Talk:Rigid body

Relativity

I think that the problems of defining a relativistic version of the rigid body should be mentioned on this page. Several paradoxes in relativity theory stem from this issue (e.g., the barn and pole-vaulter). Ty8inf (talk) 18:26, 7 April 2008 (UTC)

I agree, and more importantly since in relativity one can see that the whole rigid body concept is an ideal approximation that is incongruent with the speed of light limit. Of course without leaving out the fact/concept of rigid motion in relativity (note that rigid motion redirects here), and to mention Ehrenfest paradox. I took the liberty of adding the Relativity wikiproject banner although I'm not a member of such group. Did I do wrong? JunCTionS 00:54, 9 April 2008 (UTC)

Position and velocity

I generalized the section about position a little better than before, and removed its dependence on a pair of reference frames. I think it is easier to understand without weighing it down by presenting the definition simultaneously with an example which never gets finished. I also added a short section on velocity and angular velocity and gave the important characteristics of each.

One thing I changed was the definition of angular velocity: it is not the time derivative of "angular position". Angular velocity is a vector, but there is no such thing as an "angular position vector" that you could take the derivative of. If this article needs a better definition of angular velocity than the one I put in it, I can provide one, but it will be more technical and perhaps not appropriate for this topic. MarcusMaximus (talk) 10:38, 27 August 2008 (UTC)

I agree about the fact that the angular velocity is not the derivative of the orientation vector. However, I disagree that an orientation vector does not exist. It does (see rotation vector; as well as a rotation matrix, a rotation vector can be used to represent an orientation; in this case, you refer to them as "orientation matrix" and "orientation vector").

I disagree about using the word displacement as a synonym of position. It is used with this meaning by some authors, especially in elementary physics textbooks, but it is more advisable, when possible, to take into account the distinction between the two concepts. Notice that this distinction is not high mathematics. Even in introductory physics courses the distinction is often clearly explained (see for instance the widely used textbook by Reisnick and Holliday: Physics, Part 1, John Wiley and Sons). According to this distinction, the displacement is (even etymologically) a change in placement, i.e. a change in position, i.e. a simplified representation of a real motion. A displacement occurs in a measurable interval of time, a position exists at a given time, with no reference whatsoever to past or future instants. In other words, the tail and tip of a position vector are two points that exist at the same time (origin and particle), while the tail and tip of a displacement vector indicate the position of the particle at two different times.

Of course, you can imagine a hypothetical displacement that brings the particle from the arbitrary origin of a coordinate system (where the particle has never been) to the point where the particle is located at a given time. But still there is a difference between this hypothetical displacement (which is more properly called a position) and a real displacement. And I suggest to use a terminology which reflects and respects this distinction, as well the etymological interpretation of the word "displacement".

I also disagree about deleting the description of the two coordinate systems L and G. The rest of the article refer to these systems. The reader needs to know that it is impossible to define the orientation of a rigid body without defining two coordinate systems. For instance, how can you state that an airplane is horizontal, without first defining its longitudinal (coordinate) axis?

By the way, the angular velocity of a rigid body is explained in detail in the angular velocity article. If you like to add a better definition, you are welcome. Actually, it is possible to compute it in several (more or less intuitive) different ways. Paolo.dL (talk) 14:39, 27 August 2008 (UTC)

I understand that there is such a thing as a rotation vector. I was trying to emphasize the fact that there is no such thing as an orientation vector that is the angular analog to position, because the natural inclination of many readers will be to assume "hey, I can integrate velocity to get position, so that must mean I can integrate angular velocity to get orientation", which is, of course, not the case.

As for position and displacement, I agree with your statement. I shouldn't have equated them. I was struggling to come up with a word for "translational position" that was not merely the word "position", since this article attempts to use position as an umbrella term for position and orientation. Do you have a better word to use in this case? Or perhaps we should not use position to also refer to orientation, and keep it strictly for "translational position"?

I didn't read the rest of the article, so pI was unaware that it is necessary to establish two coordinate systems (or sets of basis vectors) for the purposes of the examples. However, I do not believe that this should be a part of the descriptions of position and velocity. Those concepts are far more general than just the position and velocity of a body relative to the "world". They can be defined relative to any arbitrary point of reference and any arbitrary reference frame. If the article needs two sets of basis vectors and/or a coordinate system for comparison, it should be introduced when it is first used, after the definitions of position and velocity. MarcusMaximus (talk) 16:35, 27 August 2008 (UTC)

You incorrectly stated that there's no orientation vector that can be differentiated. There is an orientation vector and it can be differentiated. Its derivative is not the angular velocity, but it can be differentiated.

The position of a rigid body includes the orientation. This is absolutely correct. In physics, linear and angular are the keywords that distinguish the two parts of kinematics. There is a position, a velocity and an acceleration both in angular and in linear kinematics. You say (even for the motion of a single particle):

• (linear) position and angular position
• (linear) velocity and angular velocity
• (linear) acceleration and angular acceleration

There's no single word to indicate both linear and angular velocity, except for "velocity" alone. The same is true for acceleration, and the same must be true for position.

You can mention in a footnote, if you like, that some people use the word "position" as a synonym of "linear position", and that they use "orientation" to indicate angular position. But I showed you that this is a questionable terminology, not compatible with the terminology that the whole world uses for velocity and acceleration. By the way, I recently discovered that a few bioengineers decided to use the horrible neologism "pose" to indicate position + orientation, but this is just because they didn't realize that the orientation is a position. And it is a neologism used only by a very small group of people. I am sure that no serious physics textbook will ever adopt this horrible terminology. I believe that Lagrange used the expression "generalized position" to indicate both the linear and angular position (and Lagrange's terminology, together with his equations of motion, are frequently used in the scientific community). This is the only correct way to indicate unambiguously both the linear and angular position, as far as I know.

I agree that the two coordinate systems should not be defined as global (G) and local (L). You are right: any coordinate system is ok to define the position of another coordinate system. But you need two coordinate systems (CS₁ and CS₂, if you like), and that should be made clear. I will adjust my example to stress this concept again: how can you state that an airplane is horizontal, without first defining both its longitudinal (coordinate) axis, and the horizontal (coordinate) axis? Paolo.dL (talk) 18:03, 27 August 2008 (UTC)

"You incorrectly stated that there's no orientation vector that can be differentiated. There is an orientation vector and it can be differentiated. Its derivative is not the angular velocity, but it can be differentiated." -- Hopefully my recent edit has sufficiently clarified this point to your satisfaction.

In my common experience (aerospace engineering), the word "position" means only linear position, and "orientation" means only orientation. There is not a general term in my technical vocabulary to describe the two together, though I believe there should be. While "position" may encompass both in certain contexts, I think this is awkward and unnecessarily ambiguous. I don't think it is necessarily bad to define a new word that would encompass both without the ambiguity. (We use "state" to refer to the four quantities of position, velocity, orientation, and angular velocity when speaking in 6dof, or just position and velocity when speaking in 3dof.)

There is no disagreement here that two "coordinate systems" (your terminology) are necessary to define the quantities we are interested in. In light of that, I believe it is important not to conflate coordinate systems, reference frames, and basis vectors. A reference frame is merely a set of points whose position relative to each other is fixed. A reference frame is kinematically indistinguishable from a rigid body. It has no inherent origin, coordinate axes, or basis vectors. A set of basis vectors can be fixed arbitrarily to a reference frame or rigid body, describing a set of directions in which orientations, velocities, and accelerations can be expressed. Finally, linear position requires a coordinate system or a set of coordinate axes, which requires a set of basis vectors and an origin, an arbitrary point fixed in the reference frame. A given reference frame can have any number of sets of basis vectors and coordinate systems attached to it, none of which is "preferred" or inherent. MarcusMaximus (talk) 08:18, 28 August 2008 (UTC)

I also have a problem with the section of the article containing all those equations. It is only asking for trouble to use vectors algebraically and to tie them to a specific basis. Vectors are vectors independent of their basis, and two vectors can be added, subtracted, cross-multiplied, dot-multiplied, or magnituded (if that's a word) algebraically without concerning oneself about the basis.

For this reason it is unnecessary to introduce rotation matrices (the A(t)'s) into the general equations that define the fundamental relations of position, velocity, and acceleration of a point on a rigid body. It only confuses the issue by making people ignorant of the fact that vectors often are expressed in different bases. Putting the rotation matrix into the general equation creates a very restrictive definition, and leaves the reader unprepared to handle any situation in which not all of his vectors are available in the desired basis. A huge amount of vector analysis can be performed correctly while completely ignorant of bases, and this type of analysis is much simpler than it would be if one had to drag multiple direction cosines through all the equations. The matrices are clumsy to manipulate relative to the vectors.

The fundamental equations should stand on their own, basis-less. The concept of vector bases should be introduced separately, to ingrain the concept that any vector can be expressed in any basis by the appropriate application of rotation matrices. Vectors can even be expressed in bases that are not fixed to the reference frame the vector is associated with. Once the vector analysis is done, the necessary input vectors can easily be expressed in the correct bases via simple relations ($\scriptstyle {x_{b}=_{b}R_{a}*x_{a}}$, where x is a 3x1 column matrix containing the measure numbers of the vector x in each respective subscripted basis, and R is the direction cosine matrix expressing the b basis vectors in terms of a) before doing the last plug-and-chug steps. MarcusMaximus (talk) 08:42, 28 August 2008 (UTC)

Point 1 - Linear and angular position

I believe that we need to introduce the terminology used in physics books. I mean:

• linear position (or simply position) to indicate the position of a particle (center of mass, usually)
• angular position or orientation to indicate the orientation of a local basis set.

And take away the word displacement, as we already agreed. I do not agree that there is an urgent need to use a single word to say

• "linear and angular position" or
• "position and orientation".

Both expressions are correct, but too many people forget the first one. It is not too hard to use 3 or 4 words that have been used for centuries. And if I write "the position of this body is represented by a linear position vector and an orientation matrix" everybody should consider this terminology perfectly correct. I mean that too many people (mainly engineers) forget that the orientation is a position (notice that I am an engineer as well). Since we (I agree with you) don't like to define rigid body position using the concepts of (hypothetic) translation and rotation from reference position, I suggest another approach. See if you like my next edit, please. Paolo.dL (talk) 17:26, 28 August 2008 (UTC)

I do like your edit. There is not much I would change in it other than some personal wording preferences, which is not worth the effort. Next up, let's tackle those kinematic equations by getting rid of the direction cosine matrices from the equations and making them basis-independent, and then introducing the concept of bases separately. MarcusMaximus (talk) 08:32, 30 August 2008 (UTC)

OK, give me some time. I am busy right now, and I have to read again your previous comment, trying to understand exactly what you mean (notice that there was a discussion about the distinction between reference frames (RF) and coordinate systems (CS) in Talk:Frame of reference). Paolo.dL (talk) 09:31, 30 August 2008 (UTC)

Part 2 - Local coordinate system

MarcusMaximus, I have not read with attention your previous comment (I will), but I can post a short reflexion in the meantime. We don't actually need two coordinate systems (CS) to measure the linear and angular position of a rigid body. We need only one of them, that is the CS attached to the "observer", plus the position of at least three particles of the rigid object. The observer cannot measure without a CS.

However, typically we use the three or more selected particles to define three orthogonal and directed axes attached to the rigid body (represented by three versors, or basis vectors, if you like). And this is typically the first step of the algorithm to compute an orientation matrix, or an orientation vector, or a set of Euler angles. And, if these axes are also the axes of a CS, in which you know the (local) position of other important particles of the object (for instance, the tail and tip of the ariplane, the point of application of external forces such as engine thrust, etc.) then you can complete the task. Paolo.dL (talk) 10:03, 30 August 2008 (UTC)

I agree with the first paragraph of your comment above.

Perhaps I do not use the same definition of "axis" as you do, but in my experience an axis has a fixed physical location because it is required to pass through the origin. This is distinct from a basis vector, which has no spatial location, but only describes a direction. Therefore you don't need a set of axes to describe the relative orientation of two reference frames, only two sets of basis vectors, or one set of basis vectors and three points in the second frame as you said.

I read most of your long discussion with Brews Ohare over on the other page. I generally agree with him that a coordinate system and a reference frame are definitely not equivalent ideas.

Now I will give an example of why reference frames and basis vectors (and by extension, coordinate systems) are not equivalent. Imagine two reference frames (say inertial frame N and rigid body B), each with a set of three mutually perpendicular unit vectors fixed to it, N1, N2, N3, and B1, B2, B3, respectively. Suppose the mass center P of body B has nonzero velocity is both the B frame and the N frame.

The velocity of P can be expressed in two different reference frames (N or B), and in two different bases (N1, N2, N3, or B1, B2, B3). To be explicit, the velocity of P in B can be expressed in terms of N1, N2, N3, or B1, B2, B3, and the velocity of P in N can also be expressed in terms of N1, N2, N3, or B1, B2, B3, giving us four options for an expression of the velocity of P, all of which might be useful in analysis, and none of which required a set of coordinate axes.

One common example where it is necessary to express vectors in bases not attached to their native reference frames is calculating the inertial velocity of the center of mass P of a rocket $_{\text{N}}\mathbf {v} _{P}$ based on the inertial velocity measured at its IMU fixed in B at point Q. The equation is:

$_{\text{N}}\mathbf {v} _{P}{\text{ }}={\text{ }}_{\text{N}}\mathbf {v} _{Q}{\text{ }}+{\text{ }}_{\text{B}}\mathbf {v} _{P}{\text{ }}+{\text{ }}_{\text{N}}\mathbf {\omega } _{\text{B}}\times \mathbf {r} _{QP}$.

The inputs to this calculation are typically:

• $_{\text{N}}\mathbf {v} _{Q}$, the inertial velocity (in N) of the IMU expressed in a basis fixed to the inertial frame (N1, N2, N3),
• $_{\text{B}}\mathbf {v} _{P}$, the velocity of the mass center in the body frame (in B) expressed in the body fixed basis (B1, B2, B3),
• $_{\text{N}}\mathbf {\omega } _{\text{B}}$, the angular velocity of the rocket in inertial space (in N) expressed in a body-fixed basis (B1, B2, B3), and
• $\mathbf {r} _{QP}$, the position vector from the IMU to the mass center expressed in the body-fixed basis (B1, B2, B3).

There are several methods to carry out the calculations to the final answer, but all of them require expressing vectors in bases that are not fixed to the reference frame in which the vector is defined. In other words, merely to say "the velocity of P in N" fully describes a specific physical quantity, but it does not fully define the measure numbers of your vector, since that velocity can be expressed in terms of any arbitrarily selected set of basis vectors fixed in any reference frame, not necessarily in N. This should establish definitively that reference frames are not equivalent to vector bases or coordinate systems. MarcusMaximus (talk) 09:46, 2 September 2008 (UTC)

You wrote a lot of intersting statements. I will discuss them one at a time, as soon as I can. Paolo.dL (talk) 11:53, 2 September 2008 (UTC)

Basis set and CS. As far as I know, your distinction between axis and versor (basis vector) is correct and useful. Good point. Thank you for reminding me. That allows me to better understand your distinction between basis set and CS (08:18, 28 August 2008). Here are my two cents about this distinction:

• Two axes are enough to define a Cartesian CS. Indeed, their intersection is the origin, and the third axis can be easily found according to the right-hand or left-hand rule (depending on your choose of handedness).
• Any basis set is just a collection of numbers (a matrix if you like) which have no physical meaning. This is also true for any other vector. The tail of a vector is always supposed to be at point (0,0,0) (see vector (spatial)). Paolo.dL (talk) 12:08, 2 September 2008 (UTC)

Fixed and codirectional. When you, in your previous comments, refer to "basis vectors fixed to" a RF or rigid body, you give to basis vectors a physical meaning that they don't have. Only axes or CSs can be fixed to a RF or rigid body, because they are defined using a set of points fixed to the RF or rigid body, one of which is typically the origin (see talk:Frame of reference). On the contrary, basis vectors can only be codirectional with the axes of a local CS. Paolo.dL (talk) 12:09, 2 September 2008 (UTC)

I don't agree with your statements in Fixed and codirectional. Sorry for jumping from rockets to airplanes, but an airplane provides much more obvious physical features to use as references than a rocket does. If I define a basis vector B1 as "parallel to the chord line of the wing, positive forward", I have not given it a physical location based on any particular points on the body. I have only defined its direction and specified that it is fixed to the body. I can create a second basis vector B2 as "normal to the plane of symmetry and directed positive rightward". Define B3 as completing the right-handed set and it points "down" relative to the belly of the aircraft. These vectors don't have an origin and are not fixed in location; they are only fixed in direction. MarcusMaximus (talk) 19:26, 2 September 2008 (UTC)

After rereading your comments in Fixed and codirectional I think we are in agreement that basis vectors have no fixed location. I think we disagree in that I don't believe the direction of a basis vector must be linked to a coordinate axis. MarcusMaximus (talk) 19:38, 2 September 2008 (UTC)

"Fixed in direction" means nothing. Since the body is free to rotate, their direction is time-varying. My point is that you cannot use the expression "fixed to the rigid body". The ony way to describe the behaviour of those basis vectors, if you want them to represent the orientation of a rigid body, is to say that they are just codirectional with directed axes (or directed straight-line segments, i.e relative position vectors, i.e. directed distances from a point A to a point B) fixed to the body. I called these directed axes a local CS, but you are right: theoretically, they don't need to pass through an origin (common intersection), and don't need to have units to measure coordinates, so they don't have to be a CS. However, this is just splitting hair. I have never heard an engineer referring to "a basis set codirectional with a set of directed axes fixed to the airplane" (nor to "a basis set fixed to the airplane"). That's too long and unnecessarily complex. As far as I know, they just refer, much more simply, to "a CS fixed to the airplane", or "local CS". And this CS is also used to measure the time-invariant (local) position of points of interest such as the vertices of a mesh describing the shape of the external surface of the airplane. Paolo.dL (talk) 16:24, 4 September 2008 (UTC)

"Fixed in direction" is certainly not meaningless; in context it should be apparent that I meant "fixed in direction relative to the rigid body (the airplane)".

You said:

The ony way to describe the behaviour of those basis vectors, if you want them to represent the orientation of a rigid body, is to say that they are just codirectional with directed axes (or directed straight-line segments, i.e relative position vectors, i.e. directed distances from a point A to a point B) fixed to the body.

This appears to agree with my definition of basis vectors, because in your parenthetical statement you are saying that they need not be codirectional with axes, they only have to be based on geometric features of the body such as line segments connecting points fixed in the body. I would add that you can specify them as normal to a plane, among other definitions. These geometric features are not necessarily axes, and do not form a coordinate system.

The phrase "a basis set fixed to the airplane" is no longer or more complex than "a CS fixed to the airplane". It is true, engineers don't typically say "a basis set codirectional with a set of directed axes fixed to the airplane", but that's because it is unnecessary to specify directed axes. And even if you have never heard an engineer say "a basis set fixed to the airplane", I have. I use this terminology all the time. I can show you textbooks, give you the names of several professors and lecturers at Stanford and UC-Davis as well as many of my colleagues here at Lockheed Martin, who use this terminology and make the same distinction between basis vectors and CSs. It is not splitting hairs. A CS has a single, specific origin and it measures distance along its axes. Basis vectors are merely a set of reference directions with no origin. It is possible to describe an infinite number of Cartesian coordinate systems using a single set of basis vectors, where the basis vectors are based on geometric features of the body (not axes). There is plenty of analysis that can be performed using only basis vectors and points without bothering to define any coordinate axes. MarcusMaximus (talk) 02:11, 5 September 2008 (UTC)

Kinematical equations, basis-independent

I was bold with my editing; I couldn't stand those clunky kinematical equations the way they were before. I borrowed heavily from a textbook by Kane and Levinson called Online Dynamics, which is a self-guided computer course in dynamics. There is certainly more to be done, like perhaps to add mathematical definitions of angular velocity and angular acceleration, or to give examples of how to use the many abstract equations I have just added. I also want to improve the equation about "spatial acceleration" to make basis- and coordinate system-independent like the others, but it is not something I am familiar with. MarcusMaximus (talk) 08:34, 3 September 2008 (UTC)

What you are trying to do is useful, but the article's structure was sound before your edits. Now parts are missing (we wrote that position and velocity are vector quantities, we did not write this for accelerations; you inserted addition theorems: why only for angular velocity and linear position?) and the order of the new paragraphs is not optimized. Wouldn't it be better for the readers to remove our latest edits, put them in a separate temporary page, and lock the article until you manage to obtain a completely new and publishable version of the article? I mean that an article, if possible, should not appear as a "work in progress". If it has reached a sound structure, that structure should not be substituted with another, unless the new one is equally sound, or at least reasonably sound. Paolo.dL (talk) 17:46, 4 September 2008 (UTC)

Ok, starting a separate temporary article may be the proper way to do things until we get all the issues hammered out. I don't know how to do that, however, so I would appreciate it if you took care of the set up.

By redoing the kinematical formulas I have not actually removed any important content from the article, and I haven't really changed the structure. I replaced a paragraph followed by formulas with a presentation of basically the same information in an interlocking manner, with better explanation of each equation. The formulas for velocity of two points on a rigid body and acceleration of two points on a rigid body were in the article before, but in a less general form. There previously were no equations for one point moving on a rigid body. Notice that I didn't remove the formula based on screw theory, but I think it needs to be generalized as well.

I didn't include "addition theorems" for angular acceleration, linear velocity, or linear acceleration because:

• The general formula for the addition of angular accelerations does not take on the same simple form as the addition theorem for angular velocity; it contains cross products of the angular velocities. I can add it if you believe it would be appropriate, but it actually didn't occur to me to do so because it wasn't featured in the source I was using.
• The general formulas for the addition of linear velocities and accelerations of two different points are there on the page; they are the one point moving on a rigid body formulas. Since they do not resemble the addition theorem for angular velocity in form, it would not be appropriate to call them that, because they are not truly addition theorems. The velocity formula resembles it visually, but the superscripts are not analogous. The present titles are much more descriptive of the function of these equations.

And a final point: I disagree with your revert of my edit where I stated that "the velocity of a rigid body is undefined, it is only defined for points fixed to the body." You reverted it to state that the velocity of a rigid body is the velocity of any arbitrary reference point on the body. This doesn't make any sense, because all points on a rigid body generally have different velocity at the same time (except along the axis of rotation). To state "the velocity of body B" is meaningless without specifying the reference point, so truly the velocity is only defined for points. In colloquial language, this point is typically assumed to be the center of mass or the centroid, such as when referring to a car driving, a person running, or a ball flying through the air and spinning rapidly. This does not mean that it is valid to say that a rigid body as a velocity. MarcusMaximus (talk) 01:44, 5 September 2008 (UTC)

Merge with Rigid Body Dynamics?

It occurs to me that this page consists almost entirely of rigid body dynamics content. There is little on this page that is not rigid body dynamics-related; possibly the section on geometry. Shouldn't these two pages be merged, or at least move most of the content of this page to the other? MarcusMaximus (talk) 04:38, 11 September 2008 (UTC)

angular velocity

This article contradicts the following from the article on angular velocity: "In general, the angular velocity in an n-dimensional space is the time derivative of the angular displacement tensor which is a second rank skew-symmetric tensor". I'm assuming this quote is accurate, and this article is not. Can anyone verify this? 146.6.200.225 (talk) 20:44, 16 February 2010 (UTC)

Valid or invalid generalizations of the term "body"

An interesting discussion about terminology is ongoing on Talk:Orientation (geometry)#Physical objects, bodies, figures, segments, spaces, planes, lines. In short, we all agree that unbounded lines or planes (in general N-D spaces) can be validly described as rigid collections of points. Can they be also validly called bodies? In other words, does it make sense to speak of a "body" of infinite size?

IMPORTANT NOTE: If you are interested, please post your contribution there. Paolo.dL (talk) 11:55, 15 January 2011 (UTC)

New page: Rigid-body motion?

There was a page called rigid body motion which was a stub and now redirects here. There's also Euclidean group, which barely mentions SE(3). There is a page for rotation group, but none that get into the specific mathy details of SE(3) as a Lie algebra. I think that page needed. Does anyone mind if I start it back at rigid body motion with a DaB link here? —Ben FrantzDale (talk) 00:27, 25 May 2011 (UTC)

References?

Refs 4,5,6,7 are either no longer available online or incorrectly cited. Is this a print textbook, an online course, or something else entirely? My interest is partly selfish - it looks like a rather useful reference and I would love to find it :). If it's an online course or textbook, a link would be useful. Thanks. 128.32.240.58 (talk) 21:28, 7 June 2011 (UTC)AJ

I added those references. It is a textbook that is only available in print, despite its title. The same authors have published several fantastic textbooks that are available for free from Cornell University here: [1] The source I cited draws nearly all its material from those books. It would probably be a good idea for me to go through and change the citations. MarcusMaximus (talk) 05:56, 8 June 2011 (UTC)

Coordinate system, schmoordinate system

I wanted to revert all five of Paulo.dL's most recent edits, but realized that would be inflammatory. The overall effect of Your edits, Paulo, is to make the paragraphs you touched less general by introducing extraneous "coordinate systems" everywhere. The way it was written before was better. For example, you can use any arbitrary point in a rigid body for the purpose of defining position vectors, but now the article falsely claims that this point has to be the origin of some coordinate system.

One improvement you made was to remove references to the meaningless concept of angular position of points and particles.

We had a long discussion several years ago on this very talk page where you and I discussed a similar set of concerns. Please review it and let me know what you think about my objection. MarcusMaximus (talk) 20:41, 8 July 2011 (UTC)

I have reviewed our previous discussion. I perfectly understand your point, however I think you did not consider that defining the linear and angular position of a coordinate system (or, if you like, the linear position of a reference point, and the angular position of a basis set) is not enough to define the "placement in space" (or position, in general terms) of the whole rigid body. The latter is actually defined as the position of all the particles which the object is composed of (or at least the main points which define the shape of the object, for instance the vertices of a cube, or the points which approximately define the external surface of an airplane).

To reconstruct the position of these particles, a translation and a rotation are not enough. You also need the local position of all the particles, relative to a "local coordinate system". This is the reason why I believe that defining a local coordinate system is always required if you want to define the placement in space of the whole rigid body. In other words, I agree with you that a basis which rotates with the body is enough to define the orientation of planes or axes that are fixed to the body, but these planes or axes, even together with a reference point, are not enough to describe the whole body. To measure the local position of all other particles, you need a coordinate system. And only if you have local positions, you can transform them to the global coordinate system.

However, I do not have time to explain this in the article, so I partially reverted my edits. See if you like the new text.

Paolo.dL (talk) 17:02, 11 July 2011 (UTC)