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[EMHL] Re: Minimal Eccentricity of Conics through Four Points

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  • Chris Van Tienhoven
    Dear friends, I calculated a general formula for the eccentricity of a circumscribed conic expressed in barycentric coordinates. Thanks to the direction
    Message 1 of 29 , Apr 9, 2011
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      Dear friends,

      I calculated a general formula for the eccentricity of a circumscribed conic expressed in barycentric coordinates.
      Thanks to the direction Francois Rideau appointed through Mr Wolstenholmes method by using the real or imaginary asymptotes of the conic.
      Let ABC be some triangle with extra points P(p,q,r) and U(u,v,w).

      In this situation the general equation for the conic is:
      qruwxy - prvwxy - qruvxz + pqvwxz + pruvyz - pquwyz = 0

      Let e be the eccentricity of the conic.
      Now e^2 = 2 X1/(X1-X2)
      where:
      X1 = Sqrt[a^4 QV RW + b^4 PU RW + c^4 PU QV
      - b^2 c^2 PU (-PU+QV+RW)
      - c^2 a^2 QV (+PU-QV+RW)
      - a^2 b^2 RW (+PU+QV-RW)]
      X2 = SA PU + SB QV + SC RW
      and:
      PU = pu(qw-rv),
      QV = qv(ru-pw),
      RW = rw(pv-qu).
      I checked in a fixed reference situation.
      Maybe it is helpful for someone.
      Best regards,

      Chris van Tienhoven

      --- In Hyacinthos@yahoogroups.com, Francois Rideau <francois.rideau@...> wrote:
      >
      > Yes, there is a general formula for eccentricity, known for a long time
      > (before 1886), for example in:
      > Mathematical Problems by Mr Wolstenholme.
      > The starting point is to look at the angle V of the (real or imaginary)
      > asymtots for we have the formula:
      > tan²(V) = 4(e²-1)/(e²-2)².
      > And it is very easy to write down the equation of the pair of asymptots from
      > the general equation of the conic.
      > Now if a pair of lines is given in trilinears by the equation:
      > Lx² +My² +Nz² +2Uyz+2Vzx+2Wxy = 0
      > the angle V of these lines is given by:
      > tan²(V) = -D/(R.E)
      > where R is the radius of the circumcircle;
      > E = L+M+N -2Ucos(A)-2Vcos(B)-2Wcos(C)
      > and D is the following 4x4 determinant with lines:
      > L(1) = (a, L, W, V)
      > L(2) = (b, W, M, U)
      > L(3) = (c, V, U, N)
      > L(4) = (0, a, b, c)
      > where a, b, c are the lengths of the sides BC, CA, AB
      > Friendly
      > Francois
      >
      >
    • Eisso Atzema
      Dear François et alii, now that I have looked at your formula for the relation between eccentricity and tan(V), I realize that my formula for finding the
      Message 2 of 29 , Apr 10, 2011
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        Dear François et alii,

        now that I have looked at your formula for the relation between
        eccentricity and tan(V), I realize that my formula for finding the
        minimal eccentricity of all conics circumscribing a quadrangle ABCD can
        also be written in the form.

        4*csc^2(V) = (cot(A) + cot(C))*(cot(B) + cot(D)),

        where csc(V) is a (generally completely superfluous) Anglo-Saxon
        notation for 1/sin(V) and V is your angle between the asymptotes of the
        conic of minimal eccentricity. Alternatively, if A*B*C*D* denotes the
        quadrilateral formed by the circumcenters of BCD, CDA etc., we have

        sin^2(V)=area(ABCD)/area(A*B*C*D*).

        Of course, this equation is only meaningful for non-convex, non
        self-intersecting ABCD, but the formula is easily verified in Geogebra
        for these.

        Eisso


        On 4/4/11 5:31 AM, Francois Rideau wrote:
        > Yes, there is a general formula for eccentricity, known for a long time
        > (before 1886), for example in:
        > Mathematical Problems by Mr Wolstenholme.
        > The starting point is to look at the angle V of the (real or imaginary)
        > asymtots for we have the formula:
        > tan²(V) = 4(e²-1)/(e²-2)².
        > And it is very easy to write down the equation of the pair of asymptots from
        > the general equation of the conic.
        > Now if a pair of lines is given in trilinears by the equation:
        > Lx² +My² +Nz² +2Uyz+2Vzx+2Wxy = 0
        > the angle V of these lines is given by:
        > tan²(V) = -D/(R.E)
        > where R is the radius of the circumcircle;
        > E = L+M+N -2Ucos(A)-2Vcos(B)-2Wcos(C)
        > and D is the following 4x4 determinant with lines:
        > L(1) = (a, L, W, V)
        > L(2) = (b, W, M, U)
        > L(3) = (c, V, U, N)
        > L(4) = (0, a, b, c)
        > where a, b, c are the lengths of the sides BC, CA, AB
        > Friendly
        > Francois
        >
        >
        >


        --

        ========================================
        Eisso J. Atzema, Ph.D.
        Department of Mathematics& Statistics
        University of Maine
        Orono, ME 04469
        Tel.: (207) 581-3928 (office)
        (207) 866-3871 (home)
        Fax.: (207) 581-3902
        E-mail: atzema@...
        ========================================
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