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changes in point of view

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  • Chris Van Tienhoven
    Dear friends, Here is my december contribution. One of the major theorems in Projective Geometry (see the book of P.S. Modenow and A.S. Parkmohenko [1, page
    Message 1 of 3 , Dec 8, 2012
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      Dear friends,

      Here is my december contribution.

      One of the major theorems in Projective Geometry (see the book of P.S. Modenow and A.S. Parkmohenko [1, page 24]) states:
      Given any four points A, B, C, D of a projective plane Pi, no three of which are collinear, and four points A', B', C', D' of a projective plane Pi', no three of which are collinear, there exists one and only one projective mapping T of Pi onto Pi' that takes A, B, C, D into A', B', C', D', respectively.
      I wondered what happens to the points in the triangle field of A,B,C performing this transformation.
      In short: the field is stretched. All points change position except A,B,C.
      Then I wondered how this could be constructed. I found 3 ways of construction.
      The simplest (with the use of conics):
      1. Let ABC be a random Triangle.
      2. Let D1 and D2 and P1 be random points unequal to each other and unequal A, B or C.
      3. We are looking for the point P2 being the ABCD1-ABCD2-Transformation of P1.
      4. Construct the Conic Co1 through A,B,C,D1,D2.
      5. Let S1 be the 2nd intersection point of Co1 and the line D1.P1.
      6. Construct the Conic Co2 through A,B,C,P1,S1.
      7. Now P2 = the 2nd intersection point of Co2 and the line D2.S1.
      Note: the 2nd intersection point of a conic and a line can be constructed by straightedge only.

      Then some other remarkable items:
      1. The locus of a ABCD1->ABCD2 Projective Transformation of the circumcircle is a circumscribed conic.
      2. Note that all transformation vectors are directed to/from the same point S on the circumcircle, which is the 4th intersection of the circumcircle and the mapped conic.
      3. Note that the 4th intersection point of the circumcircle and the circumscribed conic ABC.D1.D2 is collinear with D2 and S.
      Using this property S can be constructed when only D1 and D2 are known.
      4. The same properties occur when the circle is interchanged for a circumscribed conic.

      Finally:
      Let X1 be some random point.
      Let X2 be the Projective ABCD1->ABCD2-Transformation of X1.
      Let Co be the Conic through ABC and X1 and X2.
      Theorem:
      Now all transformation-vectors of the same Projective ABCD1->ABCD2-Transformation of points on this conic will point to X2.

      So the X1-X2-circumscribed conic is the locus of all points that are D1-D2-mapped with transformation direction to X2.
      Further:
      a. The Transformation vector at X2 is tangent at the conic.
      b. Let S1 = D1.X1 ^ D2.X2. This point is on the conic Co.
      c. The Transformation vector in S1 has the direction of D2.X2.

      It all might seem a bit theoretical, but when you make pictures, it comes to live.
      I can send Cabri-pictures for those interested.

      Algebraic implications
      I calculated what happens to a random point P(x0 : y0 : z0), when another random point D1 (x1 : y1 : z1) in the projective plane defined by points A, B, C, D1 is changed into D2 (x2 : y2 : z2). The algebraic outcome is amazingly simple: (x0.x2/x1 : y0.y2/y1 : z0.z2/z1).

      Note that a projective transformation actually is a projected picture as seen under perspective. Sometimes even projected twice.
      The ABCD1->ABCD2 Projection Transformation can be described as two consecutive changes of point of view (one finite and one infinite).
      There are more interesting features about this transformation.
      Probably several items already are known under some name. I am interested in references of what is described more about this subject.

      Best regards,

      Chris van Tienhoven



      [1] P.S. Modenow and A.S. Parkmohenko, Geometric Transformations, Volume 2: Projective Transformations, p.24 Two fundamental Theorems on Projective Transformations.
    • Francois Rideau
      Dear Chris May be you can look at this theorem in the other way. A projective map f of the plane and a point O are given. Then the locus of the point M such
      Message 2 of 3 , Dec 11, 2012
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        Dear Chris
        May be you can look at this theorem in the other way.
        A projective map f of the plane and a point O are given.
        Then the locus of the point M such that O, M, f(M) are on a same line is
        the conic thru O, f^{-1}(O) and the three fixed points of f.
        Merry Xmas and Happy New Year
        Francois

        On Sun, Dec 9, 2012 at 8:48 AM, Chris Van Tienhoven <van10hoven@...>wrote:

        > **
        >
        >
        > Dear friends,
        >
        > Here is my december contribution.
        >
        > One of the major theorems in Projective Geometry (see the book of P.S.
        > Modenow and A.S. Parkmohenko [1, page 24]) states:
        > Given any four points A, B, C, D of a projective plane Pi, no three of
        > which are collinear, and four points A', B', C', D' of a projective plane
        > Pi', no three of which are collinear, there exists one and only one
        > projective mapping T of Pi onto Pi' that takes A, B, C, D into A', B', C',
        > D', respectively.
        > I wondered what happens to the points in the triangle field of A,B,C
        > performing this transformation.
        > In short: the field is stretched. All points change position except A,B,C.
        > Then I wondered how this could be constructed. I found 3 ways of
        > construction.
        > The simplest (with the use of conics):
        > 1. Let ABC be a random Triangle.
        > 2. Let D1 and D2 and P1 be random points unequal to each other and unequal
        > A, B or C.
        > 3. We are looking for the point P2 being the ABCD1-ABCD2-Transformation of
        > P1.
        > 4. Construct the Conic Co1 through A,B,C,D1,D2.
        > 5. Let S1 be the 2nd intersection point of Co1 and the line D1.P1.
        > 6. Construct the Conic Co2 through A,B,C,P1,S1.
        > 7. Now P2 = the 2nd intersection point of Co2 and the line D2.S1.
        > Note: the 2nd intersection point of a conic and a line can be constructed
        > by straightedge only.
        >
        > Then some other remarkable items:
        > 1. The locus of a ABCD1->ABCD2 Projective Transformation of the
        > circumcircle is a circumscribed conic.
        > 2. Note that all transformation vectors are directed to/from the same
        > point S on the circumcircle, which is the 4th intersection of the
        > circumcircle and the mapped conic.
        > 3. Note that the 4th intersection point of the circumcircle and the
        > circumscribed conic ABC.D1.D2 is collinear with D2 and S.
        > Using this property S can be constructed when only D1 and D2 are known.
        > 4. The same properties occur when the circle is interchanged for a
        > circumscribed conic.
        >
        > Finally:
        > Let X1 be some random point.
        > Let X2 be the Projective ABCD1->ABCD2-Transformation of X1.
        > Let Co be the Conic through ABC and X1 and X2.
        > Theorem:
        > Now all transformation-vectors of the same Projective
        > ABCD1->ABCD2-Transformation of points on this conic will point to X2.
        >
        > So the X1-X2-circumscribed conic is the locus of all points that are
        > D1-D2-mapped with transformation direction to X2.
        > Further:
        > a. The Transformation vector at X2 is tangent at the conic.
        > b. Let S1 = D1.X1 ^ D2.X2. This point is on the conic Co.
        > c. The Transformation vector in S1 has the direction of D2.X2.
        >
        > It all might seem a bit theoretical, but when you make pictures, it comes
        > to live.
        > I can send Cabri-pictures for those interested.
        >
        > Algebraic implications
        > I calculated what happens to a random point P(x0 : y0 : z0), when another
        > random point D1 (x1 : y1 : z1) in the projective plane defined by points A,
        > B, C, D1 is changed into D2 (x2 : y2 : z2). The algebraic outcome is
        > amazingly simple: (x0.x2/x1 : y0.y2/y1 : z0.z2/z1).
        >
        > Note that a projective transformation actually is a projected picture as
        > seen under perspective. Sometimes even projected twice.
        > The ABCD1->ABCD2 Projection Transformation can be described as two
        > consecutive changes of point of view (one finite and one infinite).
        > There are more interesting features about this transformation.
        > Probably several items already are known under some name. I am interested
        > in references of what is described more about this subject.
        >
        > Best regards,
        >
        > Chris van Tienhoven
        >
        > [1] P.S. Modenow and A.S. Parkmohenko, Geometric Transformations, Volume
        > 2: Projective Transformations, p.24 Two fundamental Theorems on Projective
        > Transformations.
        >
        >
        >


        [Non-text portions of this message have been removed]
      • Chris Van Tienhoven
        Thanks Francois, I also received interesting personal mails from Francisco and from Eckart Schmidt. According to Eckart Schmidt there is a striking accordance
        Message 3 of 3 , Dec 21, 2012
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          Thanks Francois,

          I also received interesting personal mails from Francisco and from Eckart Schmidt.
          According to Eckart Schmidt there is a striking accordance between a projective D1-D2-Transformation and an Isoconjugate (see [1]).
          When:
          Q2 = (P1,P2) – Isoconjugate of Q1.
          then:
          Q2 = (Q1,P2) – ProjectiveTransformation of P1, or
          P2 = (P1,Q2) – ProjectiveTransformation of Q1.
          This shows:
          1. Indeed the Isoconjugate is a Projective Transformation.
          2. There is a cross-wise use of parameters for Isoconjugates and P1P2-Projective Transformations.

          Best regards,

          Chris van Tienhoven

          [1] Jean-Pierre Ehrmann and Bernard Gibert, Special Isocubics in the Triangle Plane, available at:
          http://bernard.gibert.pagesperso-orange.fr/files/isocubics.html


          --- In Hyacinthos@yahoogroups.com, Francois Rideau <francois.rideau@...> wrote:
          >
          > Dear Chris
          > May be you can look at this theorem in the other way.
          > A projective map f of the plane and a point O are given.
          > Then the locus of the point M such that O, M, f(M) are on a same line is
          > the conic thru O, f^{-1}(O) and the three fixed points of f.
          > Merry Xmas and Happy New Year
          > Francois
          >
          > On Sun, Dec 9, 2012 at 8:48 AM, Chris Van Tienhoven <van10hoven@...>wrote:
          >
          > > **
          > >
          > >
          > > Dear friends,
          > >
          > > Here is my december contribution.
          > >
          > > One of the major theorems in Projective Geometry (see the book of P.S.
          > > Modenow and A.S. Parkmohenko [1, page 24]) states:
          > > Given any four points A, B, C, D of a projective plane Pi, no three of
          > > which are collinear, and four points A', B', C', D' of a projective plane
          > > Pi', no three of which are collinear, there exists one and only one
          > > projective mapping T of Pi onto Pi' that takes A, B, C, D into A', B', C',
          > > D', respectively.
          > > I wondered what happens to the points in the triangle field of A,B,C
          > > performing this transformation.
          > > In short: the field is stretched. All points change position except A,B,C.
          > > Then I wondered how this could be constructed. I found 3 ways of
          > > construction.
          > > The simplest (with the use of conics):
          > > 1. Let ABC be a random Triangle.
          > > 2. Let D1 and D2 and P1 be random points unequal to each other and unequal
          > > A, B or C.
          > > 3. We are looking for the point P2 being the ABCD1-ABCD2-Transformation of
          > > P1.
          > > 4. Construct the Conic Co1 through A,B,C,D1,D2.
          > > 5. Let S1 be the 2nd intersection point of Co1 and the line D1.P1.
          > > 6. Construct the Conic Co2 through A,B,C,P1,S1.
          > > 7. Now P2 = the 2nd intersection point of Co2 and the line D2.S1.
          > > Note: the 2nd intersection point of a conic and a line can be constructed
          > > by straightedge only.
          > >
          > > Then some other remarkable items:
          > > 1. The locus of a ABCD1->ABCD2 Projective Transformation of the
          > > circumcircle is a circumscribed conic.
          > > 2. Note that all transformation vectors are directed to/from the same
          > > point S on the circumcircle, which is the 4th intersection of the
          > > circumcircle and the mapped conic.
          > > 3. Note that the 4th intersection point of the circumcircle and the
          > > circumscribed conic ABC.D1.D2 is collinear with D2 and S.
          > > Using this property S can be constructed when only D1 and D2 are known.
          > > 4. The same properties occur when the circle is interchanged for a
          > > circumscribed conic.
          > >
          > > Finally:
          > > Let X1 be some random point.
          > > Let X2 be the Projective ABCD1->ABCD2-Transformation of X1.
          > > Let Co be the Conic through ABC and X1 and X2.
          > > Theorem:
          > > Now all transformation-vectors of the same Projective
          > > ABCD1->ABCD2-Transformation of points on this conic will point to X2.
          > >
          > > So the X1-X2-circumscribed conic is the locus of all points that are
          > > D1-D2-mapped with transformation direction to X2.
          > > Further:
          > > a. The Transformation vector at X2 is tangent at the conic.
          > > b. Let S1 = D1.X1 ^ D2.X2. This point is on the conic Co.
          > > c. The Transformation vector in S1 has the direction of D2.X2.
          > >
          > > It all might seem a bit theoretical, but when you make pictures, it comes
          > > to live.
          > > I can send Cabri-pictures for those interested.
          > >
          > > Algebraic implications
          > > I calculated what happens to a random point P(x0 : y0 : z0), when another
          > > random point D1 (x1 : y1 : z1) in the projective plane defined by points A,
          > > B, C, D1 is changed into D2 (x2 : y2 : z2). The algebraic outcome is
          > > amazingly simple: (x0.x2/x1 : y0.y2/y1 : z0.z2/z1).
          > >
          > > Note that a projective transformation actually is a projected picture as
          > > seen under perspective. Sometimes even projected twice.
          > > The ABCD1->ABCD2 Projection Transformation can be described as two
          > > consecutive changes of point of view (one finite and one infinite).
          > > There are more interesting features about this transformation.
          > > Probably several items already are known under some name. I am interested
          > > in references of what is described more about this subject.
          > >
          > > Best regards,
          > >
          > > Chris van Tienhoven
          > >
          > > [1] P.S. Modenow and A.S. Parkmohenko, Geometric Transformations, Volume
          > > 2: Projective Transformations, p.24 Two fundamental Theorems on Projective
          > > Transformations.
          > >
          > >
          > >
          >
          >
          > [Non-text portions of this message have been removed]
          >
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