Talk:Active and passive transformation

Covariance and contravariance

Is this an accurate statement?: "Under active transformation, the components of vectors transform covariantly when the same basis is used. Under passive transformation, the basis transforms covariantly and so the components of vectors transform contravariantly so the vectors remain unchanged." —Ben FrantzDale 13:55, 11 May 2007 (UTC)

It seems that a mention (if not a brief discussion) of covariance and contravariance would fit in very well in this article. I don't understand them well enough to do so.All Clues Key (talk) 05:39, 3 September 2012 (UTC)

Alias and alibi transformations

Mathworld's article on Transformation uses "alias" for passive and "alibi" for active. Our own article on Rotation matrix also uses those same terms. Which terminology should Wikipedia standardise on? -- Cheers, Steelpillow (Talk) 15:44, 21 March 2010 (UTC)

I believe that active and passive transformation is much easier to interpret and remember. Alias and alibi are uncommon words, that I cannot easily remember. The origin of this weird terminology (that was apparently used in Berkley in the 50's) is in my opinion difficult to grasp, which makes its meaning difficult to remember. See AliasAlibi on MathPhys. Paolo.dL (talk) 08:57, 5 May 2013 (UTC)

Robotics

The current boom in robotics recalls many of the developments in algebraic kinematics in the 19th and 20th centuries. Many terms were developed and concepts of motion described. Active transformations were called alibi transformations, connoting a change of place. Passive transformations were called alias transformations, connoting a change of name. But in robotics one must remember that there are frames of reference for laboratory as well as the robot, and the use of referential terms like active and passive soon lose clarity. In geometry, the notion of an abstract motion that generates congruence became useful. Are there contributors to the project here that can build this article into an enduring reference? It looks like a job for an army.Rgdboer (talk) 19:37, 15 October 2010 (UTC)

Reqdiagram tag removed

Reqdiagram tag added in 2007, lead diagram in article added in 2009; Reqdiagram appears satisfied. Egmason (talk) 07:58, 25 January 2011 (UTC)

Notation??

Meaning of the 'a','b' and 'c' sub- and super-scripts in the Passive Transformation section are unclear, at least to me. Do they refer to different bases? Do they refer to dimensions (as in the Example section)? How does a superscript differ from a subscript? In general, notations in various Wikipedia articles are often inconsistent (unsurprisingly), so it would be kind of the author to explain conventions beyond the most basic and universal ones learned in High School or early College. Gkoulomz (talk) 16:53, 18 January 2013 (UTC)

Confusing and ambiguous terminology

The terminology active/passive alias/alibi is confusing and ambiguous. Please rename the article, or make a new one with correct title (location/base oriented observation), and link the new one from all other pages citing it. A new version of the alibi/alias or active/passive transformation page should first warn about this ambiguity, and then describe it. What is not: basis-oriented observer / location oriented observer. And this is not about physics, this is mathematics and ambiguity resolution. What transformation do you apply to the intelligent viewer (over time) One fixed to the basis, or one fixed to all other location to be represented in the basis reasoning: position vectors are contravariant, coordinate bases are covariant. The same tensor (matrix) must be used for both! Because coordinate bases need to be packed into an 1-D tensor (vector) of different covariance, so in a transposed shape vector those contain spatial location coordinates. The order of multiplying them by the same tensor of transformation will be different, of curse.

In the following we assume that column vectors are used for measured spatial coordinates (storing contravariant quantities)

   and THEREFORE row vectors contain base vectors of coordinate frames (and all covariant quantities, as lines in P^2->E^3 representation)

A consequence of this is, that if you measure a coordinate base A in a 3rd coordinate frame C, than you might substitute bases in your covariant row vector by 3 coordinate contravariant column vectors, the coordinates of Frame A's bases in FrameC. Then you get a 3x3 matrix, which is either an algebraic representation of coordinate frame A measured in coordinate frame C However, the same 3x3 matrix can be interpreted as one transforming vectors measured in frameA to frameC. Therefore, it can be written as FrameA=FrameC*T_CA, point_Pi_in_C=T_CA*point_Pi_in_A, FrameB=FrameC*T_CA*T_AB, point_Pi_in_C=T_CA*T_AB*point_Pi_in_B, therefore T_CB=T_CA*T_AB. Using column vectors for storing spatial coordinates (and not coordinate bases), interpreting a transformation chain T_0_6=T_0_1* ... *T_4_5*T_5_6 from the left to right corresponds is some motion of a coordinate frame (consider vehicle). From the right to left is to determine the coordinates of an external statig objects center in older and older positions of the vehicle. Shortly: left to right: transformation of frames, right to left: transformation of measured external coordinates.

So in this physical aspect, there is no question what is active motion, and what is writing the same entity in different frames. BUT if you consider, that the relation point_in_D=T_ED*point_in_E to describe the location of an external point in a standing reference, then this describes a physically active motion.

Conclusion: Calling the transformations corresponding to frames as alibi/alias or worse: active/passive is misleading, since it depends if the frames used for measuring points are looked as being static (and the external points moving) or moving and the external points being static. So labels active/passive exchange in the same situation depending on the aspect of the intellectual viewer: is it a frame-locked-interpretation or measured location-locked-interpretation? Suggestion: basis oriented/location oriented perspective OR basis perspective/location perspective

If the above is neglected, and coordinate bases of frame A measured in frame C are collected as being spatial = contravariant vectors, then a lot of confusion occurs. But it is a violation of co-/contra variant rules.

REMARK: it is not necessary to explain co-/contravariance. It is enough to explain the rule: A spatial, geometric location is the scalar product of a vector containing [i,j,k] bases denoting a spatial coordinate frame, and a vector containing [p_i,p_j,p_k] coordinates of the given location measured in the same [i,j,k] frame. THEREFORE, one has to be row vector, the other column vector, and this should not be messed up!

for the figure using pictures: remark on active rotation of pixel/vector 2D graphics...

 transforming pixel images: (active motion on the screen's standard coordinate frame)
     if transformer scans output and transforms output coordinates, where to read input, 

so we have to feed in a rotation matrix R: from=output, to=input

 transforming vectorgraphics: (active motion on the screen's standard coordinate frame) 

we have to feed in a rotation matrix R: from=input, to=output

so it is clear that inverse transformations has to be used respect to each other.— Preceding unsigned comment added by 91.82.168.207 (talk) 13:31, 14 August 2021 (UTC)

Picture of "alias" vs "alibi" transformations

@Davidvandebunte, thanks for making your svg image a bit simpler, that's helpful; now the SVG is in my opinion more or less equivalent to the previous PNG. There are still some features that I think are unnecessary. I don't think we need or particularly benefit from concrete coordinates or concrete angle measures to convey the message here. I think it would be more helpful to just have a variable name for the initial angle (say ⁠$\alpha$⁠) and the rotation angle (say ⁠$\beta$⁠), then we could write sums or differences of angles as ⁠$\alpha +\beta$⁠ or ⁠$\alpha -\beta$⁠ (or substitute whatever other greek letters).

I also wonder if we might get a more easily comprehensible result if we used something more concrete than an arrow as the object of rotation and a slightly more obvious visible representation of a coordinate system than just a pair of arrows at right angles.

For example, we could draw a little clipart top-view of an airplane at the origin of a square grid aligned with it (ideally a relatively low contrast grid to not be too distracting; the axes could still be emphasized), with our angle showing something like the angle of a wing. Then in the "after" picture, showing what happens when the plane turns, we can imagine either rotating the whole coordinate system to keep plane-relative coordinates so that the wing stays at the same relative angle, or just rotating the plane and leaving the coordinates fixed, relative to which the wing’s angle changes.

Or maybe there’s some even simpler to draw concrete example. –jacobolus (t) 18:54, 3 August 2024 (UTC)

Both suggestions seem OK. I wasn't quite as excited about the suggestion to reduce the concrete angles to variables, since I saw this drawing (in contrast to the one of the castle on the same page) as a more numerical example. At this point the drawings are more similar. Still, I see your point about how it may not be obvious that the particular angles add, and making that suggested change also makes all the text more homogeneous (the drawing no longer relies on a Latex renderer). Anyone who is interested in a numerical example still has the blue numbers from the original drawing and could easily work out the angles (or they could look at the history of the SVG).

I did my best with an airplane drawing (a car would also have been a good object). I left the basis vectors so it's clear how they're being rotated. Let me know what you think. Davidvandebunte (talk) 22:20, 3 August 2024 (UTC)

I like your drawing style. This one looks great. –jacobolus (t) 22:32, 3 August 2024 (UTC)