The Scimemitransformation COTf3 is a pointtopoint 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.
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.
COTf3 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 > COTf3(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 COTf3^{1} , the inverse COTf3, has the same definition, provided the factor r is replaced by its inverse r^{1}.
It appears from the definition that
COTf3 and COTf3^{1} are negative (odd) similarities.
Their fixed point is Z, their fixed lines are the COaxes.
Notice that one gets the same mappings for Zhomothetic conics CO.
Definition (Ref34, B. Scimemi, QFG#935 and #1252):
**When CO is an Ellipse:
then COTf3 is the product of the reflection in the major axis and the homothety, centered in Z, with scale factor (a^{2}  b^{2} )/ (a^{2} + b^{2}). Here a > b are the axes lengths. Equivalently r = cos[], where is the angle under which a vertex views the minor axis.
**When CO is a (nonorthogonal) Hyperbola:
call “principal” the axis which lies inside the smaller angle formed by the asymptotes (N.B. not always the principal axis is the focal axis),
then COTf3 is the product of the reflection in the principal axis and
the homothety, centered in Z, with scale factor r = (a^{2} + b^{2}) / (a^{2}  b^{2} ), where a > b
are the axes lengths. In this case r = cos^{1} [].
**When CO is a Parabola:
then COTf3 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 (nonorthogonal) lines crossing in Z:
then COTf3 is the product of the reflection in the angle bisector of the smaller angle formed by the lines; and the homothety, centered in Z, with scale factor cos [].
**When CO is degenerated into two parallel lines:
then COTf3 is the reflection in the midline (parallel, equidistant).
**When CO is an Ellipse:
then COTf3 is the product of the reflection in the major axis and the homothety, centered in Z, with scale factor (a^{2}  b^{2} )/ (a^{2} + b^{2}). Here a > b are the axes lengths. Equivalently r = cos[], where is the angle under which a vertex views the minor axis.
**When CO is a (nonorthogonal) Hyperbola:
call “principal” the axis which lies inside the smaller angle formed by the asymptotes (N.B. not always the principal axis is the focal axis),
then COTf3 is the product of the reflection in the principal axis and
the homothety, centered in Z, with scale factor r = (a^{2} + b^{2}) / (a^{2}  b^{2} ), where a > b
are the axes lengths. In this case r = cos^{1} [].
**When CO is a Parabola:
then COTf3 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 (nonorthogonal) lines crossing in Z:
then COTf3 is the product of the reflection in the angle bisector of the smaller angle formed by the lines; and the homothety, centered in Z, with scale factor cos [].
**When CO is degenerated into two parallel lines:
then COTf3 is the reflection in the midline (parallel, equidistant).
Limitcases:
One can extend COTf3 and by COTf3^{1}, as limits, to orthogonal hyperbola and circles, but the resulting mappings are not invertible.
For an orthogonal hyperbola, COTf3(P) tends the infinity point of the line Z.P, reflected in the principal axis; COTf3^{1}(P) tends to the conic center for all P.
As for circles, COTf3 maps each point P into the circle center; COTf3^{1} can’t be constructed.
Eccentricity
By introducing the eccentricity e, the homothety factor ((a^{2} −b^{2})/(a^{2} + b^{2}))^{±1} can be uniﬁed into the single formula r = e^{2}/2−e^{2} holding for all types of conics.
See Ref34, QFG#935 and #1252.
By introducing the eccentricity e, the homothety factor ((a^{2} −b^{2})/(a^{2} + b^{2}))^{±1} can be uniﬁed into the single formula r = e^{2}/2−e^{2} holding for all types of conics.
See Ref34, QFG#935 and #1252.
COTf3 Chord Lemma
• This Chord Lemma is a basic property of CoTf3 (Benedetto Scimemi, 2018, February 8, personal mail).
• For any chord P1P2 of any conic CO, let M be its midpoint. Then then COTf3(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. COTf3(P).
Conversely: For any point X, if the line through X normal to X.COTf3(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 COinscribed triangle , O its circumcenter, N its ninepoint circle. Then O lies on the circle COTf3(N).
2. Let Tr be any triangle, O its circumcenter. For any conic CO circumscribed to Tr, the point COTf3^{1}(O) lies on the ninepoint circle of Tr.
3. Let CO be a parabola, X1, X2 any points, Mi the midpoint of Xi . COTf3(Xi). Then the line M1M2 is the axis of the parabola.
4. For any quadrangle Q inscribed in any conic CO, the COTf3 image of QAP2 is QAP4.
5. For any pentangle PA = P1P2P3P4P5 let QAi the Component Quadrangle, obtained by ignoring the point Pi. Let Hi = QAiP2, Oi = QAiP4. Then the pentangles H1H2H3H4H5, O1O2O3O4O5 are negatively similar, the similarity being COTf3 : 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 COTf3.
• For any chord P1P2 of any conic CO, let M be its midpoint. Then then COTf3(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. COTf3(P).
Conversely: For any point X, if the line through X normal to X.COTf3(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 COinscribed triangle , O its circumcenter, N its ninepoint circle. Then O lies on the circle COTf3(N).
2. Let Tr be any triangle, O its circumcenter. For any conic CO circumscribed to Tr, the point COTf3^{1}(O) lies on the ninepoint circle of Tr.
3. Let CO be a parabola, X1, X2 any points, Mi the midpoint of Xi . COTf3(Xi). Then the line M1M2 is the axis of the parabola.
4. For any quadrangle Q inscribed in any conic CO, the COTf3 image of QAP2 is QAP4.
5. For any pentangle PA = P1P2P3P4P5 let QAi the Component Quadrangle, obtained by ignoring the point Pi. Let Hi = QAiP2, Oi = QAiP4. Then the pentangles H1H2H3H4H5, O1O2O3O4O5 are negatively similar, the similarity being COTf3 : 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 COTf3.
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
SchmidtConstruction for COTf3(P):
This construction is based upon Eckart Schmidt’s construction at Ref34, QFG#2811.
1. Let L be the polar COTf1 of P.
2. Let Lp be the line through P perpendicular to L.
3. Let Lc be the reflection of P.COP1 in the principal axis of CO.
4. Finally COTf3(P) = the intersection point of Lp and Lc.
SchmidtConstruction for COTf3^{1}(P):
This construction was found by Eckart Schmidt. See Ref34, QFG#2811.
1. Let L be the polar COTf1 of P.
2. Let Lr be the reflection of L in the principal axis of CO.
3. Let Lp be the line through P perpendicular to Lr.
4. Let Lc be the reflection of P.COP1 in the main axis of CO.
5. Finally COTf3^{1}(P) = the intersection point of Lp and Lc.
This construction was found by Eckart Schmidt. See Ref34, QFG#2811.
1. Let L be the polar COTf1 of P.
2. Let Lr be the reflection of L in the principal axis of CO.
3. Let Lp be the line through P perpendicular to Lr.
4. Let Lc be the reflection of P.COP1 in the main axis of CO.
5. Finally COTf3^{1}(P) = the intersection point of Lp and Lc.
KeizerConstruction for COTf3(P):
This construction is based upon Bernard Keizer’s construction at Ref34, QFG#2826.
1. Let P1 be the inverse of P in the Orthoptic Circle COCi2
2. Let P2 be the inverse of P1 in the Circle COCi1 with the foci as diameter
3. COTf3(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 (a^{2}  b^{2})^{1/2}.
This construction is based upon Bernard Keizer’s construction at Ref34, QFG#2826.
1. Let P1 be the inverse of P in the Orthoptic Circle COCi2
2. Let P2 be the inverse of P1 in the Circle COCi1 with the foci as diameter
3. COTf3(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 (a^{2}  b^{2})^{1/2}.
KeizerConstruction for COTf3^{1}(P):
This construction is based upon Bernard Keizer’s construction at Ref34, QFG#2826.
1. Let P1 be the inverse of P in the Circle COCi1 with the foci as diameter
2. Let P2 be the inverse of P1 in the Orthoptic Circle COCi2
3. COTf3^{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 (a^{2}  b^{2})^{1/2}.
This construction is based upon Bernard Keizer’s construction at Ref34, QFG#2826.
1. Let P1 be the inverse of P in the Circle COCi1 with the foci as diameter
2. Let P2 be the inverse of P1 in the Orthoptic Circle COCi2
3. COTf3^{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 (a^{2}  b^{2})^{1/2}.
Another Construction for COTf3^{1}(P):
Next construction for COTf3^{1}(P) is based upon the construction of 5PsTf3, 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 P_{i}.P_{j}.P_{k} (O_{ijk} = center circumcircle P_{i}.P_{j}.P_{k}), where (i,j,k) are different numbers from (1,2,3,4,5).
4. Let Ci_{1234} be the circle through P_{123}, P_{124}, P_{134}, having center O_{1234}.
Let Ci_{1235} be the circle through P_{123}, P_{125}, P_{135}, having center O_{1235}.
These circles have point P_{123} in common.
5. COTf3^{1}(P) will be the 2nd intersection point of Ci_{1234} and Ci_{1235}.
Next construction for COTf3^{1}(P) is based upon the construction of 5PsTf3, 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 P_{i}.P_{j}.P_{k} (O_{ijk} = center circumcircle P_{i}.P_{j}.P_{k}), where (i,j,k) are different numbers from (1,2,3,4,5).
4. Let Ci_{1234} be the circle through P_{123}, P_{124}, P_{134}, having center O_{1234}.
Let Ci_{1235} be the circle through P_{123}, P_{125}, P_{135}, having center O_{1235}.
These circles have point P_{123} in common.
5. COTf3^{1}(P) will be the 2nd intersection point of Ci_{1234} and Ci_{1235}.
Construction for COTf3^{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 Zhomothety.
2. Construct the centroid QAP1 of the quadrangle QA = A1A2A3A4.
3. Then P ~ QAP4 and COTf3^{1}(P) ~ QAP2 will be the reflection of P on QAP1 (QACentroid).
(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 Zhomothety.
2. Construct the centroid QAP1 of the quadrangle QA = A1A2A3A4.
3. Then P ~ QAP4 and COTf3^{1}(P) ~ QAP2 will be the reflection of P on QAP1 (QACentroid).
This construction is based on what can be considered the main property of COTf3 / COTf3^{1}:
they swap QAP2 and QAP4 for all quadrangles QA inscribed in CO (see the final Table).
they swap QAP2 and QAP4 for all quadrangles QA inscribed in CO (see the final Table).
Construction, when knowing COTf3(P) or COTf3^{1}(P), of their inverse:
The construction of COTf3^{1}(P) is, knowing the construction of COTf3(P), very simple.
1. Let Cix be the circle through P with center COP1.
2. COTf3R (P) is the inverse of COTf3(P) wrt Cix.
Relationship with Frégier’s Point
For each point P on any Conic COTf3(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 Ref13, keyword “Frégier’s Theorem”.
Frégier’s Point is only defined for points on a conic. COTf3 is defined for all points in the plane.
This makes COTf3 a generalization of Frégier’s Point.
For each point P on any Conic COTf3(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 Ref13, keyword “Frégier’s Theorem”.
Frégier’s Point is only defined for points on a conic. COTf3 is defined for all points in the plane.
This makes COTf3 a generalization of Frégier’s Point.
Relationship with Orthotransversal Line and Pedal Circle wrt a COinscribed Triangle
For any COinscribed triangle Tr and any P on CO:
• COTf3(P) lies on the orthotransversal line of P w.r.t. Tr
• COTf3^{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 COTf3(P), the two pedal circles of P meet in COTf3^{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 Ref59, AoPS, VU Thanh Tung, 2015, June 26 and Ref34, QFG#1233).
(2) the 4 pedal circles have a common point. See Ref59, AoPS, Luis Gonzales, 2011, August 7 and Ref34, QFG#1218).
For a description of the connection of both properties see Ref34, QFG#1252, Benedetto Scimemi, August 2015).
For any COinscribed triangle Tr and any P on CO:
• COTf3(P) lies on the orthotransversal line of P w.r.t. Tr
• COTf3^{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 COTf3(P), the two pedal circles of P meet in COTf3^{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 Ref59, AoPS, VU Thanh Tung, 2015, June 26 and Ref34, QFG#1233).
(2) the 4 pedal circles have a common point. See Ref59, AoPS, Luis Gonzales, 2011, August 7 and Ref34, QFG#1218).
For a description of the connection of both properties see Ref34, 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 COTf3[(x:y:z)] will have these barycentric coordinates:
( X (+a^{2} (x + y + z)  b^{2} (x + y  z)  c^{2} (x  y + z))  2 a^{2} (Y z + y Z) :
Y ( a^{2} (x + y  z) + b^{2} (x + y + z) + c^{2} (x  y  z))  2 b^{2} (X z + x Z) :
Z ( a^{2} (x  y + z) + b^{2} (x  y  z) + c^{2} (x + y + z))  2 c^{2} (X y + x Y) )
( X (+a^{2} (x + y + z)  b^{2} (x + y  z)  c^{2} (x  y + z))  2 a^{2} (Y z + y Z) :
Y ( a^{2} (x + y  z) + b^{2} (x + y + z) + c^{2} (x  y  z))  2 b^{2} (X z + x Z) :
Z ( a^{2} (x  y + z) + b^{2} (x  y  z) + c^{2} (x + y + z))  2 c^{2} (X y + x Y) )
The barycentric coordinates of COTf3^{1}[(x:y:z)] are much longer.
Cartesian Coordinates:
Cartesian Coordinates for both of COTf3 and COTf3^{1} are very easily written if one represents CO by P = [a cos ∂, b sin ∂] for ellipses or [a/cos ∂, b tan ∂] for hyperbolas.
Then COTf3([x,y]) = [x,  y]((a^{2} b^{2})/(a^{2} + b^{2}))^{±1}.
If CO is the parabola y = x^{2} then COTf3( [x,y] ) = [x, y+1].
Cartesian Coordinates for both of COTf3 and COTf3^{1} are very easily written if one represents CO by P = [a cos ∂, b sin ∂] for ellipses or [a/cos ∂, b tan ∂] for hyperbolas.
Then COTf3([x,y]) = [x,  y]((a^{2} b^{2})/(a^{2} + b^{2}))^{±1}.
If CO is the parabola y = x^{2} then COTf3( [x,y] ) = [x, y+1].
Examples of COTf3 and COTf3^{1} Transformations:
The action of COTf3 can be read from left to right.
The action of COTf3^{1} can be read from right to left.
*) QALx = line through QAP6 parallel to QAP2.QAP23
QALy = line through QAP23 parallel to QAL1
The action of COTf3 can be read from left to right.
The action of COTf3^{1} can be read from right to left.
Reference Conic 




Triangle 
Steiner Inellipse / Steiner Circumellipse 
X(5988) 
X(1) 

X(2) 
X(2) 
X(2) 

X(114) 
X(3) 
X(1352) 

X(98) 
X(4) 
X(6776) 

X(6036) 
X(5) 
X(182) 

X(115) 
X(6) 
X(2549) 

X(1281) 
X(8) 

X(10) 
X(3923) 

X(11) 
X(5091) 

X(6109) 
X(13) 
X(10654) 

X(6108) 
X(14) 
X(10653) 

X(6115) 
X(15) 

X(6114) 
X(16) 

X(1513) 
X(98) 


X(230) 
X(115) 



Quadrangle 
Any Conic circumscribing Reference Quadrangle 

QALx *) 

QALy *) 



Pentangle 
Circumscribed Conic of Pentangle 

5PsPx **) 
5PsPy **) 
*) QALx = line through QAP6 parallel to QAP2.QAP23
QALy = line through QAP23 parallel to QAL1
**) 5PsPx = Conic Center of the five 5Pversions of QAP2
5PsPy = Conic Center of the five 5Pversions of QAP4
Warning: In a Pentangle the Scimemi Transformation COTf3 wrt the circumscribed conic
5PsPy = Conic Center of the five 5Pversions of QAP4
Warning: In a Pentangle the Scimemi Transformation COTf3 wrt the circumscribed conic
is identical to the inverse of the 5PsTf3 Transformation.
These results were found by Eckart Schmidt, Benedetto Scimemi and the author of EQF/QPG.
There will be probably many more appealing examples.