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Affine Transformations October 18, 2011 10:25AM |
Registered: 9 years ago Posts: 72 Rating: 0 |

Re: Affine Transformations October 18, 2011 11:25AM |
Registered: 10 years ago Posts: 114 Rating: 0 |

Hmmm, interesting question... When I saw it, I thought I had the answer, and then just as quickly, I realized that I was using the term affine transformations, but with only a rough idea of what I thought its definition was...

Anyways, I haven't seen anything in the textbook regarding a definition, but after a quick search, I can across this short n sweet definition:

An affine transformation is any transformation that preserves collinearity (i.e., all points lying on a line initially still lie on a line after transformation) and ratios of distances (e.g., the midpoint of a line segment remains the midpoint after transformation). In this sense, affine indicates a special class of projective transformations that do not move any objects from the affine space R^3 to the plane at infinity or conversely. An affine transformation is also called an affinity.

It was taken from the page: http://mathworld.wolfram.com/AffineTransformation.html

Anyways, I haven't seen anything in the textbook regarding a definition, but after a quick search, I can across this short n sweet definition:

An affine transformation is any transformation that preserves collinearity (i.e., all points lying on a line initially still lie on a line after transformation) and ratios of distances (e.g., the midpoint of a line segment remains the midpoint after transformation). In this sense, affine indicates a special class of projective transformations that do not move any objects from the affine space R^3 to the plane at infinity or conversely. An affine transformation is also called an affinity.

It was taken from the page: http://mathworld.wolfram.com/AffineTransformation.html

Re: Affine Transformations October 18, 2011 12:03PM |
Registered: 10 years ago Posts: 3,496 Rating: 1 |

I think as far as our course goes there's one more point to add. Every vector in the old space will map to a vector in the new space. (Although in homogeneous coordinates that distinction falls away, doesn't it?).

Nice definition. The text book doesn't specifically note that the relative mappings "remain the same". (Midpoint is still midpoint, etc).

PRESERVES COLLINEARITY. Actually that does cover the vectors, doesn't it? It would automatically follow that "(p-q) is mapped to (p'-q')".

Need I mention again that an affine transformation is one that preserves collinearity? I suppose really it's redundant to mention that affine transformations preserve collinearity, though, so I won't mention the fact again, after all.

Nice definition. The text book doesn't specifically note that the relative mappings "remain the same". (Midpoint is still midpoint, etc).

PRESERVES COLLINEARITY. Actually that does cover the vectors, doesn't it? It would automatically follow that "(p-q) is mapped to (p'-q')".

Need I mention again that an affine transformation is one that preserves collinearity? I suppose really it's redundant to mention that affine transformations preserve collinearity, though, so I won't mention the fact again, after all.

Re: Affine Transformations October 19, 2011 08:08AM |
Registered: 10 years ago Posts: 114 Rating: 0 |

slow_eddy Wrote:

-------------------------------------------------------

> I think as far as our course goes there's one more

> point to add. Every vector in the old space will

> map to a vector in the new space.**(Although in
**

> homogeneous coordinates that distinction falls

> away, doesn't it?).

Is that last sentence correct? The reason homogeneous coordinates exist is to distinguish between points and vectors in 3 dimensions. Thus a vector in a 3 dimensional space, still remains a vector in homogeneous coordinates. The only difference is that in homogeneous coordinates, we can now see that it is a vector, and not a point...

Well, at least thats my understanding... So I do apologise if that doesn't make much sense, but I think I'm getting a little snowed under here somewhere...

-------------------------------------------------------

> I think as far as our course goes there's one more

> point to add. Every vector in the old space will

> map to a vector in the new space.

> homogeneous coordinates that distinction falls

> away, doesn't it?).

Is that last sentence correct? The reason homogeneous coordinates exist is to distinguish between points and vectors in 3 dimensions. Thus a vector in a 3 dimensional space, still remains a vector in homogeneous coordinates. The only difference is that in homogeneous coordinates, we can now see that it is a vector, and not a point...

Well, at least thats my understanding... So I do apologise if that doesn't make much sense, but I think I'm getting a little snowed under here somewhere...

Re: Affine Transformations October 19, 2011 04:56PM |
Registered: 10 years ago Posts: 3,496 Rating: 1 |

You're right. I think it's more to do with me expressing myself badly. I mean the distinction is no longer troublesome? (Actually now I forget what I meant, but I think it was something like that).

You want to map vectors to vectors. If your coordinates are homogeneous, that's just going to happen as a matter of course. No need to specify. Aargh. I think that's what I was on about when I went on in the next paragraph of whatever. Or maybe I was changing my mind. If you map p to p', q to q', then it follows that you've mapped p-q to p'-q'. So no need to specifically mention the vectors.

Anyway it's good that you're on your toes and "alert to the alerts". Thanks. I'll definitely mention that affine transformations are those that preserve collinearity if it's asked. And then I might just quickly add the little bit about vectors to vectors just to be on the safe side. Same question would've taken me hours if you hadn't shared like this.

You want to map vectors to vectors. If your coordinates are homogeneous, that's just going to happen as a matter of course. No need to specify. Aargh. I think that's what I was on about when I went on in the next paragraph of whatever. Or maybe I was changing my mind. If you map p to p', q to q', then it follows that you've mapped p-q to p'-q'. So no need to specifically mention the vectors.

Anyway it's good that you're on your toes and "alert to the alerts". Thanks. I'll definitely mention that affine transformations are those that preserve collinearity if it's asked. And then I might just quickly add the little bit about vectors to vectors just to be on the safe side. Same question would've taken me hours if you hadn't shared like this.

Re: Affine Transformations October 21, 2011 10:45AM |
Registered: 10 years ago Posts: 114 Rating: 0 |

Re: Affine Transformations October 21, 2011 12:47PM |
Registered: 10 years ago Posts: 3,496 Rating: 1 |

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