CO-Tf3 Scimemi Transformation

The Scimemi-transformation CO-Tf3 is a point-to-point mapping defined by a conic CO.This transformation and its action on pentagons was presented by Benedetto Scimemi at Feb. 2005 in Bloomington as a homage to Douglas Hofstadter for his 60th birthday.

This page has been written with the contribution of Benedetto Scimemi.

CO-Tf3 is defined wrt any conic CO, except circles and rectangular hyperbolas. Extensions to these cases, as limits, will be discussed below.For a central conic CO, the mapping P CO-Tf3(P) can be described as:

1. a reflection of P about an axis of the conic, followed (or preceded) by
2. a homothety with fixed point in the conic center Z and scale factor  r > 0. The choice of the reflection axis and the value r only depend on the shape of the conic (i.e. they are the same for similar conics; see below).

When CO is a parabola the homothety is replaced by a translation.The mapping CO-Tf3^-1 , the inverse CO-Tf3, has the same definition, provided the  factor  r  is replaced by its inverse  r^-1.

It appears from the definition that

CO-Tf3 and CO-Tf3^-1 are negative (odd) similarities.

Their fixed point is Z, their fixed lines are the CO-axes.

Notice that one gets the same mappings for Z-homothetic conics CO.

Definition

See [34], B. Scimemi, QFG#935 and #1252).

When CO is an Ellipse:

then CO-Tf3 is the product of the reflection in the major axis and the homothety, centered in Z, with scale factor (a² – b² )/ (a² + b²). Here a > b are the axes lengths. Equivalently r = cos[α], where α is the angle under which a vertex views the minor axis.

When CO is a (non-orthogonal) Hyperbola:

call “principal” the axis which lies inside the smaller angle α formed by the asymptotes (N.B. not always the principal axis is the focal axis),

then CO-Tf3 is the product of the reflection in the principal axis and

the homothety, centered in Z, with scale factor  r = (a² + b²) / (a² – b² ), where a > bare the axes lengths. In this case r = cos^-1 [α].

When CO is a Parabola:

then CO-Tf3 will be an isometry, the product of the reflection in the parabola axis and a parallel translation which amounts to twice the (oriented) distance directrix -> focus.

When CO is degenerated into two (non-orthogonal) lines crossing in Z:

then CO-Tf3 is the product of the reflection in the angle bisector of the smaller angle α formed by the lines; and the homothety, centered in Z, with scale factor cos [α].

When CO is degenerated into two parallel lines:

then CO-Tf3 is the reflection in the mid-line (parallel, equidistant).

Limit-cases

One can extend CO-Tf3 and by CO-Tf3^-1, as limits, to orthogonal hyperbola and circles, but the resulting mappings are not invertible.

For an orthogonal hyperbola, CO-Tf3(P) tends the infinity point of the line Z.P, reflected in the principal axis; CO-Tf3^-1(P) tends to the conic center for all P.

As for circles, CO-Tf3 maps each point P into the circle center; CO-Tf3^-1 can’t be constructed.

Eccentricity

By introducing the eccentricity e, the homothety factor ((a² −b²)/(a² + b²))^±1 can be unified into the single formula r = e²/|2−e²| holding for all types of conics.

See Ref-34, QFG#935 and #1252.

CO-Tf3 Chord Lemma

• This Chord Lemma is a basic property of Co-Tf3 (Benedetto Scimemi, 2018, February 8, personal mail).
• For any chord P1P2 of any conic CO, let M be its midpoint. Then then CO-Tf3(M) lies on the perpendicular bisector of P1P2 .

In particular, for any point P of CO, the normal to CO in P is the line P. CO-Tf3(P).

Conversely: For any point X, if the line through X normal to X.CO-Tf3(X) cuts CO in two (real) points P1, P2, then X is the midpoint of P1.P2.

Here are some consequences:

1. Let Tr be any CO-inscribed triangle , O its circumcenter, N its ninepoint circle. Then O lies on the circle CO-Tf3(N).
2. Let Tr be any triangle, O its circumcenter. For any conic CO circumscribed to Tr, the point CO-Tf3^-1(O) lies on the nine-point circle of Tr.
3. Let CO be a parabola, X1, X2 any points, Mi the midpoint of Xi . CO-Tf3(Xi). Then the line M1M2 is the axis of the parabola.
4. For any quadrangle Q inscribed in any conic CO, the CO-Tf3 image of QA-P2 is QA-P4.
5. For any pentangle PA = P1P2P3P4P5 let QAi the Component Quadrangle, obtained by ignoring the point Pi. Let Hi = QAi-P2, Oi = QAi-P4. Then the pentangles H1H2H3H4H5, O1O2O3O4O5 are negatively similar, the similarity being CO-Tf3 : Hi -> Oi, where CO is the conic circumscribing PA.

It was the proof of the last statement which gave origin to the whole subject of CO-Tf3.

Proofs of the above properties can be derived by synthetic arguments from the Chord Lemma. A proof of the Lemma itself is very easy if one represents CO by P = [a cos ∂, b sin ∂] or similar (see below, in the Section Coordinates).

Constructions

Schmidt-Construction for CO-Tf3(P)

This construction is based upon Eckart Schmidt’s construction at Ref-34, QFG#2811.

1. Let L be the polar CO-Tf1 of P.
2. Let Lp be the line through P perpendicular to L.
3. Let Lc be the reflection of P.CO-P1 in the principal axis of CO.
4. Finally CO-Tf3(P) = the intersection point of Lp and Lc.

Schmidt-Construction for CO-Tf3^-1(P)

This construction was found by Eckart Schmidt. See Ref-34, QFG#2811.

1. Let L be the polar CO-Tf1 of P.
2. Let Lr be the reflection of L in the main axis of CO.
3. Let Lp be the line through P perpendicular to Lr.
4. Let Lc be the reflection of P.CO-P1 in the principal axis of CO.
5. Finally CO-Tf3^-1(P) = the intersection point of Lp and Lc.

Keizer-Construction for CO-Tf3(P)

This construction is based upon Bernard Keizer’s construction at Ref-34, QFG#2826.

1. Let P1 be the inverse of P in the Orthoptic Circle CO-Ci2
2. Let P2 be the inverse of P1 in the Circle CO-Ci1 with the foci as diameter
3. CO-Tf3(P) = the reflection of P2 in the principal axis of CO.

Note: When the principal axis does not connect the foci, the orthoptic circle is not real. Replace it by the circle with radius (a² – b²)^1/2.

Keizer-Construction for CO-Tf3^-1(P)

This construction is based upon Bernard Keizer’s construction at Ref-34, QFG#2826.

1. Let P1 be the inverse of P in the Circle CO-Ci1 with the foci as diameter
2. Let P2 be the inverse of P1 in the Orthoptic Circle CO-Ci2
3. CO-Tf3^-1(P) = the reflection of P2 in the principal axis of CO.

Note: When the principal axis does not connect the foci, the orthoptic circle is not real. Replace it by the circle with radius (a² – b²)^1/2.

Another Construction for CO-Tf3^-1(P)

Next construction for CO-Tf3^-1(P) is based upon the construction of 5P-s-Tf3, which is a construction of Telv Cohl. See [33], Anopolis #1986.

1. Let P be the point to be transformed wrt some reference conic CO.
2. Span some Pentangle P1.P2.P3.P4.P5 into Conic CO.
3. Let P_ijk be the Orthopole of line P.O_ijk wrt triangle Pi.Pj.Pk (O_ijk = center circumcircle Pi.Pj.Pk), where (i,j,k) are different numbers from (1,2,3,4,5).
4. Let Ci₁₂₃₄ be the circle through P₁₂₃, P₁₂₄, P₁₃₄, having center O₁₂₃₄.
5. Let Ci₁₂₃₅ be the circle through P₁₂₃, P₁₂₅, P₁₃₅, having center O₁₂₃₅.
6. These circles have point P₁₂₃ in common.
7. CO-Tf3^-1(P) will be the 2^nd intersection point of Ci₁₂₃₄ and Ci₁₂₃₅.

Construction for CO-Tf3^-1 from Benedetto Scimemi

(a quick recipe for CABRI and Geogebra users)

1. Choose a circle Ci centered in P, such that CO and Ci have 4 intersections A1, A2, A3, A4. This is always possible when CO is a hyperbola; in other cases, it may be necessary to replace CO by a (larger or smaller) conic CO* obtained from CO by a convenient Z-homothety.
2. Construct the centroid QA-P1 of the quadrangle QA = A1A2A3A4.
3. Then P ~ QA-P4 and CO-Tf3^-1(P) ~ QA-P2 will be the reflection of P on QA-P1 (QA-Centroid).

This construction is based on what can be considered the main property of CO-Tf3 / CO-Tf3^-1:

they swap QA-P2 and QA-P4 for all quadrangles QA inscribed in CO (see the final Table).

Construction, when knowing CO-Tf3(P) or CO-Tf3^-1(P), of their inverse

The construction of CO-Tf3^-1(P) is, knowing the construction of CO-Tf3(P), very simple.

1. Let Cix be the circle through P with center CO-P1.
2. CO-Tf3R^-1 (P) is the inverse of CO-Tf3(P) wrt Cix.

The construction of CO-Tf3(P), when knowing CO-Tf3^-1(P), is identical.

Relationship with Frégier’s Point

For each point P on any Conic CO-Tf3(P) coincides with Frégier’s Point.

Frégier was a French mathematician who published several articles in the “Annales de Gergonne” around 1810. He discovered a remarkable point transformation with respect to a conic.

Let P be some point on conic CO. Draw a number of right angles having this point as their vertices. Then the intersected chords have in common a point, being called Frégier’s Point. See Ref-13, keyword “Frégier’s Theorem”.

Frégier’s Point is only defined for points on a conic. CO-Tf3 is defined for all points in the plane.

This makes CO-Tf3 a generalization of Frégier’s Point.

Relationship with Orthotransversal Line and Pedal Circle wrt a CO-inscribed Triangle

For any CO-inscribed triangle Tr and any P on CO:

• CO-Tf3(P) lies on the orthotransversal line of P w.r.t. Tr
• CO-Tf3^-1(P) lies on the pedal circle of P w.r.t. Tr.

Therefore, given any two triangles inscribed in the same conic CO, the two orthotransversal lines of P meet in CO-Tf3(P), the two pedal circles of P meet in CO-Tf3^-1(P). In particular, this holds when we deal with Component Triangles of a Quadrangle and it proves (by choosing CO circumscribed to the Quadrangle and passing through P) that

(1) the 4 orthotransversal lines have a common point. See Ref-59, AoPS, VU Thanh Tung, 2015, June 26 and Ref-34, QFG#1233).

(2) the 4 pedal circles have a common point. See Ref-59, AoPS, Luis Gonzales, 2011, August 7 and Ref-34, QFG#1218).

For a description of the connection of both properties see Ref-34, QFG#1252, Benedetto Scimemi, August 2015).

Barycentric Coordinates

Let Reference Conic CO be defined by 5 points (1:0:0), (0:1:0), (0:0:1), (p:q:r), (u:v:w),

then CO has equation X y z + Y z x + Z x y,

where X = p u (q w – r v), Y = q v (r u – p w), Z = r w (p v – q u).

and CO-Tf3[(x:y:z)] will have these barycentric coordinates:

( X (+a² (x + y + z) – b² (x + y – z) – c² (x – y + z)) – 2 a² (Y z + y Z) :

Y ( -a² (x + y – z) + b² (x + y + z) + c² (x – y – z)) – 2 b² (X z + x Z) :

Z ( -a² (x – y + z) + b² (x – y – z) + c² (x + y + z)) – 2 c² (X y + x Y) )

The barycentric coordinates of CO-Tf3^-1[(x:y:z)] are much longer.

Cartesian Coordinates

Cartesian Coordinates for both of CO-Tf3 and CO-Tf3^-1 are very easily written if one represents CO by P = [a cos δ, b sin δ] for ellipses or [a / cos δ, b tan δ] for hyperbolas.

Then CO-Tf3([x,y]) = [x, – y]((a² -b²)/(a² + b² ))^±1.

If CO is the parabola y = x² then CO-Tf3( [x,y] ) = [-x, y+1].

Examples of CO-Tf3 and CO-Tf3^-1 Transformations

The action of CO-Tf3 can be read from left to right.

The action of CO-Tf3^-1 can be read from right to left.

*)QA-Lx = line through QA-P6 parallel to QA-P2.QA-P23

QA-Ly = line through QA-P23 parallel to QA-L1

**)5P-s-Px = Conic Center of the five 5P-versions of QA-P2

5P-s-Py = Conic Center of the five 5P-versions of QA-P4

Warning: In a Pentangle the Scimemi Transformation CO-Tf3 wrt the circumscribed conic is identical to the inverse of the 5P-s-Tf3 Transformation.

These results were found by Eckart Schmidt, Benedetto Scimemi and the author of EPG.

There will be probably many more appealing examples.

Estimated human page views: 293