Is there a formula which describes *all* conic sections, which will generate a particular class of same...
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Problem
Is there a formula which describes *all* conic sections, which will generate a particular class of same (e.g., circles, hyperbolas) when certain coefficients are plugged into it?
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>> 2 2
>> F(x,y) = Ax + Cy + Dx + Ey + F
>>
>>F(x,y) = 0 is the general equation representing a conic section.
> ...
>This only produces thoses ellipses, parabolas and hyperbolas that have an
>axis parallel to the X and Y axis. If you want rotated versions you have to
>add in the Bxy term giving the equation:
>
> Ax**2 + Bxy + Cy**2 + Dx + Ey + F = 0
>
(Thus the family of conic sections in the plane is five-dimensional, since
one can always eleminate one of the real parameters A ... F by rescaling
the equation.)
This is essentially the answer everyone came up with. It's a good one,
but suffers on two grounds: it does not generalize to higher dimensions very
nicely, and it does not reveal the geometry which inspired the original
intuition that fundamentally all the conic sections are "the same". I offer
two ways to remedy this, one very well-known, the other fairly obscure.
One could parametrize the ellipses, parabolae, and circles in the plane by
intersecting the standard cone in R**3 (given by x**2 + y**2 = z**2) with
various planes not passing through the origin, and then translating and
rescaling the various curves which result. A better, but equivalent way, is
to move the cone and keep the plane fixed.
Briefly, one moves the cone via any nonsingular linear transformation of R**3.
The lines through the origin are mapped to lines through the origin, so that
the image of the cone becomes an 'elliptic' cone. The curve given by intersec-
ting this cone with the plane z=1 is the image of the standard circle
x**2 + y**2 = 1. Thus one could uniformly parametrize all of the conic
sections by the interval [0,2*pi] beginning with the standard parametrization
of the unit circle. The result can be written down immediately by anyone at
all familiar with projective coordinates; the result is,
t ----> (a3cos(t)+b3sin(t)+c3)**(-1) * (a1cos(t)+b1sin(t)+c1,
a2cos(t)+b2sin(t)+c2)
where (a1 b1 c1
a2 b2 c2
a3 b3 c3) is any nonsingular real matrix.
There is a redundancy in this parametrization of the conic sections;
four of these parameters are really unnecessary.
The result is computationally messy, but is distinguished from the others in
that the conic sections are given explicit parametrizations. (Incidentally,
these parametrizations are singular in general; for a small, finite number
of values of t, one or both of the coordinates become infinite. These are
the 'points at infinity' on the conics.)
The other method is nicer in many ways. A more detailed description can be
found in M. Spivak's Differential Geometry, Vol. 2, pp 52 - 58. There he
shows that the conic sections are precisely those parametrized curves
c: R -----> R**2
whose special affine curvature k is constant. There is a one-parameter
family of representatives:
A) c(t) = ((1/k)sin(kt), -(1/k**2)cos(kt)), whose image
is the curve (kx)**2 + (k**2y)**2 = 1, an ellipse
(k>0);
B) c(t) = ((1/k)sinh(kt), -(1/k**2)cosh(kt)), whose image
is the curve (kx)**2 - (k**2y)**2 = 1 (k>0);
C) c(t) = (t, .5t**2), whose image is the curve y=x**2/2.
(NB. These equations differ from his. I have hastily tried to correct
his solutions to a simple O.D.E. I might be wrong too.)
k thus amounts to one real parameter, equal to the square of the special affine
curvature. (The hyperbolae have the negative curvature.)
The other five parameters are those of the special affine group.
In short, every conic section is the image of one of these after applying a
translation (two real parameters) and then an element of SL(2,R), which is
the group of 2x2 real matrices of determinant one. The beautiful thing about
this particular group of transformations is that it preserves the special
affine curvature. (Each standard conic is preserved by a one-dimensional
subgroup of this group.)
The relationship between the solutions so far is that of subgroup: the largest
group around is PGL(3,R), a real 8-dimensional group which acts transitively
on the conic sections in R**2. The special affine group is a five-dimensional
subgroup whose orbits on the conic sections are parametrized by the special
affine curvature. The euclidean group for R**2 is the 3-dimensional group of
translations and rotations of the plane. Its orbits on the conic sections
are parametrized by the family of all ellipses, hyperbolae, and parabolae
which are 'centered at the origin' and 'straight up and down' (it would be
too pedantic to describe these more fully; the language is not meant to be
technical). Each member of this family is sent into itself only by a finite
subgroup of the euclidean group. The elements of this family are determined
by an eccentricity and a 'size'.
One way to understand all this is to look at invariants of each group. The
smaller the group, the more invariants it has. Here's a table.
Group Invariants in the family of conic sections
____ __________________________________________
PGL(3,R) None. All are equivalent.
R^2 + SL(2,R) Special affine curvature.
Similarity group S.A. curvature.
Euclidean group SA curvature, eccentricity.
Translation group SA curvature, eccentricity, orientation.
Trivial group SA curvature, eccentricity, orientation,
position (two parameters there).
(The similarity group includes homogeneous rescalings, as well as translations
and rotations.)
The natural way, then, to generalize the study of conics, is to look for in-
variants of families of hypersurfaces (hyperquadrics, say) in R**n with respect
to subgroups of PGL(n+1,R). For one interesting very incomplete beginning,
see Spivak vol. 3 pp 113-194.
Incidentally, if one replaces 'hyperquadrics' above with 'submanifolds'
then what I have written is one interpretation of Klein's Erlangerprogram (?),
c. 1880.
-- Bill Huber, St Joseph's University, Philadelphia <bhuber@sjuvax.UUCP>
B Huber's fine discussion parallels the first chapter of C H Clemens' nice
little book _A Scrapbook of Complex Curve Theory_, published by Plenum Press,
and well worth studying.
-- Matthew P Wiener <ucbvax!brahms!weemba>, UCB Math Dept