- Dec 11, 2012Dear 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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