Conic_Instrument

Conic Instrument

This is a device allowing the drawing of the  generic conic given in cartesian coordinates by an equation of the form
(1)                                                                     f(x,y) = ax²+2hxy+by²+2gx+2fy + c = 0,
The instrument has 6 parameters {a, h, b, g, f, c} which can be set arbitrarily and define the conic.

Remark
If the intercepts with the axes are real, then:
[1] By changing the coefficients {a, g} only, the intercepts {A,B} with the y-axis remain the same. The center G of the conic
     moves on a line (e) passing through the middle of AB.
[2] By changing the coefficients {b, f} only, the intercepts {C,D} with the x-axis remain the same. The center G of the conic
     moves on a line (f) passing through the middle of CD.
[3] By changing the coefficient {h} only, all four intercepts with the axes remain the same.
[4] By changing the coefficient {c} only, all four intercepts change proportionally, the center remains constant and the axes
of the conic maintain constant directions. The conics for variable c's are homothetic with respect to their center. In the case
of parabolas, varying c we obtain translates of the parabola.

The proofs of the remarks follow from the formulas discussed in Conic_Equation.html . For example, [1] follows from
the formulas giving the center in terms of the coefficients (ibid, section-8). In fact, setting x=0 in the equation follows:
                                                                    by² + 2fy + c = 0,
whose roots determine {A,B} on the y-axis, with their middle defined by y=-f/b. Since these do not depend on {a,g} they
remain the same if only these two coefficients change. Regarding the motion of G on line (e) consider the aforementioned
equations of the center (x₀,y₀) and show that they satisfy the line equation:
                                                                   hx₀ + by₀ +  f = 0.
Analogously the center is also on the line (f):
                                                                   ax₀ + hy₀ +  g = 0.
(G is the intersection of these two lines). The first does not change by modifying {a,g} and the second does not change
by modifying {b,f}.

Remark There results a "simple" proof (modulo Chasles-Steiner generation, see Chasles_Steiner.html ) that the center
G describes a conic as h only varies. In fact, the dependence of the two lines {e,f} is a projective correspondence between the
pencils E^*, F^* and the aforementioned principle applies to show that G describes a conic as h only varies.
See [SalmonConics, p. 153] for an alternative proof using a similar analysis.

Bibliography

[SalmonConics] Salmon, G. A treatise on conic sections Longmans, London 1855

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Conic Instrument

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Bibliography