Dtjytyk

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This word with a lot of past tenses

Name for ratio of three co-linear points in affine geometry

Ellipse: Name for the ratio $a/b$?Motion in affine geometryAffine geometry about parallelogramAffine set and linear equationAffine geometry textbookAffine hull of two points in R4Rotation of a hyperbola in affine geometryAffine geometry book for physicistResources studying affine geometry.Affine geometry and simply transitive action

0

$begingroup$

Given three points $a_1$, $a_2$, $a_3$ in an affine space such that $dim langle a_1, a_2, a_3rangle = 1$ (i.e. they are co-linear) and $a_1ne a_2$, my professor defined the "simple ratio" (making a literal translation) as the unique scalar
$$
(a_1a_2a_3) := lambda inmathbb{K} qquadtext{such that}qquad overrightarrow{a_1a_3}=lambda, overrightarrow{a_1a_2},.
$$

I haven't found any equivalent term in English sources. Is such a concept defined in other places, or does it happen to be "popular" where I live?

For example, this notation is used in "Álgebra Lineal i Geometria" by M. Castellet and I. Llerena.

notation terminology affine-geometry

share|cite|improve this question

edited Mar 11 at 9:23

Anakhand

asked Feb 16 at 18:04

AnakhandAnakhand

259114

$endgroup$

add a comment | 

0

$begingroup$

Given three points $a_1$, $a_2$, $a_3$ in an affine space such that $dim langle a_1, a_2, a_3rangle = 1$ (i.e. they are co-linear) and $a_1ne a_2$, my professor defined the "simple ratio" (making a literal translation) as the unique scalar
$$
(a_1a_2a_3) := lambda inmathbb{K} qquadtext{such that}qquad overrightarrow{a_1a_3}=lambda, overrightarrow{a_1a_2},.
$$

I haven't found any equivalent term in English sources. Is such a concept defined in other places, or does it happen to be "popular" where I live?

For example, this notation is used in "Álgebra Lineal i Geometria" by M. Castellet and I. Llerena.

notation terminology affine-geometry

share|cite|improve this question

edited Mar 11 at 9:23

Anakhand

asked Feb 16 at 18:04

AnakhandAnakhand

259114

$endgroup$

add a comment | 

$begingroup$

Given three points $a_1$, $a_2$, $a_3$ in an affine space such that $dim langle a_1, a_2, a_3rangle = 1$ (i.e. they are co-linear) and $a_1ne a_2$, my professor defined the "simple ratio" (making a literal translation) as the unique scalar
$$
(a_1a_2a_3) := lambda inmathbb{K} qquadtext{such that}qquad overrightarrow{a_1a_3}=lambda, overrightarrow{a_1a_2},.
$$

I haven't found any equivalent term in English sources. Is such a concept defined in other places, or does it happen to be "popular" where I live?

For example, this notation is used in "Álgebra Lineal i Geometria" by M. Castellet and I. Llerena.

notation terminology affine-geometry

share|cite|improve this question

edited Mar 11 at 9:23

Anakhand

asked Feb 16 at 18:04

AnakhandAnakhand

259114

$endgroup$

share|cite|improve this question

edited Mar 11 at 9:23

Anakhand

asked Feb 16 at 18:04

AnakhandAnakhand

259114

share|cite|improve this question

edited Mar 11 at 9:23

Anakhand

asked Feb 16 at 18:04

AnakhandAnakhand

259114

asked Feb 16 at 18:04

AnakhandAnakhand

259114

asked Feb 16 at 18:04

AnakhandAnakhand

259114

3 Answers
3

active

oldest

votes

2

$begingroup$

The Wikipedia Projective harmonic conjugate article uses the term "division ratio" which is rare in English unfortunately. More precisely, it states "C divides AB externally" and gives the ratio $AC:BC$ which corresponds to your $lambda$ and where $ABC$ are three points in order on a line. A third point can divide a line segment "internally" where the ratio $lambda$ is negative. Further it also states:

In terms of a double ratio, given points a and b on an affine line, the division ratio of a point x is $$ t(x)=frac {x-a}{x-b}.$$

It seems to me that the term "division ratio" is what you want, although "simple ratio" is simpler.

share|cite|improve this answer

edited Feb 17 at 14:49

answered Feb 17 at 5:57

SomosSomos

14.5k11336

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Please see my reference to Coxeter's book. I must admit that I am more familiar with Russian terminology, where the adjective "simple" is used instead.
  $endgroup$
  – GReyes
  Feb 17 at 9:14
  

  
  
  
  

add a comment | 

3

$begingroup$

I don't think I've ever seen this affine version named before, but I have seen the projective geometry concept that it's a case of: the cross-ratio
$$(a_4,a_1;a_2,a_3) = frac{a_1a_3cdot a_4a_2}{a_1a_2cdot a_4a_3}$$
If we choose $a_4$ to be the point at $infty$ on the line, then $frac{a_4a_3}{a_4a_2}=1$ and this becomes $frac{a_1a_3}{a_1a_2}=lambda$.

share|cite|improve this answer

answered Feb 16 at 19:36

jmerryjmerry

14.4k1630

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The cross ratio is an invariant in Projective Geometry (simple ratio is not invariant there, and you need a ratio of ratios). Observe that the cross ratio is a ratio of simple ratios.
  $endgroup$
  – GReyes
  Feb 16 at 19:37
  
  
  

  
  
  
  

add a comment | 

1

$begingroup$

Ratios between segments are preserved in Affine Geometry. The simple ratio is not just popular, it has been widely used. Sometimes they define it a bit differently (instead of $A_1A_3/A_1A_2$ you take, for example, $A_1A_3/A_2A_3$. All of those are related, and what is more important, all these are preserved under affine transformations. BTW, Castellet and Llerena is a great book!

share|cite|improve this answer

answered Feb 16 at 19:37

GReyesGReyes

2,04815

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  For example, the midpoint divides the segment in a simple ratio =2, according with your definition. The midpoint is an affine invariant.
  $endgroup$
  – GReyes
  Feb 16 at 19:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you have a reference for it being named in English?
  $endgroup$
  – jmerry
  Feb 16 at 20:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I was a bit surprised not to find the term "simple ratio" as often as expected. But you can check, for example, Coxeter's book (a classic) books.google.com/…
  $endgroup$
  – GReyes
  Feb 17 at 6:21
  
  
  

  
  
  
  

add a comment | 

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2

$begingroup$

The Wikipedia Projective harmonic conjugate article uses the term "division ratio" which is rare in English unfortunately. More precisely, it states "C divides AB externally" and gives the ratio $AC:BC$ which corresponds to your $lambda$ and where $ABC$ are three points in order on a line. A third point can divide a line segment "internally" where the ratio $lambda$ is negative. Further it also states:

In terms of a double ratio, given points a and b on an affine line, the division ratio of a point x is $$ t(x)=frac {x-a}{x-b}.$$

It seems to me that the term "division ratio" is what you want, although "simple ratio" is simpler.

share|cite|improve this answer

edited Feb 17 at 14:49

answered Feb 17 at 5:57

SomosSomos

14.5k11336

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Please see my reference to Coxeter's book. I must admit that I am more familiar with Russian terminology, where the adjective "simple" is used instead.
  $endgroup$
  – GReyes
  Feb 17 at 9:14
  

  
  
  
  

add a comment | 

2

$begingroup$

The Wikipedia Projective harmonic conjugate article uses the term "division ratio" which is rare in English unfortunately. More precisely, it states "C divides AB externally" and gives the ratio $AC:BC$ which corresponds to your $lambda$ and where $ABC$ are three points in order on a line. A third point can divide a line segment "internally" where the ratio $lambda$ is negative. Further it also states:

In terms of a double ratio, given points a and b on an affine line, the division ratio of a point x is $$ t(x)=frac {x-a}{x-b}.$$

It seems to me that the term "division ratio" is what you want, although "simple ratio" is simpler.

share|cite|improve this answer

edited Feb 17 at 14:49

answered Feb 17 at 5:57

SomosSomos

14.5k11336

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Please see my reference to Coxeter's book. I must admit that I am more familiar with Russian terminology, where the adjective "simple" is used instead.
  $endgroup$
  – GReyes
  Feb 17 at 9:14
  

  
  
  
  

add a comment | 

$begingroup$

The Wikipedia Projective harmonic conjugate article uses the term "division ratio" which is rare in English unfortunately. More precisely, it states "C divides AB externally" and gives the ratio $AC:BC$ which corresponds to your $lambda$ and where $ABC$ are three points in order on a line. A third point can divide a line segment "internally" where the ratio $lambda$ is negative. Further it also states:

In terms of a double ratio, given points a and b on an affine line, the division ratio of a point x is $$ t(x)=frac {x-a}{x-b}.$$

It seems to me that the term "division ratio" is what you want, although "simple ratio" is simpler.

share|cite|improve this answer

edited Feb 17 at 14:49

answered Feb 17 at 5:57

SomosSomos

14.5k11336

$endgroup$

share|cite|improve this answer

edited Feb 17 at 14:49

answered Feb 17 at 5:57

SomosSomos

14.5k11336

answered Feb 17 at 5:57

SomosSomos

14.5k11336

answered Feb 17 at 5:57

SomosSomos

14.5k11336

3

$begingroup$

I don't think I've ever seen this affine version named before, but I have seen the projective geometry concept that it's a case of: the cross-ratio
$$(a_4,a_1;a_2,a_3) = frac{a_1a_3cdot a_4a_2}{a_1a_2cdot a_4a_3}$$
If we choose $a_4$ to be the point at $infty$ on the line, then $frac{a_4a_3}{a_4a_2}=1$ and this becomes $frac{a_1a_3}{a_1a_2}=lambda$.

share|cite|improve this answer

answered Feb 16 at 19:36

jmerryjmerry

14.4k1630

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The cross ratio is an invariant in Projective Geometry (simple ratio is not invariant there, and you need a ratio of ratios). Observe that the cross ratio is a ratio of simple ratios.
  $endgroup$
  – GReyes
  Feb 16 at 19:37
  
  
  

  
  
  
  

add a comment | 

3

$begingroup$

I don't think I've ever seen this affine version named before, but I have seen the projective geometry concept that it's a case of: the cross-ratio
$$(a_4,a_1;a_2,a_3) = frac{a_1a_3cdot a_4a_2}{a_1a_2cdot a_4a_3}$$
If we choose $a_4$ to be the point at $infty$ on the line, then $frac{a_4a_3}{a_4a_2}=1$ and this becomes $frac{a_1a_3}{a_1a_2}=lambda$.

share|cite|improve this answer

answered Feb 16 at 19:36

jmerryjmerry

14.4k1630

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The cross ratio is an invariant in Projective Geometry (simple ratio is not invariant there, and you need a ratio of ratios). Observe that the cross ratio is a ratio of simple ratios.
  $endgroup$
  – GReyes
  Feb 16 at 19:37
  
  
  

  
  
  
  

add a comment | 

$begingroup$

I don't think I've ever seen this affine version named before, but I have seen the projective geometry concept that it's a case of: the cross-ratio
$$(a_4,a_1;a_2,a_3) = frac{a_1a_3cdot a_4a_2}{a_1a_2cdot a_4a_3}$$
If we choose $a_4$ to be the point at $infty$ on the line, then $frac{a_4a_3}{a_4a_2}=1$ and this becomes $frac{a_1a_3}{a_1a_2}=lambda$.

share|cite|improve this answer

answered Feb 16 at 19:36

jmerryjmerry

14.4k1630

$endgroup$

share|cite|improve this answer

answered Feb 16 at 19:36

jmerryjmerry

14.4k1630

answered Feb 16 at 19:36

jmerryjmerry

14.4k1630

answered Feb 16 at 19:36

jmerryjmerry

14.4k1630

1

$begingroup$

Ratios between segments are preserved in Affine Geometry. The simple ratio is not just popular, it has been widely used. Sometimes they define it a bit differently (instead of $A_1A_3/A_1A_2$ you take, for example, $A_1A_3/A_2A_3$. All of those are related, and what is more important, all these are preserved under affine transformations. BTW, Castellet and Llerena is a great book!

share|cite|improve this answer

answered Feb 16 at 19:37

GReyesGReyes

2,04815

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  For example, the midpoint divides the segment in a simple ratio =2, according with your definition. The midpoint is an affine invariant.
  $endgroup$
  – GReyes
  Feb 16 at 19:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you have a reference for it being named in English?
  $endgroup$
  – jmerry
  Feb 16 at 20:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I was a bit surprised not to find the term "simple ratio" as often as expected. But you can check, for example, Coxeter's book (a classic) books.google.com/…
  $endgroup$
  – GReyes
  Feb 17 at 6:21
  
  
  

  
  
  
  

add a comment | 

1

$begingroup$

Ratios between segments are preserved in Affine Geometry. The simple ratio is not just popular, it has been widely used. Sometimes they define it a bit differently (instead of $A_1A_3/A_1A_2$ you take, for example, $A_1A_3/A_2A_3$. All of those are related, and what is more important, all these are preserved under affine transformations. BTW, Castellet and Llerena is a great book!

share|cite|improve this answer

answered Feb 16 at 19:37

GReyesGReyes

2,04815

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  For example, the midpoint divides the segment in a simple ratio =2, according with your definition. The midpoint is an affine invariant.
  $endgroup$
  – GReyes
  Feb 16 at 19:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you have a reference for it being named in English?
  $endgroup$
  – jmerry
  Feb 16 at 20:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I was a bit surprised not to find the term "simple ratio" as often as expected. But you can check, for example, Coxeter's book (a classic) books.google.com/…
  $endgroup$
  – GReyes
  Feb 17 at 6:21
  
  
  

  
  
  
  

add a comment | 

$begingroup$

Ratios between segments are preserved in Affine Geometry. The simple ratio is not just popular, it has been widely used. Sometimes they define it a bit differently (instead of $A_1A_3/A_1A_2$ you take, for example, $A_1A_3/A_2A_3$. All of those are related, and what is more important, all these are preserved under affine transformations. BTW, Castellet and Llerena is a great book!

share|cite|improve this answer

answered Feb 16 at 19:37

GReyesGReyes

2,04815

$endgroup$

share|cite|improve this answer

answered Feb 16 at 19:37

GReyesGReyes

2,04815

answered Feb 16 at 19:37

GReyesGReyes

2,04815

answered Feb 16 at 19:37

GReyesGReyes

2,04815