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21315Re: [EMHL] changes in point of view

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  • Francois Rideau
    Dec 11, 2012
      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.
      >
      >
      >


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