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Locus of the images of a point given by certain affine transformations

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$begingroup$

In the affine plane, let P, Q, R be three aligned points. Now let's consider all the affine transformations that keep R as a fixed point, transform P into Q, and have a unique invariant line. Which is the locus of the images of a point given by these affine transformations?

I have thought about considering a reference Ref $=R;vecRP,vec??$, with $vecRP$ eigenvector with eigenvalue $fracQ$, but I'm not sure this is the right way to do the exercise, as I don't know how to continue... Could you help me? Thanks in advance!

geometry affine-geometry locus

share|cite|improve this question

asked Mar 16 at 21:40

GibbsGibbs

137111

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  Yes, that is a good approach. You can translate $R$ to the origin and assume that the transformations are linear. The invariant line must be the line $PQ$, since the vector $RQ$ is an eigenvalue because it got sent to $RP$. Now pick a point $x$. We need to distinguish cases. If $x$ is in the line $PQ$, then the image is just another point $x'$ such that $xRx'$ are in the same proportion as $PRQ$. If $x$ is outside the line $PQ$, then $x'$ can be anywhere except on the lines $PQ$ and $Rx$.
  $endgroup$
  – user647486
  Mar 16 at 21:52
  
  

  
  
  
  


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  $begingroup$
  By “aligned” do you mean colinear?
  $endgroup$
  – amd
  Mar 16 at 21:56
  

  
  
  
  


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  @amd That is what aligned means.
  $endgroup$
  – user647486
  Mar 16 at 21:58
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @user647486 Your explanation is quite good! When you say "$x'$ can be anywhere excepte on the lines $PQ$ and $Rx$", how can we determine where does the image go? I suppose this will clarify which is the locus of the images...
  $endgroup$
  – Gibbs
  Mar 16 at 22:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That sentence that I said is that 'location'. Pick any of those points and there is a single affine (linear after translating $R$ to the origin) that sends $P$ to $Q$ and $x$ to $x'$, and which only has the line $PQ$ and invariant line.
  $endgroup$
  – user647486
  Mar 16 at 22:27
  
  

  
  
  
  

add a comment | 

0

$begingroup$

In the affine plane, let P, Q, R be three aligned points. Now let's consider all the affine transformations that keep R as a fixed point, transform P into Q, and have a unique invariant line. Which is the locus of the images of a point given by these affine transformations?

I have thought about considering a reference Ref $=R;vecRP,vec??$, with $vecRP$ eigenvector with eigenvalue $fracQ$, but I'm not sure this is the right way to do the exercise, as I don't know how to continue... Could you help me? Thanks in advance!

geometry affine-geometry locus

share|cite|improve this question

asked Mar 16 at 21:40

GibbsGibbs

137111

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, that is a good approach. You can translate $R$ to the origin and assume that the transformations are linear. The invariant line must be the line $PQ$, since the vector $RQ$ is an eigenvalue because it got sent to $RP$. Now pick a point $x$. We need to distinguish cases. If $x$ is in the line $PQ$, then the image is just another point $x'$ such that $xRx'$ are in the same proportion as $PRQ$. If $x$ is outside the line $PQ$, then $x'$ can be anywhere except on the lines $PQ$ and $Rx$.
  $endgroup$
  – user647486
  Mar 16 at 21:52
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  By “aligned” do you mean colinear?
  $endgroup$
  – amd
  Mar 16 at 21:56
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @amd That is what aligned means.
  $endgroup$
  – user647486
  Mar 16 at 21:58
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @user647486 Your explanation is quite good! When you say "$x'$ can be anywhere excepte on the lines $PQ$ and $Rx$", how can we determine where does the image go? I suppose this will clarify which is the locus of the images...
  $endgroup$
  – Gibbs
  Mar 16 at 22:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That sentence that I said is that 'location'. Pick any of those points and there is a single affine (linear after translating $R$ to the origin) that sends $P$ to $Q$ and $x$ to $x'$, and which only has the line $PQ$ and invariant line.
  $endgroup$
  – user647486
  Mar 16 at 22:27
  
  

  
  
  
  

add a comment | 

$begingroup$

In the affine plane, let P, Q, R be three aligned points. Now let's consider all the affine transformations that keep R as a fixed point, transform P into Q, and have a unique invariant line. Which is the locus of the images of a point given by these affine transformations?

I have thought about considering a reference Ref $=R;vecRP,vec??$, with $vecRP$ eigenvector with eigenvalue $fracQ$, but I'm not sure this is the right way to do the exercise, as I don't know how to continue... Could you help me? Thanks in advance!

geometry affine-geometry locus

share|cite|improve this question

asked Mar 16 at 21:40

GibbsGibbs

137111

$endgroup$

share|cite|improve this question

asked Mar 16 at 21:40

GibbsGibbs

137111

share|cite|improve this question

asked Mar 16 at 21:40

GibbsGibbs

137111

asked Mar 16 at 21:40

GibbsGibbs

137111

asked Mar 16 at 21:40

GibbsGibbs

137111