|Feb16-11, 08:29 PM||#1|
This isn't a homework question, but I'm having trouble understanding something about rotations conceptually. While reading about the Euler Identity online, I keep running into a few things that I can't wrap my head around and never come with an explanation.
Here are the concepts I can't understand:
A half-turn is the square of a quarter turn.
Rotations cannot be added, only multiplied.
Why is this? Intuitively, if I'm thinking about rotating by a quarter turn, then another quarter turn, I add them together to get a half turn. Instead, it appears I have to multiply the quarter turn by itself. I don't understand this.
pi equals 2(pi/2), not (pi^2)/4
Basically, I keep running across descriptions of rotations as exponential growth, whereas all I can see when looking at them is linear growth. How are rotations exponential?
|Feb16-11, 08:42 PM||#2|
Consider the rotation matrix R that does you're quarter turn.
In a nutshell what it is saying is that your half turn is represented by
R(R(X)) where X is your point assuming you are rotating the point about the origin.
Consider your quarter turn to be R x X = X(0)
Apply that same rotation R x X(0) applies the rotation to the result of the 1st rotation.
So essentially R^2(X) is applying the same rotation "twice" hence the half turn result.
If you want to prove this rigorously, create the rotation matrix R using the definition of your rotation and calculate R^2 and see what you get for each individual entry in your resultant matrix R^2.
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