Affine Theory of Triangle Conics
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This movie shows how a circumconic and its center change as the perspector moves. If the perspector is in a colored region, the center is in the other region of the same color. |
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These pages tell the properties of conics most relevant to triangle geometry. While conics in general are well known, those particularly related to the triangle seem ill studied. Clark Kimberling's list of points is over three thousand. Bernard Gibert's list of cubics is over 400. But the two best lists of conic I have found list 20 or so. Here is Wilson Stothers' which lists 8 inconics and 5 circumconics. MathWorld lists 19, although it is a very strange list. Named conics, naming being an indication of importance, are few.
It is strange omission. I think that conics have major structural significance in the plane of the triangle. This webpage attempts to show this.
A point is taken as known when its method of construction is known. If the method of construction is symmetric for all permutations of the triangle vertices, the point is called a center.
Notation:
If P = ( l : m : n ) is a general point. mP = ( m+n : n+l : l+m ) and dP = ( m+n–l : n+l–m : l+m–n ) represent the point P constructed to the medial and antimedial (or "dilated") triangle respectively. If a point is constructed symmetrically in its coordinates, we often show only the second coordinate. For example, mP = ( : n+l : ) or even mP = : n+l : .
Note: m, the medial operation on points, is constructed by reflecting P through G and then taking half the distance. It is the similarity with ratio –1/2. The dilated operation d has the similarity with ratio –2. This pictures shows the relationship between P, G, mP and dP.
tP = ( 1/l : 1/m : 1/n ) is the isotomic conjugate of P.
Note: t is self-inverse, a special operation known as an involution, which divides the plane into distinct regions. Here is a picture of the triangle plane divided into regions by the isotomic conjugate t.
P2– = ( : m2 – nl : ) is the Steiner inverse of P, the inverse in the Steiner circumellipse.
nP is the refection of P through G. If P is on the Steiner ellipse, nP is its antipode.
All these operations preserve affine symmetry in that the stated relationships survive an affine transformation.
The term "conjugate" will always refer to "isotomic conjugate."
The conjugate is defined as an operation on points. The conjugate of a line is taken point by point, and unless the line goes through a vertex, it is a circumconic.
(l:m:n) represents the barycentric coordinates of a point. [l:m:n] = lx+my+nz the barycentric coordinates of a line. Barycentric coordinates are built on affine symmetry and are essential to this investigation. A complete listing of notation is here.
Constructive notation:
A—B indicates the line between points A and B. a•b indicates the point of intersection of lines a and b. These, along with the affine invariant operations, give us an invariant and constructive way of specifying points. For example t(∞•(G—P)) indicates the isotomic conjugate of the intersection of the GP line with the line at infinity. As such it represents a point on the Steiner ellipse.
The constructive name of a point is affine invariant.
Duals:
The dual of (l:m:n) is the line [l:m:n] with the same coordinates. The dual of P is notated as ~P. It is the tripolar line of the isotomic conjugate. The dual relation is affinely invariant whereas the tripolar relation is projectively invariant. The dual relation is a proper dual.
The dual of a vertex is the corresponding edge. The dual of a point in any of the 4 projective regions lies in the other three and correspondingly the dual of a line through 3 of them is in the 4th.
The dual of G is the line at infinity. In constructive notation this is ~G = ∞.
The dual of a line through G is a point on the line at infinity.
In constructive notation, the properties of duality are ~(A—B) = ~A•~B, that the dual of a line between two points is the intersection of the duals of the points; and ~(a • b) = ~a — ~b, or that the dual of an intersection is the line between the duals.
A nice example of the duality property: ~(G—P) = ∞•~P; i.e., that the dual of a line through the centroid is the endpoint of the dual of P.
The dual of a line is a point and vice-versa. The dual of a point on a conic is a tangent line to another conic. We say that the dual of a point conic is a line conic, defined as the envelope of tangent lines.
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G
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centroid
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~G = ∞
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[1:1:1]
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dual of centroid is the line at infinity and vice-versa
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A = ~a
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a = ~A
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the dual of a point is a line with the same coordinates
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l/x + m/y + n/z = 0
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dual of point on inconic = tangent to circumconic
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Dual of P-circumconic
= tP inconic |
√lx + √my+√nz = 0
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dual of point on circumconic = tangent to inconic
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~(A—B) = ~A•~B
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(:n1l2–l1n2:)
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statement of duality for points
A = (l1:m1:n1); B = (l2:m2:n2) |
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~(a • b) = ~a — ~b
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[:n1l2–l1n2:]
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statement of duality for lines
a = [l1:m1:n1]; b = [l2:m2:n2] |
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How to understand the affine plane of the triangle.
In the affine plane we get one line for free, the line at infinity, to which we add 3 points, the triangle vertices. We give the triangle vertices coordinates (1:0:0), (0:1:0), and (0:0:1). Each point in the plane then has a unique triple of constant coordinates. The centroid is (1:1:1). (1:4:1) is the midpoint of a median. This structure of points is rigid and the same for each non-degenerate triangle.
The affine structures we will describe each belong to each affine point; e.g., a point has a dual line, a line connecting it to G, an inscribed and circumscribed conic for which it is perspector, an hyperbola with perspector at infinity, an inparabola with perspector on the Steiner ellipse, and more. Metric points move on top of the affine background of points. When they are over an affine point, they "borrow" the line, point, and conic structures of that point.
When we say "the circumconic with perspector I," the incenter, we really mean the circumconic of the affine point on which the incenter currently resides.
In a sense the Euclidean triangle plane is "locally affine."
Using the above operations, consider P,
and then its dilated (or antimedial): dP,
and then the conjugate of this: tdP,
and finally the medial: mtdP.
Four points; six pairs of points, each pair a significant triangle and/or conic relationship.
One pair (dP, tdP) is conjugate, defining a Mineur conic (explained below);
two pairs (P, dP and mtdP, tdP) have the medial/dilated relation and are each collinear with G, the centroid;
three have the perspector/center relation, one for a circumconic (P, mtdP) and two (P, tdP and dP, mtdP) for an inconic.
But most significant of all, all four points are on the Mineur conic defined by the conjugate pair. The four points are also on, and define, the affine parallelogram for the conjugate pair.
I do not think it is possible for four points to carry more structure.
Figure: The pattern for a generic point P is given, then applied to two specific points P = K, the symmedian point, and P = Ix (x = o,a,b,c), one of the four incenters. Red is the color of weak points and blue that of strong points. Points on the K schema are on the Jerabek hyperbola; those of the Ix schema are on the four Feuerbach hyperbolas.
Point Relationships:
Here are these points in their geometrical context, showing the affine parallelogram , the Mineur conic on which they lie, and the points on the Steiner ellipse with which they are collinear. This is a very important picture.
The center of a circumconic is the medial of the conjugate of the dilation of the conic perspector, or C = mtdP. mtd has the special property that it is an involution with mtd mtd = 1. An involution divides the triangle plane into regions; here is a picture for this one.
This picture shows the plane divided into paired regions, indicated by color. The boundary lines are the sides of ABC, its medial triangle mAmBmC, the inner Steiner ellipse and the line at infinity. If P is in a colored region, mtdP is in the other one of the same color. The inner Steiner ellipse is the mtd of the line at infinity. Later there is a movie illustrating the relation between the perspector and center of a circumconic.
I learned from François Rideau that mtd is the isotomic with respect to the medial triangle, as the following picture shows.
Figure: The plane is divided into regions separated by the edges of a triangle and its medial triangle. The regions are colored and numbered as corresponding. If the perspector of a conic is in a region of a certain color, the center of the conic is in the other region of that same color (and number). If the perspector is inside the inner Steiner ellipse or the center inside the medial triangle, the conic is an ellipse. If the perspector is inside the medial triangle, so is the center. Each numbered point maps to the other point with the same number.
Algebraic considerations; algebraic isolation
Center, perspector, axes, asymptotes, foci, directrix.
The triangle vertex coordinates exist in a certain computational field, often the rationals since we tend to choose A, B, C to have integer coordinates. But computations often require the field to be extended. For example edgelengths are square roots, so that coordinates that use edgelengths in their construction must exist in a different and larger field, the rationals extended by those 3 distances. Similarly if a circle intersects a line, coordinates of the intersections are likely to have additional square roots, requiring another field extension. Of course many constructions require circles.
Point and line coordinate computations are done in certain algebraic fields, in many cases the field of the triangle vertices. When circles or conics are present, these fields often have to be extended. Points computed in algebraically extended fields often have properties different from points computed in a different field.
Perspector and Center: these exist in an algebraically harmonious world. Whatever field extensions are required to state the perspector are those for the center.
Asymptotes require the roots obtained from the intersection of a line with the Steiner ellipse, which requires a root field extension incommensurate with other points. For this reason asymptotes are seldom on or concurrent with any other structures.
Axes also require a new, different field extension.
Vertices, Foci, and directrices. The vertices are the intersection of the axes with the conic. Each axis requires a field extension. The foci are on the axes these require a second square root. It is very difficult for these to communicate with the general run of points.
The projective generation of conics can be found here.
Of circumconics
The isotomic conjugate of points on a line not through a vertex lies on a circumconic whose perspector is the dual of the line. The center is the mtd of the perspector.
The center of a conic is the pole of the line at infinity. Dependence on the line at infinity makes the center an affine property, preserved under affine transformations.
For circumconic with perspector P, the center is the point mtdP = (: m(n+l–m) :) This operation, the medial of the conjugate of the dilation (antimedial) is an involution, discussed above.
The circumparabola is the intermediate case between ellipses and hyperbolas. The center of a parabola is at infinity, so that the perspector is on the medial of the conjugate of the line at infinity, which is the Steiner inellipse. If the perspector is inside the Steiner inellipse, the conic is elliptical; if outside, hyperbolic.
Axes and asymptotes of circumconics
Algebraic considerations: The axes of a conic and asymptotes of an hyperbola occur in pairs. Algebraically these objects include an adjoined square root. This gives them a different algebraic nature from the points, such as the perspector and center, that create them. The axes and asymptotes of conics generated from known points and lines only rarely go through more than one of these known points (the center).
Partly due to the work of Wilson Stothers, this topic has recently advanced.
See here for an affine invariant construction of the asymptotes of a circumconic. The asymptotes are isotomic lines (one of Wilson's discoveries). Go here for a nice picture.
If P = (l:m:n) is the perspector of a circumhyperbola, then the equations of the asymptotes are
( : m ( n2 – lm + l2 – mn ± (n–l) Z )2 : ) where Z2 = l2+m2+n2–2mn–2nl–2lm.
Note that Z is imaginary when the perspector is inside the Steiner inellipse whose equation is x2+y2+z2–2yz–2zx–2xy = 0.
See here for a maximally affine construction of the axes of a circumconic. The axes of a P-circumconic are parallel to the asymptotes of a circumhyperbola with perspector ~dP • ~D, on the Polar axis ~D.
Just as an inconic can be constructed from one focus, a circumconic can be constructed from one asymptote.
A very special circumconic, the Steiner ellipse
The affine plane comes with one very special line, the one at infinity. If we add a triangle to the plane, other special objects are created. We get the centroid and the Steiner ellipse. These three objects are related. The dual of the centroid is the line at infinity; the conjugate of the line at infinity is the Steiner ellipse. The center of the ellipse is the centroid.
Special points
The conjugate of a line is a conic. A general line crosses each edge of ABC. These three points map to the vertices under conjugation. Hence that the conjugate of a line is a circumconic.
The conjugate of the line at infinity is the Steiner ellipse so that a line endpoint maps to the fourth intersection of the conic generated by the line with the Steiner ellipse.
The points where a line crosses the Steiner ellipse are particularly important. The conjugates of points on the Steiner ellipse are at infinity. These points are the ends of the asymptotes of the conic. Hence if a line intersects the Steiner ellipse twice, the generated conic is an hyperbola. If once, a parabola.
The dual of a circumconic is an inconic
The dual of a circumconic of perspector P is an inconic of perspector tP, center mP.
Figure: Point Q is on a circumconic, perspector P and center mtdP. Its dual ~Q is a line tangent to an inconic, perspector tP and center mP. The inconic is the envelope of the tangent lines.
There is a nice formula relating a point on a circumconic to the corresponding point (the point of tangency) on its dual. Let the circumconic, perspector P = (l:m:n) have equation l/x + m/y + n/z = 0. Its dual inconic has equation √( lx ) + √( my ) + √( nz ) = 0
If Q = ( : y1 : ) is on this conic then l/x1 + m/y1 + n/z1 = 0. The dual of Q is the line x1 x + y1 y + z1 z = 0. The point (: m/y12 :) is manifestly on this line as well as the inconic. So this must be the point of tangency.
Hence the transformation : y : –> : m/y2 : takes a point on a circumconic to the corresponding point of tangency on its dual inconic.
As an aside (: m/y1 :) is on the line at infinity.
The circumconic and the inconic with the same perspector can never intersect.
Proof:
(l2x2 + m2y2 + n2z2 – 2 mn yz – 2nl zx – 2lm xy) + ( 2 lmn (yz/l + zx/m + xy/n) = l2x2 + m2y2 + n2z2
dual inconic circumconic imaginary diagonal conic
which show that the intersections of the dual inconic and circumconic lie on a conic with only imaginary points.
The fourth intersection for circumconics
The isotomic of the point of intersection of two lines is on the two circumconics generated by the lines. The dual of this intersection is the fourth tangent (other than the edges a, b, c) to the dual inconics generated by the lines. The fourth tangent of a circumconic to the Steiner ellipse is the dual of the infinite point on the generating line.
The line between corresponding points on two circumconics goes through their 4th intersection. Here is a picture of the Kiepert, Jerabek, and Feuerbach hyperbolas, all of which have a fourth intersection at H, with corresponding points joined by lines through H. The pages dedicated to these conics have more detail.
Each collinearity in the circumconic world implies a concurrence in the dual inconic world. We deal with that next.
Perspector and center of inconics
An inconic with perspector P has center mtP. If the perspector is inside/on/outside the Steiner circumellipse, the inconic is an ellipse, parabola, hyperbola. If a parabola its focus is on the circumcircle. Inparabolas are special cases which are always associated with circumconics whose perspectors are at infinity.
The foci of an inconic are isogonally conjugate. The midpoint of the foci is the conic center, from which the perspector is known. Hence an inconic is known if a single focus is known.
The isogonal conjugate of line from a point to its isogonal conjugate produces a special type of circumconic that I call a Mineur conic. Each focal conic thus has an associated circumconic formed as the dual of its axis. The MacBeath inconic, which has O and H as foci, is thus related to the Jerabek hyperbola.
The asymptotes of this circumconic intersect the focal inconic at its vertices.
If the perspector of an inconic is weak, there are four versions of the conic. If these points do not lie on a line or conic, there is a second, mated inconic related to the first. Here is an example of the mated structure for the incircles and the Mandart inconics.
For the incircle the original Gergonne point Go is the central perspector, but the other three Ga, Gb, and Gc are together a central object, being in perspective at the Nagel point. Repeating this process for each region and each Gergonne point, we generate the Gergonne-Nagel desmic system.
For the original Mandart conic, the original Nagel point No is the central perspector, but the other three Na, Nb, Nc are together a central object, being in perspective at the Gergonne point. Repeating this process for each region and each Nagel point, we generate the same Nagel-Gergonne desmic system.
The intersection of two inconics produces desmic structure, see here for the incircle-Mandart example.
The fourth tangent for inconics
Just as two circumconics meet at the three vertices and on other point, the fourth intersection, so two inconic are each tangent to the three edges of ABC and to one other, the fourth tangent.
In the special case of weak inconics there are 4 conics and 6 tangents, which John Conway colorfully calls "the missing tangents," produce very special structures. See here for an example using the Mandart inconics. Also see here for a description of inconics that share the same 4th tangent.
The four intersections of two inconics
Inconics meet four times. One of these points is a centrally defined point. The other three are centrally defined as a group and are in perspective to ABC at the desmic mate of the central point leading to a desmic system of quartile points. Note that the formulas involve square roots. This means that often these points will have a different algebraic nature and note that there are four of them, taking all possible signs of the square roots. A more detailed description can be found here.
If (l:m:n) and (L : M : N ) are the perspectors of two inconics then the two central points are
of which the first is the central intersection and the second is the perspector of the other three. A picture follows.
Figure: The four intersections are of two inconics form a mated desmic system. The four intersections are green and labeled as a quartile set. "o" is the central point and the red "o" is the central mated point found as the perspector of the other three points. The desmic system is shown as a projective cube. The blue point is the desmon of the system. The picture here is taken from the tFx inconics webpage.
Switched circumconics
The operation mtd takes the perspector of a circumconic to its center. This operation is an involution in that if it is done twice, the result is unity; i.e., mtdmtd = 1. Hence given a circumconic with perspector P and center Q, there is another circumconic with perspector Q and center P. I call this the switched circumconic. The original circumconic is the conjugate of the dual of its perspector. The switched circumconic is the conjugate of the dual of the center of the original circumconic.
If the circumconic is an hyperbola,
there is a nice relationship between the dual of the center of a circumhyperbola, its dual inconic, and the switched circumconic.
1. The duals of the asymptotes of the circumhyperbola are on the line joining the center of the hyperbola and are conjugate.
Proof: Call the asymptotes a1 and a2. They meet at the center, so we write center = a1•a2. The dual of this is ~center = ~a1— ~a2. They are conjugate which is shown above.
2. These points are on the dual inconic; the lines from G to these points are tangent to the dual inconic.
Proof: ∞•a1 is the endpoint of an asymptote and is on the hyperbola. Hence its dual is tangent to the dual inconic. We have ~(∞•a1) = ~∞— ~a1 = G— ~a1.
3. The switched conic goes through these conjugate points and is thus a Mineur conic.
Proof: The switched circumconic is generated as the isotomic conjugate of the dual of the center of the original hyperbola. Since these points are isotomic, they lie on the switched circumconic. The points m~a1 and m~a2 are also on the conic.
Two circumconics, two inconics for a point
Given a point P, not at a vertex or G, there are two natural lines: G—P and ~P, the dual of P. The isotomic of each line give one of the two types of conics. The isotomic of ~P is a circumconic with perspector P. The circumconic of G—P is an hyperbola with perspector the infinite point on ~P. The two inconics consist of one with perspector tP, center mP, and a parabola with perspector t(∞•(~P)) on the Steiner ellipse. The following picture shows this situation with P = Io.
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circumhyperbola (analogous to Kiepert)
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inparabola, dual of circumhyperbola
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Classifying Circumconics and inconics
The affine method of generating conics creates them as conjugates of lines where the perspector of the conic is the dual of the line. Hence differences in conics originate with the lines used to generate them.
Type I. From a line through G:
circumconics through G and with infinite perspector;
inscribed parabolas;
circumscribed parabolas
Examples: The Kiepert hyperbola and parabolas, weakened Kiepert hyperbola and Yff parabola, the Eulerian hyperbola and parabolas.
Type I is always generated as the conjugate of a line through the centroid G. The conic will also go through G.
Let this line be G—P = [ : n–l : ] for some P = (l:m:n). It intersects the Steiner ellipse twice so the generated conic meets the line at infinity in two different directions. These are asymptote directions so this conic will always be an hyperbola. Its perspector, the dual of the line through G, is at infinity.
If P = K, this conic is the Kiepert hyperbola, a model for the properties of this whole class.
The line G—P is associated with the line ~P in several ways. The dual of G—P is the endpoint of ~P. In constructive notation this is written ~(G—P) = ∞•~P. The dual of any point on G—P is parallel to ~P. Hence the conic generated by each lines will interact and should be considered together. Along with the hyperbola will include the P-circumconic. If P is inside the Steiner ellipse, this circumconic will be an ellipse.
1. Definition: This hyperbola is the isotomic conjugates of the line G—P. Its perspector is the dual of G—P and the infinite point on ~P. In constructive notation this is ~(G—P) = ∞•~P = ( : n–l : ). The equation of this hyperbola is
(m–n)/x + (n–l)/y + (l–m)/z = 0.
The affine approach used here derives a great number of properties with little effort, as well as puts this hyperbola in a larger context of triangle conics.
Since the perspector is at infinity, its conjugate is on the Steiner ellipse.
2. G—P and ~P meet at mP2– = ( : n2 + l2 – mn – lm : ), whose conjugate tmP2– is the fourth intersection of the hyperbola with the P-circumconic. This point is the opposite in the P-circumconic center of the conjugate of the perspector, which is on the Steiner ellipse. In constructive notation, this is tmP2– = nt(∞•~P).
These properties are analogous to those of the Steiner and Tarry points with respect to the circumcircle and the Kiepert hyperbola.
3. Asymptotes: The directions of the asymptotes a1 and a2 are the conjugates of the meets of G—P with the Steiner ellipse. The tripolars of these directions are parallel to the asymptotes through G.
An easy construction of the asymptotes is to connect the hyperbola center, which is on the Steiner inellipse, to the meets of G—P with the Steiner inellipse.
The asymptotes of a circumconic are conjugate (discovered by Wilson Stothers). I think this fact is simply amazing.
The intersections of G—P with the Steiner ellipse are (: n–l ± Q :), where Q = √(l2 + m2 + n2 – mn – nl – lm). Their conjugates are the endpoints of the asymptotes. The asymptotes are derived here.
The asymptote equations are [ :(n–l)( n+l–2m ± Q)2: ]. We can verify from this that these are conjugate, so the that asymptotes of the circumconic are isotomic.
4. The axes are parallel to the asymptotes of the rectangular hyperbola with perspector ~dP•~D, generated by the line through D, parallel to ~P, where D = tH = dK is known as the "Desmon" point. The equations for the axes are decently ugly.
5. The line G—P contains G, P, mP, dP so that this hyperbola goes through G, tP, tmP, and tdP. A chart with more points is given below.
For the Kiepert hyperbola (P = K), these points are G, R = tK, tmK, and H.
6. The dual of a point on G—P is parallel to ~P. Hence the tripolar (the dual of the isotomic conjugate) of any point on the hyperbola is parallel to ~P.
7. Center: The center for a circumconic is the point mtdP and, for this hyperbola, is (∞•~P)2 = (: (n–l)2 :) which is on the Steiner inellipse. The center is the medial of t(∞•~P) = (: 1/(n–l) :), the fourth intersection of the P-circumconic with the Steiner ellipse.
8. The dual inparabola: The dual of a circumhyperbola with perspector at infinity is an inparabola with perspector t(∞•~P) = (: 1/(n–l) :), the isotomic conjugate of the hyperbola perspector, which is on the Steiner ellipse. This point is the analogue of the Steiner point for this system. The parabola axis is parallel to ~P.
Note: As the Steiner point is the intersection of the Steiner ellipse and the circumcircle, so t(∞•~P) = (: 1/(n–l) :) is the intersection of the Steiner ellipse and the P-circumconic.
There is a nice formula relating a point on a circumconic to the corresponding point (the point of tangency) on its dual. Let the circumconic, perspector P = (l:m:n) have equation l/x + m/y + n/z = 0. Its corresponding inconic has equation √( lx ) + √( my ) + √( nz ) = 0
If Q = ( : y1 : ) is on this conic then l/x1 + m/y1 + n/z1 = 0. The dual of Q is the line x1 x + y1 y + z1 z = 0. The point (: m/y12 :) is manifestly on this line as well as the inconic. So this must be the point of tangency. Hence the transformation (: y :) –> (: m/y2 :) takes a point on a circumconic to the corresponding point of tangency on its dual inconic. For the Kiepert hyperbola to the Kiepert parabola this is (:y:) -> (: (c2 – a2)/y2 :)
9. Focus, axis, and directrix: The focus of the dual inparabolas is the "pro" operation of its perspector. The pro operation takes the Steiner ellipse to the circumcircle. In this case it takes the perspector to the focus (: b2/(n–l) :). The axis is the line parallel to ~P though this point. The directrix is the line through H perpendicular to ~P.
10. The switched circumparabola: The center and perspector of a circumconic can be switched, the center becoming the perspector and vice-versa. Since its perspector is on the inner Steiner ellipse, this conic is a circumparabola with perspector mtdP, and center P at infinity. Its axis is also in the direction of ~P, the endpoint of which is the perspector.
Wison Stother has a nice page on circumparabolas.
The two parabolas meet on the dual of the hyperbola center. The duals of these two intersections are the corresponding hyperbola asymptotes and are conjugate. The lines from G to the two intersections are tangent to the dual inparabola.
Proof: ∞•a1 is the endpoint of an asymptote and is on the hyperbola. Hence its dual is tangent to the dual inconic. We have ~(∞•a1) = ~∞— ~a1 = G— ~a1.
Since the circumparabola goes through the conjugate dual points, it is a Mineur conic (see below). This means that the medials of these dual points are also on the parabola.
11. Fourth intersections: The isotomic conjugates of the intersection of ~P and G—P is tmP2– and is the fourth intersection of the hyperbola with the P-circumconic. This is the analogue of the Tarry point for this configuration. The fourth intersection with the Steiner ellipse is the isotomic of the infinite point on G—P: t(∞•(G—P)) = ( : 1/(n+l–2m) : ), the reflection of the inparabola perspector in G.
12. The dual of the inparabola perspector is tangent to Steiner inellipse at the hyperbola center and goes through center of the Mineur conic defined as the isotomic of the line dP—tdP. (If P = K, this would be the Jerabek hyperbola).
13. G tdP U tmP2–, all on the conic, form a parallelogram, where U is the hyperbola intersection with the Steiner ellipse, and tmP2– the intersection with the P-circumconic. The sides of this parallelogram are parallel to the G—tdP line or to second axis of the affine parallelogram of tdP.
Figure: This figure shows conics derived from the natural lines of Io: ~Io, its dual, and G—Io (shown as dark, bold lines). The circumconics are in blue, one with perspector Io and the other, an hyperbola having its perspector the infinite point ~Io•∞. The dual inconics is in red, a parabola (the Yff parabola) with perspector t(~Io•∞). The isotomics of Q1 and Q2, the intersections of G—Io with the Steiner ellipse, give the directions of the asymptotes of the circumhyperbola. The blue parallelogram is described in the item above.
14. Tangents at important points: G—P is tangent to the hyperbola at G. The line parallel to G—P through t(∞•(G—P)) is tangent at t(∞•(G—P)). This line meets the line parallel to ~P through t(∞•~P) meets on the Steiner ellipse at the barycentric product of t(∞•~P) and t(∞•(G—P)). The parabola in 10 goes through this point.
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point
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tangent
|
coordinate of tangent line
|
|
G
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G—P
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[ : (n–l): ]
|
|
tdP
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[ : smm (n–l) : ]
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|
|
tP
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[ : m2 (n–l) : ]
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|
|
A
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C-edge of preCevian triangle of perspector
A—(CtdP—BG•CG—BtdP) |
[ 0 : (l–m) : (n–l) ]
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|
B
|
B-edge of preCevian triangle of perspector
B—(CtdP—AG•CG—AtdP) |
[ (l–m) : 0 : (m–n) ]
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|
C
|
C-edge of preCevian triangle of perspector
C—(BtdP—AG•BG—AtdP) |
[ (n–l) : (m–n) : 0 ]
|
The line P—(∞•~P)2 (parallel to G—tmP2–) intersects the hyperbola at the analogue of the Fermat points. The tangents at these points are parallel to the analogue of the Euler line G—tdP.
Tangents to the hyperbola at A, B, C are the ex-Cevian lines of the perspector but see below for another nice construction of the tangent lines. The tangent at G is G—P.
A second way to get the tangents is from the point table given directly below. A third way is by linearizing at the point of tangency. A fourth way is Pascal's Theorem (see example here).
Points on the hyperbola and its dual
As shown above the transformation (: y :) –> (: m/y2 :) takes a point on a circumconic to the corresponding point of tangency on its dual inconic.
Example: our hyperbola (perspector (: n– l :) and its dual inparabola (perspector (: 1/(n– l) :). In the following chart sm = (n+l–m)/2
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Point on G—P
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Point on circumhyperbola
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Point on dual inparabola
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tangent line to parabola
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( : y : )
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( : (n-l)/y2 : )
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[ : y : ]
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|
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G
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G
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: n – l : , its infinite point
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line at infinity is tangent line
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P
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tP = ( : 1/ m : )
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: m2 (n–l) :
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~tP = [ : 1/ m : ] is tangent line.
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mP
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tmP = ( : 1/(n+l) : )
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: (n+l)2 (n–l) :
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~tmP = [ : 1/(n+l) : ]
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dP
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tdP = ( : 1/ sm : )
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: smm (n–l) :
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~tdP = [ : 1/ sm : ] is tangent line.
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1/(sm + constant)
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: (sm + const)2 (n–l) :
this is general point; |
[ : sm + constant : ] is general tangent line (some with imaginary contact point).
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|
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1/(m + constant)
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: (m + const)2 (n–l) :
another way to write general point |
[ : m + constant : ] is general tangent line (some with imaginary contact point).
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|
Type II. From a line through D:
the rectangular hyperbola, circumconics through H, perspector on Polar axis,
The conjugate of a line through the desmon point D = tH, will produce a rectangular hyperbola. Well known examples: The Jerabek hyperbola, the Kiepert hyperbola.
There is only one rectangular hyperbola with perspector at infinity. It is generated by the line D—G and is the Kiepert hyperbola.
1. Let Q = (l:m:n), then the line D—Q = [ : SA n – SC l : ] so that the equation of the general rectangular circumhyperbola is
( SC m – SB n )/x + ( SA n – SC l )/y + ( SB l – SA m )/z = 0.
with perspector P = ( : SA n – SC l : ), which is on the Polar axis [ : SB : ] .
Note: This hyperbola can be generated as the isogonal conjugate of O—pP, where p = gt is the "pro" operation introduced by John Conway.
2. The points where D—Q meets the rectangular hyperbola are conjugate, and the duals of the asymptotes of another hyperbola (see item 9). Since they are conjugate and on a line through D, they are on the Lucas cubic.
This is one of the classical ways of generating the Lucas cubic.
2. Center = mtdP = ( : (SA n – SC l )(c2(m + n) – a2(l + m)) : ), which is on the nine point circle.
Proof: tdP = (: 1/(a2 l + (a2 – c2)m – c2 n) :) which can be seen to be on the circumcircle, so the mtdP is on the nine point circle.
3. Goes through H, the conjugate of D.
4. The asymptotes a1 and a2 are the Simpson lines of the meets of O—pP with the circumcircle. Their directions are the conjugates of the meets of D—Q with the Steiner ellipse. See here for a construction.
5. Axes. The axes are parallel to the asymptotes of the rectangular hyperbola with perspector
(: 2 S2(n + l) + b2(a2 l + b2 m + c2 n) : )
6. Dual inconic: perspector tP = [ : 1/(SA n – SC l ): ]; center mP. This conic meets the dual of the hyperbola center at two conjugate points ~a1 and ~a2, the duals of the hyperbola asymptotes.
6. Switched conic: Perspector and center are switched. The switched conic has center P and perspector mtdP. This conic is the conjugate of ~mtdP. This conic goes through the two conjugate points mentioned in the last item which makes it a Mineur conic so that m(~a1) and m(~a2) are on it. The asymptotes of this hyperbola are the duals of the intersections of D—Q with its conjugate, the rectangular hyperbola.
Figure: The blue hyperbola has perpendicular asymptotes, and hence, is a regular hyperbola. Its dual inconic in the inscribed ellipse on the left, perspector tP and center mP. The Green conic is the switched circumconic. The dual inconic and the switched circumconic go through the duals of the hyperbola asymptotes, which are on the dual of the hyperbola center.
Corresponding points on different rectangular hyperbolas are collinear with H.
Points on circumconics can be considered to have originated on the line at infinity. A very neat property comes from this. We will call points on different circumconics originating from the same infinite point by the term corresponding points. The line joining corresponding points goes through the fourth intersection of the two conics. Since every rectangular conic goes through H, this is the fourth intersection for every pair of them, hence all these corresponding points are on the same line through H.
Example1, showing second coordinate only.
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infinite point on ~K
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Kiepert
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Jerabek
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Feuerbach
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:c2-a2:
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: 1 :
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:b2 SB:
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:b sb/(c+a)
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|
tS
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G
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O
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Schiffler, Ho
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Example2, showing second coordinate only.
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infinite point
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Kiepert
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Jerabek
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Feuerbach
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:b2(c2–a2):
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: 1/b2 :
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:SB:
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:sb/b(c+a):
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|
infinite pt on Lemoine line
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R = tK
76 |
D = tH
69 |
X646
|
Figure: The Jerabek hyperbola is red; Kiepert blue, both rectangular hyperbolas. Some pairs of corresponding points are circled and lines drawn between them, all going through the orthocenter H, "4" in the picture.
Type III. From a line through a point and its conjugate: the Mineur conic
inconics tangent to many known lines;
circumconics with known asymptotes
This is one of the most important conic types. The best known examples are the Jerabek and Feuerbach hyperbolas. There are several reasons for its importance. First is that significant centers are likely to be on it. Significant centers on conics are in short supply. Second is that the significant centers on a Mineur conic organize other significant conics. This is a very powerful result. Details here.
Figure: The affine parallelogram and its relation to the Mineur conic. The Mineur line m is one diagonal of the parallelogram. The second diagonal goes through the center of the hyperbola. The four important points P, tP, mP, mtP are shown. The duals of P and tP go through M, the perspector.
1. Definition: This conic is generated as the isotomic conjugate of the line m from P = ( l : m : n ) to its isotomic conjugate tP. It has perspector M = (: m(n2–l2) :), the dual of the Mineur line m = P—tP. Its equation is
l(m2–n2)/x + m(n2–l2)/y + n(l2–m2)/z = 0
I usually follow John Conway's naming conventions. Note that the Mineur point and the Mineur line are invariant under the interchange of P with its conjugate.
2. Perspector: The perspector is M = ~P•~tP, called the Mineur point, the intersection of the duals of P and tP. The tripolars of points on the conic go through the perspector. If the point P resides inside the triangle, so will tP so that the perspector will be outside the Steiner ellipse, making this conic, in this circumstance, an hyperbola. A Mineur conic could be an ellipse, parabola (you will see one of these below), or an hyperbola.
3. Points on Conic: P, tP, mP, mtP are on the conic. This guarantees that interesting points reside on it, or at least points as interesting as the P we begin with. Since conics are often deficient in interesting points, this is a great advantage. The tripolars of these points meet at the perspector.
4. Lines through special points. A point P is naturally associated with two lines, its dual ~P, and G—P. The conjugates of the infinite points (directions) of these lines are antipodal on the Steiner ellipse.
The four special points P, tP, M, and mtdP, the conic center thus give 8 points on the Steiner ellipse. Lines connecting them go through the four special points on the conic and define two new points on the conic.
Figure: lines connect important points on the conic and the Steiner ellipse. In this picture P' = tP.
Using the Jerabek conic as an example, the isotomic points are D and H. The four Jerabek special points K, D, H, and O turn to be on the lines connecting the antipodal points on the Steiner ellipse, as seen in the figure.
Figure: Lines and points. Antipodal pairs of points on the Steiner ellipse related to the directions of lines related to the points D and H. Lines through these points go through G and the special Jerabek points. -H and -D are the reflections in the conic center of H and D. k and e are the G—K line (the symmedian track) and the Euler line.
5. 4th intersections: The fourth intersection of this conic with the circumcircle is the isogonal of the Mineur endpoint. The Steiner intersection is the isotomic of the Mineur endpoint.
6. Affine parallelogram: These points and this conic define the affine parallelogram, which orient the hyperbola and the points on it to the triangle and other known points. It also gives a construction of the conic center as the medial of the 4th intersection of the mP-circumconic and the mtP-circumconic.
The affine parallelogram is formed by drawing two lines from P: P—mtP and P—dtP and two from tP: tP—mP and tP–dP. Surprisingly these form a parallelogram which explains much of the distribution of related points in the triangle plane. See here for the affine parallelogram for the weak points and here for the strong ones.
7. The asymptotes a1 and a2 have the directions related to the isotomics of the meets of the Mineur line, which is the dual of the perspector, with the Steiner ellipse. The equations of the asymptotes are here (soon).
8. The duals of the asymptotes are conjugate, which Wilson Stother seems to have been the first to discover. They are both on the dual of the center of the Mineur conic.
9. The axes are the asymptotes of the rectangular circumhyperbola whose perspector is the point where the ~dP line meets the ~D line, the Polar axis.
10. The center: mtdM = (: m(n–l)2 :). The center is on the second diagonal of the affine parallelogram as the medial of the 4th intersection of the mP and mtP circumconics.
11. The dual inconic: The dual of the circumconic is an inconic with perspector tP and center mP. Duals of points on the Mineur conic are tangent to the dual inconic. Hence while the Mineur conic generally has lots of known points, its dual contains lots of known lints. In particular its intersections with the dual of the Mineur conic center are at two points which are duals of the asymptotes. The lines from G to these two points are tangent to the inconic.
Note: this gives a new way of constructing the asymptotes for a circumhyperbola.
12. The switched circumconic: has perspector and center switched. It is generated by the dual of the center: ~mtdM. If this is an hyperbola its asymptotes are ~P and ~tP, thus giving a way to construct an hyperbola with known asymptotes. This conic goes through the duals and the medials of the duals of the two asymptotes.
13. The last two conics meet on ~mtdM, the dual of the original conic center, at two istotomic points which are on the dual of the original conic and the switched circumconic. This makes the switched circumconic itself a Mineur conic generated from the line ~mtdM = ~a1— ~a2 (note: ~a2 = t ~a1). By the properties of a Mineur conic, the medials of these two points will be on the conic.
The other two intersections of the dual inconic with the switched circumconic are where ~mP and ~mtP are tangent to the inconic.
Figure: The Mineur conic in blue, its dual inconic in red, and the switched circumconic in a lighter blue. The dual of the center goes between the intersections of the latter two conics, going through interesting points on the way. The four points in blue on the dual inconic are the points of tangency of the four important points on the original hyperbola. Two of the points are the last two intersections of the switched circumconic with the dual inconic.
14. Other associated conics derive from the defining points of the affine parallelogram, mP, mtP, dP, and dtP.
The circumconics of mP and mtP have eachother as centers and meet at the dilated hyperbola center.
The inconics of P and tP which have mtP and mP as centers, meet at the Mineur conic center.
Type IV. From general line:
general circumconics, general inconics
This is the archtypical circumconic, but it comes with the least built-in structure of them all. Three nice examples are the circumconic, generated from the dual of K; the Steiner ellipse, from the dual of G; and the Io-circumconic, generated from the dual of Io.
1. The perspector is P = (l:m:n), the dual of P is ~P = [l:m:n], and the equation of its conjugate is
l/x + m/y + n/z = 0
The position of the perspector compared to the Steiner inellipse determines the nature of the conic: inside gives an ellipse, on gives a parabola, outside gives an hyperbola.
2. Its center is mtdP = ( : m (n + l – m) : ). If the perspector is on the Steiner inellipse, then the center is on the medial triangle. So we can characterize the type of conic by the position of the center to the medial triangle: inside implies ellipse, on implies parabola, and outside implies hyperbola.
3. Tangents to the conic at the vertices are the ex-Cevian lines of P.
Proof: lyz + mzx + nxy = z(ly+mx) + nxy = 0. Since the last term intersects the conic at C twice and z = 0 does not, the factor ly+mx must have a double intersection and hence be the tangent. This is an ex-Cevian line of P.
4. Axes and or asymptotes. The axes of a P-circumconic are the asymptotes of a circumhyperbola with perspector where ~dP meets the ~D, the Polar axis. If the conic is an hyperbola, then here are the asymptotes.
If P = (l:m:n) is the perspector of a circumhyperbola, then the equation of the asymptotes are
( : m ( n2 – lm + l2 – mn ± (n–l) Z )2 : ) where Z2 = l2+m2+n2–2mn–2nl–2lm.Note that Z is imaginary when the perspector is inside the Steiner inellipse whose equation is x2+y2+z2–2yz–2zx–2xy = 0.
G—P and ~P meet at mP2–. It's isotomic tmP2– is the opposite in the conic center of the isotomic of the perspector.
5. Projection from Steiner ellipse using the fourth intersection. These conics are the projective transformation ABCG –> ABCP of the Steiner ellipse, implemented geometrically as shown in the figure and algebraically as barycentric multiplication by P. Corresponding points can be found by projecting from t(∞•~P), the fourth intersection of the conic with the Steiner ellipse. Draw a line from this point to any point on the Steiner ellipse. The intersection of this line with the P-circumconic will be the projected point.
Figure: The circumcircle and Steiner ellipse are shown. Points are paired with the line from each pair going through S. Bold points are strong. Weak ones are given with their extraversion index. 100o stands for the original version (of 4) of point X100 listed in ETC.
6. Its dual is the tP inconic with center mP.
7. Its switched conic, an ellipse, changes the center and perspector, and has perspector mtdP and center P. Its equation is
lsl/x + msm/y + nsn/z = 0.
where sm = (n+l–m)/2, etc. It is generated as the isotomic conjugate the dual of center of the P-circumconic.
Figure: This figure shows conics derived from the natural lines of Io: ~Io, its dual, and G—Io. The circumconics are in blue, one with perspector Io and the other, an hyperbola having its perspector the infinite point ~Io•∞. The inconics are in red. One is an ellipse with perspector tIo and center So. The other is a parabola with perspectors t(~Io•∞). The isotomics of Q1 and Q2, the intersections of G—Io with the Steiner ellipse give the directions of the asymptotes of the circumhyperbola.
Points on the P-circumconic
This chart of points shows points on the P-circumconic and their relation to points on the line at infinity and the Steiner ellipse.
P = ( l : m : n ).
|
line at infinity
|
Steiner ellipse
|
P-Circumconic
|
|
|
(: y :)
|
(: 1/y :)
|
(: m/y :)
|
|
|
n–l |
1/(n–l)
t(∞•~P) Steiner analogue |
m/(n–l)
analogue of KP focus |
|
|
n+l–2m
∞•(G—P) |
1/(n+l–2m)
t(∞•(G—P)) |
m/(n+l–2m)
|
|
|
m(n–l)
∞•~tP |
1/m(n–l)
t(∞•~tP) |
1/(n–l)
t(∞•~P) Steiner analogue |
|
|
(n–l)(n+l–2m)
|
1/(n–l)(n+l–2m)
|
m/(n–l)(n+l–2m)
|
|
|
(n–l)(m2–nl)
|
1/(n–l)(m2–nl)
|
m/(n–l)(m2–nl)
|
|
|
(n+l)(m2–nl)
∞•(P—tP) |
1/(n+l)(m2–nl)
t(∞•(P—tP)) |
m/(n+l)(m2–nl)
|
|
|
(l2–mn+n2–lm)(n–l)
|
|||
|
m(n2+l2–lm–mn)
|
? 1/m(n2+l2–lm–mn)
|
1/(n2+l2–lm–mn)
Tarry analogue |
|
|
m(n–l)smm,
|
? 1/m(n–l)smm,
|
1/(n–l)smm,
|
|
|
(n–l)sm
∞•~tdP |
1/(n–l)sm
|
m/(n–l)sm
|
|
|
nsn+lsl–2msm
∞•(G—tdP) |
1/(nsn+lsl–2msm)
t(∞•(G—tdP)) |
m/(nsn+lsl–2msm)
|
|
|
n2–l2
= ∞•~P2 |
1/(n2–l2)
t(∞•~P2) |
m/(n2–l2)
|
|
|
n2+l2–2m2
∞•(G—P2) |
1/(n2+l2–2m2)
t(∞•(G—P2)) |
m/(n2+l2–2m2)
|
|
|
m2(n2–l2)
∞•~tP2 |
1/m2(n2–l2)
t(∞•~tP2) |
1/m(n2–l2)
t of P–Mineur conic perspector |
|
|
m2SM2–
|
1/m2SM2–
|
1/mSM2–
|
|
|
(n2–l2)SM
|
1/(n2–l2)SM
|
m/(n2–l2)SM
|
|
|
SMN+SLM–2SNL
∞•(G—P2) analogue of Euler endpoint |
1/(SMN+SLM–2SNL)
|
m/(SMN+SLM–2SNL)
|
|
|
? m2(n2–l2)SMM
|
? 1/m2(n2–l2)SMM
|
1/m(n2–l2)SMM
|
|
|
? m(n–l)sm(4SMM–n2l2)
|
? 1/m(c–l)sm(4SMM–n2l2)
|
1/(n–l)sm(4SMM–n2l2)
|
|
|
m/(n–l)(n2+l2+ln)
|
Transformations among conics
Each circumconic can be derived from the Steiner ellipse by a projective transformation. Peter Moses has listed many of these here.