Fundamental Theorem of Affine Transformations in 3D

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Fundamental Theorem of Affine Transformations states that "Given two ordered sets of three non-collinear points each, there exists a unique affine transformation $f$ mapping one set onto the other."



There are several proof of the same but all of them consider collinear points in 2D. By that I mean that both the ordered sets of non-collinear points lies in the same plane.



I am wondering if things will change if the the ordered sets of non-collinear points lies in different planes Or the theorem is true for all pair of three non-collinear points.




linear-algebra group-theory geometry linear-transformations




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    In 3D you need 4 non coplanar points.
    – Jens Schwaiger
    Aug 24 at 6:32











  • @JensSchwaiger Can you provide some reference?
    – Zahir
    Aug 24 at 6:53















up vote
1
down vote

favorite
1












Fundamental Theorem of Affine Transformations states that "Given two ordered sets of three non-collinear points each, there exists a unique affine transformation $f$ mapping one set onto the other."



There are several proof of the same but all of them consider collinear points in 2D. By that I mean that both the ordered sets of non-collinear points lies in the same plane.



I am wondering if things will change if the the ordered sets of non-collinear points lies in different planes Or the theorem is true for all pair of three non-collinear points.




linear-algebra group-theory geometry linear-transformations




share|cite|improve this question


















  • 1




    In 3D you need 4 non coplanar points.
    – Jens Schwaiger
    Aug 24 at 6:32











  • @JensSchwaiger Can you provide some reference?
    – Zahir
    Aug 24 at 6:53













up vote
1
down vote

favorite
1









up vote
1
down vote

favorite
1






1





Fundamental Theorem of Affine Transformations states that "Given two ordered sets of three non-collinear points each, there exists a unique affine transformation $f$ mapping one set onto the other."



There are several proof of the same but all of them consider collinear points in 2D. By that I mean that both the ordered sets of non-collinear points lies in the same plane.



I am wondering if things will change if the the ordered sets of non-collinear points lies in different planes Or the theorem is true for all pair of three non-collinear points.




linear-algebra group-theory geometry linear-transformations




share|cite|improve this question














Fundamental Theorem of Affine Transformations states that "Given two ordered sets of three non-collinear points each, there exists a unique affine transformation $f$ mapping one set onto the other."



There are several proof of the same but all of them consider collinear points in 2D. By that I mean that both the ordered sets of non-collinear points lies in the same plane.



I am wondering if things will change if the the ordered sets of non-collinear points lies in different planes Or the theorem is true for all pair of three non-collinear points.





linear-algebra group-theory geometry linear-transformations





share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 24 at 6:30

























asked Aug 24 at 5:59









Zahir

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464







  • 1




    In 3D you need 4 non coplanar points.
    – Jens Schwaiger
    Aug 24 at 6:32











  • @JensSchwaiger Can you provide some reference?
    – Zahir
    Aug 24 at 6:53













  • 1




    In 3D you need 4 non coplanar points.
    – Jens Schwaiger
    Aug 24 at 6:32











  • @JensSchwaiger Can you provide some reference?
    – Zahir
    Aug 24 at 6:53








1




1




In 3D you need 4 non coplanar points.
– Jens Schwaiger
Aug 24 at 6:32





In 3D you need 4 non coplanar points.
– Jens Schwaiger
Aug 24 at 6:32













@JensSchwaiger Can you provide some reference?
– Zahir
Aug 24 at 6:53





@JensSchwaiger Can you provide some reference?
– Zahir
Aug 24 at 6:53











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Let $V$ be a vector space of dimension $n$ and assume that $a_0,a_1,ldots,a_n$ are affine Independent, i.e., the vectors $a_1-a_0,ldots,a_n-a_0$ are linearly independent. Then given arbitrary points $b_0,b_1,ldots,b_nin V$ there is a unique linear mapping $gcolon Vto V $ such that $g(a_i-a_0)=b_i-b_0$ for all $1leq ileq n$. Then $f$ defined by $f(x):=g(x-a_0)+b_0=g(x)+b_0-g(a_0)$ is a (in fact the unique) affine mapping satisfying $f(a_i)=b_i$ for all $0leq ileq n$.






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    Let $V$ be a vector space of dimension $n$ and assume that $a_0,a_1,ldots,a_n$ are affine Independent, i.e., the vectors $a_1-a_0,ldots,a_n-a_0$ are linearly independent. Then given arbitrary points $b_0,b_1,ldots,b_nin V$ there is a unique linear mapping $gcolon Vto V $ such that $g(a_i-a_0)=b_i-b_0$ for all $1leq ileq n$. Then $f$ defined by $f(x):=g(x-a_0)+b_0=g(x)+b_0-g(a_0)$ is a (in fact the unique) affine mapping satisfying $f(a_i)=b_i$ for all $0leq ileq n$.






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      Let $V$ be a vector space of dimension $n$ and assume that $a_0,a_1,ldots,a_n$ are affine Independent, i.e., the vectors $a_1-a_0,ldots,a_n-a_0$ are linearly independent. Then given arbitrary points $b_0,b_1,ldots,b_nin V$ there is a unique linear mapping $gcolon Vto V $ such that $g(a_i-a_0)=b_i-b_0$ for all $1leq ileq n$. Then $f$ defined by $f(x):=g(x-a_0)+b_0=g(x)+b_0-g(a_0)$ is a (in fact the unique) affine mapping satisfying $f(a_i)=b_i$ for all $0leq ileq n$.






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        Let $V$ be a vector space of dimension $n$ and assume that $a_0,a_1,ldots,a_n$ are affine Independent, i.e., the vectors $a_1-a_0,ldots,a_n-a_0$ are linearly independent. Then given arbitrary points $b_0,b_1,ldots,b_nin V$ there is a unique linear mapping $gcolon Vto V $ such that $g(a_i-a_0)=b_i-b_0$ for all $1leq ileq n$. Then $f$ defined by $f(x):=g(x-a_0)+b_0=g(x)+b_0-g(a_0)$ is a (in fact the unique) affine mapping satisfying $f(a_i)=b_i$ for all $0leq ileq n$.






        share|cite|improve this answer














        Let $V$ be a vector space of dimension $n$ and assume that $a_0,a_1,ldots,a_n$ are affine Independent, i.e., the vectors $a_1-a_0,ldots,a_n-a_0$ are linearly independent. Then given arbitrary points $b_0,b_1,ldots,b_nin V$ there is a unique linear mapping $gcolon Vto V $ such that $g(a_i-a_0)=b_i-b_0$ for all $1leq ileq n$. Then $f$ defined by $f(x):=g(x-a_0)+b_0=g(x)+b_0-g(a_0)$ is a (in fact the unique) affine mapping satisfying $f(a_i)=b_i$ for all $0leq ileq n$.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Aug 24 at 8:30

























        answered Aug 24 at 7:34









        Jens Schwaiger

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