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VCE Stuff => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematics => Topic started by: pi on December 14, 2010, 04:19:27 pm

Title: Rotating graphs
Post by: pi on December 14, 2010, 04:19:27 pm: My GMA teacher touched on this in class when going over rotational matrices

ie. $\begin{bmatrix}<br /> cos(\theta) & -sin(\theta) \\<br /> sin(\theta) & cos(\theta)<br />\end{bmatrix}$

Any way to rotate a graph (maybe even a function?) using matrices, or is there another method?

Not part of any homework, just wondering if there is a way...
Title: Re: Rotating graphs
Post by: TrueTears on December 14, 2010, 04:22:18 pm: Do you really want to learn about this in detail? I would try to explain but starting from the basics until I get to this stuff but that would take pages of explanations since I don't know how much you know about linear algebra. If you want to see how these are derived or learn more about rotational matrices (or linear transformations of matrices in general) check out Elementary Linear Algebra by Anton and Rorres, I can send you the ebook if you want.
Title: Re: Rotating graphs
Post by: pi on December 14, 2010, 04:26:45 pm: I only know stuff from GMA and methods (yr11)...

Probably too complex for me :(
Title: Re: Rotating graphs
Post by: TrueTears on December 14, 2010, 04:28:09 pm: I suggest for now just memorise that rotational matrix and don't think about where it comes from, often in linear algebra you'll see that you need to know the HOW before the WHY, it is a branch of mathematics where if you get more confident in applying the techniques you will find the proofs later on much easier. Also there are alot of definitions and fundamental skills you should get clear before analyzing these rotational matrices. Again, the best way is to read the textbook I suggested :P
Title: Re: Rotating graphs
Post by: pi on December 14, 2010, 04:30:25 pm: Would it be too big for an e-mail?

I'll PM you my e-mail address (same one for MSN if it is too big for e-mail).

Thanks TT (and you need to get your stars back...)!
Title: Re: Rotating graphs
Post by: TrueTears on December 14, 2010, 04:33:12 pm: Actually that's right, too big for email, I shall upload it for you.

Done: http://ifile.it/245h9t6/text.rar

Have fun reading, you can start from the start but if you're just interested in the transformations, go to chapter 4 :)
Title: Re: Rotating graphs
Post by: pi on December 14, 2010, 07:58:11 pm: Thanks TT!

Edit: The book starts simple, which is awesome!
Title: Re: Rotating graphs
Post by: kamil9876 on December 14, 2010, 08:44:27 pm: I don't agree, you can learn the techniques and their proofs simultaneously without trouble and it makes it easier to understand. However you would only learn more concrete instead of abstract versions first (e.g: real number matrices only, no general vector spaces).

e.g: If you know the compound angle formula from trig then you can understand why the matrix is the way it is easily (only the more abstract, and imo quite elegant, approach with linear transformations would require more study beforehand as TT suggested)

Anyway back to the ORIGINAL question (note OP didn't even ask for a proof):

Quote
Any way to rotate a graph (maybe even a function?) using matrices, or is there another method?

Not part of any homework, just wondering if there is a way...

Too lazy to think of an example that isn't too trivial but also not too messy but I remember this one probably fits that description: http://vce.atarnotes.com/forum/index.php/topic,27705.0.html
Title: Re: Rotating graphs
Post by: brightsky on December 14, 2010, 09:31:52 pm: Disprove me if I'm wrong, but I guess since every graph is made up on points, then all you have to do is know how to rotate a certain given point, and then repeat the process for every point on the line (i.e. the equation) to get the full rotation of the graph. So grab a point (x, y). Let's say we want to rotate it counterclockwise $\theta$ degrees. Call this new point (a, b). Join points (0,0) and (a,b); and (0,0) and (x,y). The distance from (x,y) to (0,0) is $\sqrt{x^2 + y^2}$ and so is the distance between (a,b) and (0,0). Now the angle between (x,y) and the x-axis is $\arctan\left(\frac{y}{x}\right)$ and so the angle between (a,b) and the x-axis would be $\arctan\left(\frac{y}{x}\right) + \theta$ . Drop a vertical line down from (a,b) to the x-axis to form a triangle. Using trig, we know that $\sin\left(\arctan\left(\frac{y}{x}\right) + \theta\right) = \frac{b}{\sqrt{x^2 + y^2}}$ and so $b = \sin\left(\arctan\left(\frac{y}{x}\right) + \theta\right)\sqrt{x^2 + y^2} = \boxed{ x\sin(\theta) + y\cos(\theta)}$ seeing that they are all positive. Similarly, $\cos\left(\arctan\left(\frac{y}{x}\right) + \theta\right) = \frac{a}{\sqrt{x^2 + y^2}}$ so $a = \cos\left(\arctan\left(\frac{y}{x}\right) + \theta\right) \sqrt{x^2 + y^2} = \boxed{x\cos(\theta) - y\sin(\theta) }$ . Hence after rotation, we have the point $\left(x\cos(\theta) - y\sin(\theta) , x\sin(\theta) + y\cos(\theta) \right)$ , which is precisely the matrix formula. Again, I'm not sure of the veritability of this 'proof'.
Title: Re: Rotating graphs
Post by: kamil9876 on December 14, 2010, 09:39:52 pm: Yeah that's pretty much the idea in the proof of what I was referring to except that I forgot to mention that it is more convenient to express $x,y$ in terms of their polar co-ordinates and then just use compound angle formula.

edit: ok let me do that explicity. Suppose that $x=rcos\phi$ , $y=rsin\phi$ , now after rotating by $\theta$ we get:

$x'=rcos(\theta+\phi), y'=rsin(\theta+\phi)$

now just use compound angle formulas and you get the matrix formula.
Title: Re: Rotating graphs
Post by: brightsky on December 14, 2010, 09:40:39 pm: This might be easier to understand and is more general (I don't think my way works for all cases). http://myyn.org/m/article/derivation-of-rotation-matrix-using-polar-coordinates/

Quote from: kamil9876 on December 14, 2010, 09:39:52 pm
Yeah that's pretty much the idea in the proof of what I was referring to except that I forgot to mention that it is more convenient to express $x,y$ in terms of their polar co-ordinates and then just use compound angle formula.

Indeed. :p
Title: Re: Rotating graphs
Post by: kamil9876 on December 14, 2010, 09:47:40 pm: Though I like your approach, it suggests another way to prove the compound angle formula. (i've said this a few times, it is quite cool that when you find two ways(say A and B) of deriving something, that usually gives you some cool new proof of B using A).
Title: Re: Rotating graphs
Post by: pi on January 06, 2011, 09:38:40 pm: Rotating graphs is actually pretty fun and worth the working! Just start rotating a random graph when getting bored during methods... (maths is fun). Thanks TT (the book is handy too!), kamil9876 and brightsky.

EDIT: Is there any easy way to rotate a graph clockwise?
Title: Re: Rotating graphs
Post by: /0 on January 07, 2011, 07:42:25 am: make $\theta$ negative

i.e. the clockwise rotation matrix for an angle $\phi$ is $\left[\begin{matrix} \cos\phi & \sin\phi \\ -\sin\phi & \cos\phi\end{matrix}\right]$ .

As a sidenote that was drilled into me from group theory, the rotation matrices through angles of $\frac{2\pi}{n}$ along with the reflection matrix $\left[\begin{matrix} 1 & 0 \\ 0 & -1\end{matrix}\right]$ form the symmetry group of regular n-gons.
Title: Re: Rotating graphs
Post by: QuantumJG on January 07, 2011, 09:31:44 am: Quote from: /0 on January 07, 2011, 07:42:25 am
make $\theta$ negative

i.e. the clockwise rotation matrix for an angle $\phi$ is $\left[\begin{matrix} \cos\phi & \sin\phi \\ -\sin\phi & \cos\phi\end{matrix}\right]$ .

As a sidenote that was drilled into me from group theory, the rotation matrices through angles of $\frac{2\pi}{n}$ along with the reflection matrix $\left[\begin{matrix} 1 & 0 \\ 0 & -1\end{matrix}\right]$ form the symmetry group of regular n-gons.

Hey /0 in group theory last year I had a lot of trouble with group actions and was wondering if you could recommend a textbook on group theory.

@Rohitpi: If this kind of maths interests you, may I recommend you also consider a pure maths degree.
Title: Re: Rotating graphs
Post by: /0 on January 07, 2011, 11:53:40 am: Quote from: QuantumJG on January 07, 2011, 09:31:44 am
Quote from: /0 on January 07, 2011, 07:42:25 am
make $\theta$ negative

i.e. the clockwise rotation matrix for an angle $\phi$ is $\left[\begin{matrix} \cos\phi & \sin\phi \\ -\sin\phi & \cos\phi\end{matrix}\right]$ .

As a sidenote that was drilled into me from group theory, the rotation matrices through angles of $\frac{2\pi}{n}$ along with the reflection matrix $\left[\begin{matrix} 1 & 0 \\ 0 & -1\end{matrix}\right]$ form the symmetry group of regular n-gons.

Hey /0 in group theory last year I had a lot of trouble with group actions and was wondering if you could recommend a textbook on group theory.

@Rohitpi: If this kind of maths interests you, may I recommend you also consider a pure maths degree.

The textbook we used was 'Algebra' by Michael Artin. I thought it was really good and if you want it just pm me. Last year we covered chapters 2,5,6 and 10, but there's a lot more advanced material as well. Apparently a new version was recently published too.

A lecturer also told me that for Algebra 2 this year we might be using the notes here http://jmilne.org/math/CourseNotes/index.html

Just out of curiousity, which text did you use?
Title: Re: Rotating graphs
Post by: QuantumJG on January 07, 2011, 01:26:54 pm: Quote from: /0 on January 07, 2011, 11:53:40 am
Quote from: QuantumJG on January 07, 2011, 09:31:44 am
Quote from: /0 on January 07, 2011, 07:42:25 am
make $\theta$ negative

i.e. the clockwise rotation matrix for an angle $\phi$ is $\left[\begin{matrix} \cos\phi & \sin\phi \\ -\sin\phi & \cos\phi\end{matrix}\right]$ .

As a sidenote that was drilled into me from group theory, the rotation matrices through angles of $\frac{2\pi}{n}$ along with the reflection matrix $\left[\begin{matrix} 1 & 0 \\ 0 & -1\end{matrix}\right]$ form the symmetry group of regular n-gons.

Hey /0 in group theory last year I had a lot of trouble with group actions and was wondering if you could recommend a textbook on group theory.

@Rohitpi: If this kind of maths interests you, may I recommend you also consider a pure maths degree.

The textbook we used was 'Algebra' by Michael Artin. I thought it was really good and if you want it just pm me. Last year we covered chapters 2,5,6 and 10, but there's a lot more advanced material as well. Apparently a new version was recently published too.

A lecturer also told me that for Algebra 2 this year we might be using the notes here http://jmilne.org/math/CourseNotes/index.html

Just out of curiousity, which text did you use?

I would love that book! As for texts we only used bound lecture notes. It was brilliant in one sense but sometimes lacked comprehension in major topics.

I'm inbetween picking Algebra (carries on from Group Theory) or Numerical & Symbolic Maths (A programming subject for Maths students).
Title: Re: Rotating graphs
Post by: pi on January 07, 2011, 02:58:40 pm: Quote from: QuantumJG on January 07, 2011, 09:31:44 am
@Rohitpi: If this kind of maths interests you, may I recommend you also consider a pure maths degree.

Just out of curiosity, what career paths lead from doing maths in uni (other than teaching/lecturing)?
Title: Re: Rotating graphs
Post by: TrueTears on January 07, 2011, 03:16:28 pm: Quote from: Rohitpi on January 07, 2011, 02:58:40 pm
Quote from: QuantumJG on January 07, 2011, 09:31:44 am
@Rohitpi: If this kind of maths interests you, may I recommend you also consider a pure maths degree.

Just out of curiosity, what career paths lead from doing maths in uni (other than teaching/lecturing)?
Could lead to some highly paid commerce degrees, but it'd be better to do those degrees straightforward.
Title: Re: Rotating graphs
Post by: Gloamglozer on January 07, 2011, 04:19:30 pm: Quote from: TrueTears on January 07, 2011, 03:16:28 pm
Quote from: Rohitpi on January 07, 2011, 02:58:40 pm
Quote from: QuantumJG on January 07, 2011, 09:31:44 am
@Rohitpi: If this kind of maths interests you, may I recommend you also consider a pure maths degree.

Just out of curiosity, what career paths lead from doing maths in uni (other than teaching/lecturing)?
Could lead to some highly paid commerce degrees, but it'd be better to do those degrees straightforward.

It could lead to commerce-related jobs but you'd have to really prove yourself and show the prospective employer that they should hire you and not a person with a commerce degree.
Title: Re: Rotating graphs
Post by: pi on January 07, 2011, 04:39:55 pm: Hmmm... Thanks for the info guys. I'm pretty interested in that sort of maths, but I really don't think commerce is my thing (I'm a more of a mathsy/sciency person).

Just looking at TT's sig, what is 'Actuarial'? (googling it after I post though)? All I've heard is that it has lots of stats, has a high fail rate at uni, and as a career is pays lots (money wise)...
Title: Re: Rotating graphs
Post by: /0 on January 07, 2011, 07:34:48 pm: Artin
Title: Re: Rotating graphs
Post by: TrueTears on January 09, 2011, 07:53:13 pm: Quote from: Rohitpi on January 07, 2011, 04:39:55 pm
Hmmm... Thanks for the info guys. I'm pretty interested in that sort of maths, but I really don't think commerce is my thing (I'm a more of a mathsy/sciency person).

Just looking at TT's sig, what is 'Actuarial'? (googling it after I post though)? All I've heard is that it has lots of stats, has a high fail rate at uni, and as a career is pays lots (money wise)...
I was going to do just pure maths however commerce has better job prospects which also includes alot of fun mathematics and has higher paid jobs than just doing pure maths.

Also Actuarial studies contains alot of mathematics, lots of probability, but a very high paid job.