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Math Concept Help Needed

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MrDiamondMind
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17 Dec 2010, 3:24 pm

I would love it if someone could explain to me what an affine transformation is. I think I know what it is, but am still greatly uncertain. I’m trying to use it as a creative metaphor for something I am writing, and I really don’t want to screw it up.
So, it would be very nice if someone could, in human laymen language, explain to this non-mathematically-minded person this concept.
Thanks a lot!



Chronos
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17 Dec 2010, 4:53 pm

MrDiamondMind wrote:
I would love it if someone could explain to me what an affine transformation is. I think I know what it is, but am still greatly uncertain. I’m trying to use it as a creative metaphor for something I am writing, and I really don’t want to screw it up.
So, it would be very nice if someone could, in human laymen language, explain to this non-mathematically-minded person this concept.
Thanks a lot!


An affine transformation is a transformation in which all points remain co-linear

http://mathworld.wolfram.com/AffineTransformation.html



Fuzzy
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17 Dec 2010, 5:41 pm

The participants in water ballet exhibit affine transformations of a sort, correct? While their independent positions are different, The transformations (translate, rotate) they make follow the same rule.


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ruveyn
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17 Dec 2010, 6:00 pm

MrDiamondMind wrote:
I would love it if someone could explain to me what an affine transformation is. I think I know what it is, but am still greatly uncertain. I’m trying to use it as a creative metaphor for something I am writing, and I really don’t want to screw it up.
So, it would be very nice if someone could, in human laymen language, explain to this non-mathematically-minded person this concept.
Thanks a lot!


It is a linear transformation followed by a translation.

Let A be a matrix and B be a vector.

the following is an affine transformation:

Y = AX + B where Y and X are vectors.

An affine transformation (or mapping) preserves collinearity of points and preserves the ratio of distances between points on a straight line.

ruveyn



MrDiamondMind
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17 Dec 2010, 6:01 pm

Hmm, Can someone perhaps offer me an example of something that has underwent an affine transformation from it’s previous state? And what exactly happened in that transformation. Thanks.



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18 Dec 2010, 1:14 am

The wolfram site that Chronos linked has good pictorial examples. All the following count as affine:

dilation:
Image

shear:
Image

reflection:
Image

things like rotation or translation.

The key is trying to visualize combinations of the above transformations, one after another, any combination will also be affine. Any basic shape (triangle, square, polygon) will keep the same number of sides, sides that use to be parallel have to remain parallel, yet all angles can change. Curved lines remain lines as well, yet are less obvious to visualize.



MrDiamondMind
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18 Dec 2010, 2:59 pm

Okay, thanks huntedman for the examples. The concept is not that difficult to grasp afterall. The tricky part is using this mathematically precise concept as an example for something that is currently vaguely understood (which is how I want to use it).
Also, thanks to Chronos for the link, which I was too lazy to use to find the visual examples. :P And to Fuzzy and ruveyn for their additional comments.