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

Expand Messages
• 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
• 0 Attachment
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
>
>
• 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
• 0 Attachment
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@...
========================================
Your message has been successfully submitted and would be delivered to recipients shortly.