sławomir Biały (talk) 13:28, 10 January 2011 (UTC)

## Physical objects, bodies, figures, segments, spaces, planes, lines

Sławomir, good job, thank you. By the way, your definition of the term "rigid body" aroused my curiosity. I agree that lines and planes are rigid. But I am not sure they can be regarded as (rigid) bodies. As far as I know, "body" is used only for a solid object. This also implies that nothing can be called "body" in 2-D (where we have rigid figures). Anyway, I believe the generic term most commonly used for both bodies and other "sets of points" is object or space. — Paolo.dL (talk) 18:58, 10 January 2011 (UTC)

It's not especially relevant to the article, but in mechanics the term "rigid bodies" refers to a collection of points such that the distance between any two points remains unchanged under the application of any external forces. This definition obviously includes lines and planes. Sławomir Biały (talk) 18:31, 14 January 2011 (UTC)
Physics textbooks (or mechanics textbooks) typically first define Newton's laws of motion for "point masses" (particles), i.e points endowed with mass, not plain points. Then they generalize to "rigid bodies" (by means of Newton-Euler equations), defined as collections of particles (not plain points). Without mass, mechanics does not make sense. So, rigid bodies in mechanics are endowed with mass. For didactical purposes, we often generalize the concept to 1-D or 2-D objects having finite length or area, but never to lines and planes.
The most common generalization is to 1-D line segments (e.g. representing a bar) or 2-D figures (such as the classic drawing of a potato). These segments or figures, however, are always treated as a collection of particles (not plain points). Indeed, the formula to compute the center of mass of a body is first explained for a 2-D drawing. A generalization to lines and planes is improper, as these are boundless objects, so they cannot be thought as a collection of particles (otherwise, they would have infinite mass and hence would be useless for modeling physical objects).
Indeed, even Wikipedia defines rigid body as a solid physical object, and I agree with that definition.
Paolo.dL (talk) 21:14, 14 January 2011 (UTC)
This seems to rely on a peculiar interpretation of the word "particle" and a fairly strict reading of the word "solid" in our own article on the subject. I don't think you will find any serious source that excludes e.g., line segments from being rigid bodies. It's worth correcting you also that one can have an unbounded line of particles that still has finite mass; see improper integral for examples. Anyway, as long as we present what reliable sources have to say about it, our philosophical speculations here about the nature of "rigid bodies" is totally irrelevant. Sławomir Biały (talk) 22:22, 14 January 2011 (UTC)
You did not read with attention: I am not excluding line segments. I explained they are a commonly used model for some physical objects. On the contrary, a "unbounded line of particles that still has finite mass" is not a valid model for a physical object. My point is that, as stated in our article, a rigid body is a model of a physical object. Also, without mass Newton's and Newton-Euler's laws are useless. So, a rigid body as defined in mechanics is not a set of points, but a set of point masses. Paolo.dL (talk) 23:52, 14 January 2011 (UTC)
"unbounded line of particles that still has finite mass" is not a valid model for a physical object
is extraordinary. A Gaussian mass distribution is probably the most widely used continuous mass distribution, and yet is certainly not compactly supported. Also, our article certainly doesn't state that "rigid body" must correspond in some platonic sense to some real-world thing. It seems that this is your own personal philosophical requirement.
Also, one can define a rigid body without any reference to the laws of physics. A collection of points, depending on a parameter t, is rigid if the distances between every pair of points is independent of t. Mass plays absolutely no role. Sławomir Biały (talk) 03:04, 15 January 2011 (UTC)
Just my thoughts on the original query. I have no problem with lines and planes, as in forms with extent only one or two dimensions, being treated as rigid bodies: I have in the past written rigid body dynamics software which included flat discs and rectangles. Lines were avoided because with zero angular momentum they could in theory gain infinite angular velocity about their long axis. They were much better modelled by long thin rectangles or cylinders (and in a computer simulation made of polygons lines with no thickness are difficult to represent). But flat shapes were perfectly good and well behaved objects in the simulation. And even lines might be used in more complex objects, to constrain or link two parts of an object.
So although 'body' is usually something with volume when talking about 'rigid bodies' you are often concerned with more than just shapes with volume, especially flat shapes with 6DOF and a non-vanishing MOI.--JohnBlackburnewordsdeeds 03:39, 15 January 2011 (UTC)

### Valid or invalid generalizations of the term "body"

John, I agree that solid figures such as flat discs or rectangles can be validly called and treated as (rigid) "bodies". In other words, they are a valid idealization (mathematical model) of quasi-rigid and flat "bodies". In my opinion, even a line segment can be considered a valid model of a body, when the null moment of inertia about its longitudinal axis is not a problem, as we focus, for didactical purposes, on rotations about its transverse axis. But in my opinion this is not true for unbounded planes or lines (N-D spaces).

Sławomir, your definition of a rigid collection of points is flawless. We perfectly agree on the meaning of the term rigid, also used in mathematics to denote a special kind of affine transformations. We only disagree on the meaning of the word body. In your opinion, a body is just a collection of points. In my opinion, "body" means physical object, and is properly used only in the context of physics. Thus, a rigid body is:

• an idealization (i.e. a mathematical model)
• of a solid physical object (i.e. quasi-rigid body)
• of finite size.

Some bodies are quasi-rigid, so the ideal assumption that they are rigid is valid and useful. But no physical object has infinite size. Hence, an unbounded line or plane or space is a collection of points, but not a valid idealization of a body. I am not saying that the concept of unbounded collection of points is useless. I just maintain it makes no sense to call it a physical object, or body. In this case, in my opinion the terms object and space are more appropriate. In other words, planes and lines are geometrical objects, but not physical objects.

Paolo.dL (talk) 12:02, 15 January 2011 (UTC)

Most sources do not impose that bodies be bounded in the definition. There is probably a good reason for this: quite often unbounded continua are used to model physical bodies. For instance, in fluid simulations, usually only one set of boundary conditions in a body of water (or gas) are relevant. I know a biophysicist who models an axon, which is certainly a physical object, as a semiinfinite line. The restriction is simply not helpful to anyone that wants to do physics. Also, the personal philosophical assertion that a body must be bounded doesn't seem to be one that is widely stated in the literature. The standard one seems to be "collection of material points" or sometimes "collection of particles" without any reference to the boundedness of that collection. Sławomir Biały (talk) 12:38, 15 January 2011 (UTC)
You confuse a terminological discussion with a guideline for scientists. Scientists may use whatever model they like and by no means the name we or they give to their models will limit their freedom. The problem here is terminological, not methodological: is it appropriate to call "bodies" unbounded collections of points such as lines or planes? Isn't it more appropriate to call them "spaces"? That's it.
So, let's discuss about terminology. You gave examples of partially bounded (not totally unbounded) objects which can be validly called bodies. The axon is a tube and its external membrane is a boundary. Also, your examples are valid in the context of fluid mechanics and possibly electromagnetism (?). Some textbooks include a chapter or section called Rigid body dynamics. It seems to me that this is the context where the idea of "rigid body" makes sense. Can you provide examples of rigid bodies with infinite size in this context? We cannot compute the center of mass or study the motion of an object with infinite infinite size, mass and moment of inertia. There's a crucial distinction between external and internal forces in this context, which implies the existance of boundaries and a finite size. In this context, where the expression "rigid body" makes sense, it makes no sense to generalize the term "body" to lines and planes. Does it make sense in other contexts?
Paolo.dL (talk) 16:30, 15 January 2011 (UTC)
As I have repeatedly said, no treatment that I have seen imposes boundedness. Our own article defines a rigid body as a "collection of particles". This generally agrees with the definitions I have seen in googling: either "collection of particles" or "collection of material points". There is never any reference to the boundedness of the collection. So I should think this should more or less settle any debate about the terminological issue, unless sources are brought to bear that actually do make this assumption explicit.
Also, it is not I that have confused a terminological issue with a philosophical one, but you yourself. Your unwillingness to consult the literature, and preference of personal philosophical opinions (body = physical object, "no physical object can have infinite size") is becoming tiresome. Now we need to distinguish between "partially unbounded" and "totally unbounded"? Seriously? Give me a break. The only relevant datum here is what reliable sources have to say.
Finally, regarding your (irrelevant) challenge, yes I can. Take the interior of the surface of formed by revolving around the x axis the exponential curve ${\displaystyle y=e^{-x}}$, ${\displaystyle x\geq 0}$. Give this the uniform mass distribution. Then it has finite total mass, and all moments are well-defined. QED Sławomir Biały (talk) 19:15, 15 January 2011 (UTC)
I am not interested in a stupid fight, so I am not going to reply in detail. You prefer to attack, rather than simply ignoring or giving an answer to the main question, which everybody is free to ignore or answer based on his knowledge of the literature or just on his experience and opinion. We are discussing whether a line or plane should be called a (rigid) body. I defined bodies as collections of particles or point masses much before you repeatedly gave the same definition. By the way, planes and lines are not collections of particles or point masses, but I am not satisfied by this simple conclusion, and asked everybody whether, in their opinion, (unbounded) lines or planes could be validly used as a "model" for some kind of body which I simply cannot imagine. If you are tired by this discussion, feel free to do something else. Paolo.dL (talk) 22:59, 16 January 2011 (UTC)
Then I'm not sure what the issue is. I have responded to various remarkable assertions that you apparently have made: (1) that bodies are by definition bounded, (2) that no physical object can have infinite size, (3) that the equations of physics make no sense for an object of infinite size.
Also, to respond to your latest claim that "planes and lines are not collections of particles" [you said "point masses", but these are not relevant for continuum mechanics anyway], my retort is: Why not? Plenty of physicists consider lines and planes of material points, even in mechanics. So I don't see this as a valid concern. See for instance, [1], where rods are regarded as infinite lines of material points. Sławomir Biały (talk) 23:41, 16 January 2011 (UTC)

As I already wrote, someone might use lines or planes to model collections of particles (bodies), for purposes I cannot imagine. Examples are welcome, if they exist, but for sure these models are useless in rigid body dynamics, as their mass would be infinite. The reference you provided seems to model elastic rings. In which page is the infinite line model defined? Paolo.dL (talk) 00:52, 17 January 2011 (UTC)

"Useless in rigid body dynamics": Wrong. Put a Gaussian mass distribution on the x axis. The mass and all moments are well-defined, and the equations of physics make sense. Sławomir Biały (talk) 19:04, 17 January 2011 (UTC)
An infinitely long rigid body with gaussian mass distribution, used in dynamics. Are you joking? You wrote that there's "plenty of physicists" who "consider lines and planes of material points, even in mechanics". If there are so many, then you have a lot to teach to those who are interested in this discussion. Can you give us some examples, explaining what is the body and body property modeled with a line or plane, and what is the purpose and possibly the application of the model? That would be interesting. And if you have an example in rigid body dynamics, that would surprise me, and make your contribution much more interesting. Otherwise, you will never convince me that planes or lines can be validly used to model bodies in rigid body dynamics. Paolo.dL (talk) 21:47, 17 January 2011 (UTC)
You're moving the goalposts. You just said that an infinitely long rigid body was useless because it had infinite mass. I just produced a counterexample. Take that for what it is. Sławomir Biały (talk) 04:04, 18 January 2011 (UTC)
I see what you mean, and I am sure you are in good faith, but in my opinion your counterexample simply ignored my main point, which was already and more clearly explained much before your counterexample (see my comment posted at 16:30, on 15 January 2011).
And yes, I confirm that in my opinion, in the context of rigid body dynamics (RBD), an infinitely long rigid body would have an infinite mass. That's because as far as I know a rigid body with Gaussian mass distribution is a model of an object which simply does not exist. Hence it is useless and invalid in the context of RBD. So, your counterexample was brilliant, but you already proposed it before, and I already discarded it as an invalid example, even more invalid in RBD than an infinitely long rigid body with uniform mass distribution.
But then again, since you wrote that "plenty of physicists consider lines and planes of material points, even in mechanics", can you explain the purpose of some of these models? Although I doubt you can find a valid example in the field of RBD, I suppose you can find examples that will prove your point in other fields. (Writing that someone modeled "an axon ... as a semiinfinite line" is not enough, as I can't imagine the purpose; and hence I suspect it was, perhaps, a tube of "semi-infinite length", rather than a semi-infinite line; moreover, it would be more interesting if you provided examples of unbounded lines and planes)
Paolo.dL (talk) 21:09, 18 January 2011 (UTC)