(1) f(x,y) =

The instrument has 6 parameters {a, h, b, g, f, c} which can be set arbitrarily and define the conic.

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:

by

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 (x

hx

Analogously the center is also on the line (f):

ax

(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}.

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

See [SalmonConics, p. 153] for an alternative proof using a similar analysis.

Chasles_Steiner.html

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