AbridgedNotation

1. Abridged notation, line formularium

Here I review the abridged notation of our ancestors following closely the beautiful Treatise on Conics of Salmon but drawing some more figures than he does. This notation is essentially the representation of entities in projective coordinates with respect to some basis ( ProjectiveBase.html ) whose coordinate axes are lines M=0, R=0, L=0.

Somewhere (arbitrary but fixed) in the plane is the fourth unit point D defining the coordinate system.
(1) aM + bR + cL = 0, represents an arbitrary line in the plane.
A point P is preferably represented as intersection of two lines:
(2) M-tR = 0, R-t'L=0 represents an arbitrary point. We speak of point (t,t'). M-tR=0 is a line through C and R-t'L=0 is a line through A.
(3) att' + bt' + c = 0 is the condition of point (t,t') to belong to line aM+bR+cL=0.
(4) {att' + bt' + c = 0 , ass' + bs' + c = 0}
Is the system satisfied by a line passing through the two points (t,t') and (s,s'). Thus, solving for (a,b,c):
(5) (t'-s', ss'-tt', t's'(t-s)) are (up to multiplicative constants) the coefficients of the line passing through points (t,t') and (s,s').
Thus replacing in (3) we get the nice contition of collinearity of three points (t,t'), (s',s'), (u,u'):
(6) (t'-s')uu' + (ss'-tt')u' + t's'(t-s) = 0.
(7) {att' + bt' + c = 0 , a'tt' + b't' + c' = 0}
Is the system satisfied by the intersection point of two lines. Thus, solving for (t,t'):
(8) t = (bc'-b'c)/(a'c-ac'), t' = (a'c-ac')/(ab'-a'b) is the intersection point of the two lines.
Thus the condition of concurrency of three lines results by replacing this into (3) for a third line, which after simplifying gives:
(9) a'''(bc'-b'c) + b'''(a'c-ac') + c'''(ab'-a'b) = 0.
This is the determinant of the coefficients of the three lines.
The connexion with the contemporary notation, which prefers to write a point as
(10) P = xA + yB + zC
results by replacing this into the system (2) (the correspondence being (M=0 to x=0, R=0 to y=0 and L=0 to z=0) and getting:
(11) t = x/y, t' = y/z , and also
(12) (x,y,z) = k(tt', t', 1) , with arbitrary constant k.
The cross ratio of four lines {M-t_iR=0, i=1,2,3,4} passing through C can be defined by the expression
(13) cr = [(t₁-t₃)/(t₂-t₃)]:[(t₁-t₄)/(t₂-t₄)].
This conforms with the definition of the cross ratio of four points on a line.
(14) M-tR=0 , M+tR=0 are harmonic conjugate lines with respect to the line-pair {M=0, R=0}, all lines passing through C.

2. Abridged notation, conics

Here the two lines are taken to be tangent to the conic and the third is the chord of contacts. The equation of the conic obtains the form
(1) LM-R² = 0.

(2) LM - kR² = 0, with variable k represents all conics passing through {A,C} and being tangent there to {L, M} correspondingly. The particular one is obtained by requiring from the conic to pass through D, which since L(D) = M(D) = R(D) = 1 gives k=1.
Every point P on the conic can be determined as intersection of two lines passing through A and B:
(3) M-tR = 0, M-t²L=0, R-tL=0 determines a point P on the conic : intersection of the lines R-tL=0 (through A) and M-t²L=0 (through B) etc.

Any pair out of the three can be used to determine the position of P through the position of the lines.
The jargon is to identify points P on the conic with the corresponding parameter t, thus, speaking of the point t on the conic. One has to get used to this symbolism. Then it is quite handy and effective. Plus, it reflects the fact that conics are bijective images of (projective) lines.
(4) Thus point t (on the conic) means the intersection point of lines : {M-tR=0, R-tL=0}, equivalently the line-pair: {M-t²L=0, R-tL=0} etc..
For an arbitrary line aM + bR + cL = 0 the intersection points with the conic are points t found through the solution of the system:
(5) {aM + bR + cL = 0, R-tL=0, M-tR=0}, which leads to the quadratic equation:
at² + bt + c = 0.
(6) (tt')L-(t+t')R+M=0.
Is the equation of the line (chord) through points P(t) and P(t'), since it is satisfied by P(t): {R-tL=0, M-tR=0} and also by P(t') : {R-t'L=0, M-t'R=0}. If all these chords are to pass through a fixed point Q then replacing into this equation the coordinates of the point we see that
(tt')p-(t+t')q+s=0 with constants (p,q,s) is the necessary and sufficient condition for chords tt' to pass through a fixed point.
Allowing complex numbers for t,t' previous equation can represent any line of the plane, since every line in this case has two (complex) intersection points with the conic. For real conics t, t' are complex conjugate hence (tt') and (t+t') are real.
(7) t²L-2tR+M=0. is the (equation of the) tangent at P(t), since it represents the limit position of the previous line when t' tends to coincide with t.
The converse is also true, since one can reverse the arguments:
(8) Theorem Any one-parameter line-equation which can be put into the form t²L-2tR+M=0 , represents a tangent to the conic LM-R²=0. The theorem has many applications in finding a conic enveloping a one parameter family of lines.
The cross ratio of four points P(t_i) i=1,2,3,4 on the conic can be defined by using directly the values of t_i as for the cross ratio of lines:
(9) cr = [(t₁-t₃)/(t₂-t₃)]:[(t₁-t₄)/(t₂-t₄)]..
It can be shown that this is independent of the representation of the lines and the conic and equal to the cross ratio of four points Q_i on a line, resulting by projecting P(t_i) from a point P₀ on the conic.

The tangent (7) passes through a point P(s,s') (M=sR, R=s'L) implies t² - 2ts + ss' = 0. Point t being on the conic implies t=R/L, t²=M/L hence
(10) M - 2sR + ss'L = 0 is the equation of the polar of P(s,s').

3. Some applications

Theorem-1 Given three fixed chords of a conic {AA',BB',CC'} a fourth chord such that the cross section (A,B,C,D)=(A',B',C',D') is always tangent to a conic having double tangency (two common tangents) with the given one.

The figure displays such an example. The chords {AA',BB',CC'} are fixed, D is free to move on the conic, and D' is calculated so that the cross ratios (A,B,C,D) and (A',B',C',D') are equal. The cross ratios of the points are calculated using the line coordinates of the projections of the points on two lines (could use also one line only). The proof (after Salmon p. 253) is easy. Assuming that A is given by a number a (i.e. the intersection of lines M-aR=0, R-aL=0) and analogous use of small letters for the other points, the condition on the cross ratio is:
[(a-c)/(b-c)]:[(a-d)/(b-d)] = [(a'-c')/(b'-c')]:[(a'-d')/(b'-d')].
This obtains the form of a homographic relation (*)
pdd' + qd + rd' + s = 0, for certain constants (p,q,r,s) depending on the (fixed) a,b,c, ...etc..
Solving for d' and substituting in equation (2.6) of the chord DD' gives the beautiful expression
d(qd + s)L + R(d(pd+r)-(qd+s)) - M(pd+r) = 0.
As expected, this is a one-parameter family of lines (in parameter d) which can be written
d²(qL+pR) + d(sL+(r-q)R-pM) - (sR+rM) = 0.
From theorem (2.8) follows that this line envelopes the conic with equation:
(sL+(r-q)R-pM)² + 4(qL+pR)(rM+sR) = 0. (**)
The nice thing is that this conic (its expression rather) can be put in the form:
4(qr-ps)(LM-R²) + (sL+(q+r)R+pM)² = 0. (***)
Since LM-R² = 0 is our conic and (sL+(q+r)R+pM)=0 is a line, the conic (***) belongs to the family generated by our conic and a (double) line. This is a bitangent family of conics, all members of which are tangent to the conic LM-R² = 0 at the points where the line (sL+(q+r)R+pM)=0 intersects it. Note that the intersection points can be imaginary as is for example the case with the family of concentric circles, which is a bitangent family all of which members are tangent at the same two imaginary points at the same two imaginary lines.
Remark The homographic relation becomes involutive (see HomographicRelation.html ) when q=r and (*) takes the form
pdd' + q(d+d') + s = 0 ,
showing that all chords DD' pass through a common point (see 2.6). Thus, we should exclude this case from the beginning since it shows a totally different behaviour.

Theorem-2 If a polygon of n sides is inscribed in a conic c and its n-1 sides pass through corresponding fixed points, then its n-th sides envelope another conic c' bitangent to c.

The figure displays such an example for triangles. Triangle ABC moves having all the time its vertices on the conic c and two of its sides passing through two corresponding fixed points {I, J}. Then the third side (AB) envelopes another conic c' which is bitangent to the given one. The proof reduces to the previous theorem by taking three different positions of the triangle and defining the corresponding chords {A₁B₁, A₂B₂, A₃B₃}. Then for the moving fourth triangle ABC we have the preservation of cross ratio (A₁,A₂,A₃,A) = (B₁,B₂,B₃,B). This because the central correspondence (involution at) from J : B_i --> C_i preserves the cross ratio. This is due to the homographic relation (2.6) ptt' + (t+t')q + s = 0 with constants {p,q,s} and the simple fact that such relations preserve the cross ratio. The same is true for the correspondence from I : C_i --> A_i. Thus composing the two correspondences we have the map B_i --> A_i preserving the cross ratio. Hence, theorem-1 applies for the moving side AB. The proof is easily generalized for any n.

Theorem-3 If a quadrangle is inscribed in a conic and three of its sides pass through three corrsponding points lying on the same line e , then its fourth side passes also through a fixed point on line e.

The proof of this belongs to another context but I put it here since it gives an exceptional case of the previous theorem. The theorem follows directly from the theorem of Desargues (see DesarguesInvolution.html ) stating that all conics pertaining to a family passing through four points intersect a line in points in involution. Here the coordinates of the intersection points A₁(a), B₍b), ... will satisfy a relation of the form pac+q(a+c)+s=0, pbd+q(b+d)+s=0, with constant {p,q,s}. Thus if b (B) is constant then the coordinate d of D will also be constant.

Next figure shows that the same property is not valid for pentagons inscribed in conics. The pentagon shown has its four sides passing through fixed points {A,B,C,D} correspondingly. The fifth side (XY) though passes not through a fixed point but envelopes a bitangent conic as stated in the general theorem-2.

The figure shows further that in this case the enveloping conic passes through the intersection points of the line e carrying the fixed points. This follows from the fact that when a vertex X of the polygon tends to e then all of them tend to this line, and line XY tends to the tangent of the conic at its intersection with the line. Thus the enveloping conic belongs to the conic family generated by the initial conic and the (double) line e, having an equation of the form c + ke² =0, if c=0 and e=0 are the equations of the conic and the line, with k a constant.
Next figure shows another case in which the line does not intersect the initial conic. It shows also another member c'' of the bitangent family c + ke².

Finally last figure shows that theorem-3 is again true for hexagons. It shows also how to prove this: divide in quadrangles and apply theorem-3 to each one of these quadrangles successively. In the figure first quadrangle VWYZ has its side YZ passing through a fixed point A and then UXYZ has the final side XY passing through the fixed B. Obviously one can generalize to:

Theorem-4 If a polygon of 2n sides is inscribed in a conic and has its 2n-1 sides passing through corresponding points lying on a line e, then its 2n-th side passes also through a fixed point on line e.

References

Salmon, G. A treatise on conic sections London 1855, Longmans, pp. 232-275.
Todhunter, Isaac A treatise on Plane co-ordinate geometry New York, Macmillan and Co. 1888, pp. 272-307.
Puckle, G. Hale Conic Sections and Algebraic Geometry London, Macmillan and Co. 1884, pp. 322-363.

1. Abridged notation, line formularium

2. Abridged notation, conics

3. Some applications

References

See Also