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1. Glide plane, composite isometries
Hi, loving the isometries at the mo I must say Hopefully it is me being simple again.

Here's the question

Let be the transformation that reflects the plane in the -axis and let be the transformation that rotates the plane clockwise about the origin through a quarter of a turn. Algebraically these have the forms

Determine the composite isometries and , and interpret them geometrically.

Now I can work out how

But I cannot understand how

Don't understand how this bit works

not clicking
2. Re: Glide plane, composite isometries
Perhaps using different symbols might help:

q((a,b)) = (a,-b)

And just replace a with x, and b with -y.
3. Re: Glide plane, composite isometries
Well, if I understand it correctly, then

is just a sign flip of the "-coordinate.
4. Re: Glide plane, composite isometries
Hmm, but I thought that by doing a similar thing like I did with the first thing I would end up with q(y,-x)=(-x,-y), daft I know, but it doesn't click.
5. Re: Glide plane, composite isometries
(Original post by SubAtomic)
doing a similar thing like I did with the first thing
You'd have to elaborate on what you mean by that.

Another way to look at it is:

q simply changes the second coordinate to the negative of itself.

r flips the order of the two coordinates and then forms the negative of the new second coordinate.
6. Re: Glide plane, composite isometries
I think you're suffering from the irritatingly ambiguous terminology that people (including myself) use when referring to the x-axis and y-axis, whereby they still call them the 'x-axis' and 'y-axis' when is used to denote a point. So when you follow the image of a point through a succession of transformations, the 'x-axis' refers always to the 1st coordinate and not to the letter x, and the 'y-axis' refers always to the 2nd coordinate and not to the letter y.

So for instance and not , even though in the definition of it is the component labelled 'y' which changes sign. Likewise, , and , and so on.
Last edited by nuodai; 21-07-2012 at 16:10.
7. Re: Glide plane, composite isometries
(Original post by ghostwalker)
...

(Original post by nuodai)
...
Yep, that cleared it up, think my book is also a bit ambiguous at times, tells me to make sure I always do something a certain way then the very next page it completely blows the 'always do something a certain way' proposition out the water. Lol and not lol at the same time.

Could visualise it just couldn't work out what was going on with the algebra.

Cheers.
8. Re: Glide plane, composite isometries
(Original post by SubAtomic)
Yep, that cleared it up, think my book is also a bit ambiguous at times, tells me to make sure I always do something a certain way then the very next page it completely blows the 'always do something a certain way' proposition out the water. Lol and not lol at the same time.

Could visualise it just couldn't work out what was going on with the algebra.
I can see why you'd be confused: you're taught that if is a function then is what you get when you substitute in place of each occurrence of in the expression for .

The key here is that is defined geometrically: that is, it's defined in terms of the basis vectors of the space (i.e. the coordinates themselves) rather than the labelling of the coordinates. You can get around this by considering it as a transformation of vectors instead. If are unit vectors in the direction of the x- and y-axes, respectively, then you can say that and . Then since it becomes clearer what happens, namely . Likewise and , and so on.
Last edited by nuodai; 21-07-2012 at 16:33.
9. Re: Glide plane, composite isometries
(Original post by nuodai)
I can see why you'd be confused: you're taught that if is a function then is what you get when you substitute in place of each occurrence of in the expression for .

The key here is that is defined geometrically: that is, it's defined in terms of the basis vectors of the space (i.e. the coordinates themselves) rather than the labelling of the coordinates. You can get around this by considering it as a transformation of vectors instead. If are unit vectors in the direction of the x- and y-axes, respectively, then you can say that and . Then since it becomes clearer what happens, namely . Likewise and , and so on.

Thanks, much appreciated for that different take on things. This is one of only two questions in the book on composite isometries, takes some getting used to that's for sure.

All the best