Concise Representations for Plane Rotations - And Generalizing Their Applications

In his excellent paper, ‘A Missing Piece: Early Elementary Plane Rotations’,. Notices of the American Mathematical Society, Vol. 61, No. 2) Bob Palais uses a simple plane geometry of the unit circle to arrive at very concise representations of plane rotation, including in the complex plane. (For example, a quarter turn rotation behaves as the square root of (-1) or the complex number, i)

The unfamiliarity with the approach is because most students are still thinking in terms of rotations of lines, as opposed to planes. This is why the diagram as shown is so useful. One begins then with the plane defined in the position: [(0,0), (x, 0), (x,y), (0,y)] which is to be rotated into the oblique position as shown. As Palais observes, “if the image of (1,0) is (X,Y) then the plane rotation formula derived expresses the image of any point (x, y) as the multiple x of:

(X,Y) plus the multiple y of (-Y, X) or the image of (0,1).

Readers can check this is so by careful inspection of the diagram and following the arc from (1,0) counter-clockwise to the end of the arrow at (X,Y). In so doing they will also see that:

(x,0) or x is now identified at (xX, xY)

(x,y) is now at: (xX – yX, xY + yX)

(0,y) or y is now at (-yY, yX) and

(0,0)remains invariant (center of the plane rotation)

The image of (0,1) then is obtained from*:

x [X,Y] + y [-Y,X] = [xX- yX, yX + xY]

And it’s important to note that the operations of addition and scaling call for vector notation (brackets, or [ ])* instead of simple point notation ( ,  ).  As Palais notes:

“The key to the plane rotation formula is the observation that when you turn your head 90 degrees clockwise what used to look like (X, Y) now looks like (-Y, X). Applying this twice, the math maven should be able to show that the image of [-Y, X] is [-X, -Y] )so a quarter turn rotation behaves as i)

Palais notes that this notation beautifully interweaves the meanings of complex products, dot and cross products, and circular addition formulas.

Consider, for example:

[x,y] =  (cos s, sin s), [X,Y] = [cos t, sin t]

In which case the original rotation formula becomes:

[cos (s + t), sin (s + t)] =

 [cos s cos t – sin s sin t, sin s cos t + cos s sin t]

As he observes, “this relates two circular addition formulas that are usually treated separately”

It is also much more economical than standard derivations (say, in college algebra texts which usually require two pages of algebraic manipulation to arrive at the cosine subtraction formula by itself. )

Palais most notable observation and point probably is:

“Any two rectangular coordinate systems may be related by a combination of shift of origin, scaling rotation and reflection, corresponding to complex addition, multiplication and conjugation, respectively.”

He goes on to write that “it seems a shame for our students to be deprived of the one missing transformation – rotation – when providing access to it is so elementary and permits so much utility and insight.”

This may be so but when I taught calculus physics and astronomy  I often found that circular motion gave the most grief to students, and that included visualizing rotation even in two dimensions! When one teaches simple harmonic motion, for example, one generally employs the unit circle via projection to illustrate how the oscillating object appears to behave.

http://www.animations.physics.unsw.edu.au//jw/SHM.htm

This approach can find further application in calculus physics, for example, by noting that the quarter turn rotation formula, [-y, x] expresses the physics of harmonic motion in Hooke’s law, e.g. F = -kx as well as uniform circular motion. In that case the velocity v = dx/dt is perpendicular to displacement so: [x, y]’ = [-y, x].  Acceleration would then be [x,y]’ applied again or [x, y]” = [-x, -y]. (From the differential equations perspective, [-y, x] denotes a change or variables while the linear combination:

X [x,y] + Y[-y, x] would be a linear combination of solutions

See e.g.

http://brane-space.blogspot.com/2013/06/solution-of-linear-homogeneous.html

Problem for Math Mavens:

Palais claims that the basic plane rotational formula, e.g.

x [X,Y] + y [-Y,X] = [xX- yX, yX + xY]

is “also the origin of the Cauchy –Riemann equations that characterize an analytic function of a complex variable”

That is:

¶ u/ ¶ x  = ¶ v/ ¶ y   And:    ¶ v/ ¶ x  = - ¶ u/ ¶ y

Can you show that this is so? 

 For reference, see:

http://brane-space.blogspot.com/2013/10/the-cauchy-riemann-equations-gateway-to.html

and:

http://brane-space.blogspot.com/2013/10/looking-at-harmonic-conjugates-and.html

(* Notation change note: Because the blogger field will not accept arrow brackets, 'square' ones, [],  have been substituted.)