rab@well.UUCP (Bob Bickford) (01/22/86)
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?
This came up at a party, when one person maintained that the
various conic sections are fundamentally different, and the rest of us
that they are all 'special cases' of the same thing. Finally, he
challenged us to write an equation as above. Nobody could do it!
(But then, nobody there had a degree in math [if that means anything].)
Please reply on net.math
Robert Bickford (rab@well.uucp)bs@linus.UUCP (Robert D. Silverman) (01/23/86)
> > 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? > > This came up at a party, when one person maintained that the > various conic sections are fundamentally different, and the rest of us > that they are all 'special cases' of the same thing. Finally, he > challenged us to write an equation as above. Nobody could do it! > (But then, nobody there had a degree in math [if that means anything].) > > Please reply on net.math > > > Robert Bickford (rab@well.uucp) This is a simple high school analytic geometry problem. A general quadratic form in two variables is: 2 2 F(x,y) = Ax + Cy + Dx + Ey + F F(x,y) = 0 is the general equation representing a conic section. The following conditions hold: If A or C is zero the conic is a parabola or in special cases two parallel lines which may be distinct, coincident or imaginary. If A and C have the same sign we have an ellipse or in special cases it can degenerate into a single point or imaginary ellipse. It is a circle when A = C. If A and C have different signs we have a hyperbola or in degenerate cases two intersecting lines. I hope this is what you were looking for. Bob Silverman
gwyn@brl-tgr.UUCP (01/23/86)
> 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? > > This came up at a party, when one person maintained that the > various conic sections are fundamentally different, and the rest of us > that they are all 'special cases' of the same thing. Finally, he > challenged us to write an equation as above. Nobody could do it! > (But then, nobody there had a degree in math [if that means anything].) Ax^2 + By^2 + Cxy + Dx + Ey + F = 0 This can be simplified, but why bother. Look it up in any analytic geometry text, math handbook, etc.
lhl@lanl.ARPA (01/23/86)
In article <532@well.UUCP> rab@well.UUCP (Bob Bickford) writes: > > 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? > All conics have the form 2 2 Ax + Bxy + Cy + Dx + Ey + F = 0 2 where either (A+C) or (B - 4AC) is not equal to zero (this degenerate case is either a line, the entire plane, or the null set.) By rotation and translation of axes, all other combinations reduce to one of two forms; either 2 Ax + Ey + F = 0 with A nonzero; this is a parabola with E nonzero, otherwise (depending on AF) a line, two parallel lines, or the null set. The other case is 2 2 Ax + Cy + F = 0 with AC nonzero. If AC < 0 this is a hyperbola or two intersecting lines, depending on F. If AC > 0 this is an ellipse, a point, or the null set, depending on AF. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - He's got feet of clay clear up to his eyebrows. A little of something else at the very top though. R. A. Lafferty
batu@hound.UUCP (S.PETERSON) (01/23/86)
The question is a "formula" for "all" the conic sections
For simplicity assume that they are in R**2 and centered at the origin,
(note to have centers not at the origin is a simply translation)
Let a & b be nonzero real numbers, r a non negative real number, x and y
unknowns as usual.
Then we have
2 2
(P) ax + by = r
i) if a=b and both are > 0 the P is a circle.
ii) if a is not equal to b and both are > 0 the P is an ellipse, and there
is a formula for finding axes, focii etc.
iii) if a is not b and one is > 0, the other < 0, then we have a hyperbola,
again there are formulas (in terms of a,b and r) for asymptopes, etc.
2 2
(Q) to get a parabola, e.g. y = ax or x = by
In general an equation of the form
2 2
ax + bx + cxy + dy + ey = r
gives a conic where a,b,c,d,e,r are any real numbers.
( if b=c=e=r=0, then we have a point, which is a conic section)
Consult any High school algebra text or a college Analyitical geometry text.ark@alice.UucP (Andrew Koenig) (01/23/86)
> 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? I seem to recall something like Ax^2 + By^2 + Cxy + Dx + Ey + F = 0
wdh@faron.UUCP (Dale Hall) (01/23/86)
In article <532@well.UUCP> rab@well.UUCP (Bob Bickford) writes: > > 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? > The general quadratic equation in two real variables: 2 2 a x + b xy + c y + d x + e y + f = 0 yields (possibly degenerate) conic sections as solution sets. The way to determine which of the three types (ellipse, parabola, hyperbola) emerges with a given set of coefficients is to turn the equation into a homogeneous equation in three variables: 2 2 2 a x + b xy + c y + d xz + e yz + f z = 0. This new equation represents a curve in the (real) projective plane, essentially a completion of the affine plane which preserves limits of curves at infinity. To be strictly honest, points are triples: [x,y,z], in which [x,y,z] ~ [tx,ty,tz] for any non-zero number t; the subspace of points for which z is non-zero (equivalently z = 1) is the standard (affine) plane, and the "line at infinity" is the set of points for which z = 0. So what? Well, take a look at this equation "at infinity", z=0: 2 2 a x + b xy + c y = 0, and solve for x in terms of y: _____________ + / 2 -b - \/ b - 4 a c x = ---------------------------- , 2 a using the quadratic formula. Notice that the "discriminant" determines whether the intersection consists of zero, one, or two points. In fact: zero : ellipse -- curve doesn't reach infinity in x or y one : parabola -- curve just barely reaches infinity (it's tangent to the line at infinity) two : hyperbola - curve intersects the line at infinity at two points (these will turn out to yield the slopes of the asymptotes) To discriminate circles from ellipses, note that a circle will always have equal coefficients for the x**2, y**2 terms, and that there will be no xy term at all, while an ellipse will have either different coefficients for x**2 and y**2, or a non-zero xy term or both. To determine whether the conic is degenerate, you need to know three things: (i) if the quadratic factors into two linear factors (redundant: if it factors, not too much else can happen), then the curve is the union of two lines (either parallel or not): 2 2 example: 2x + xy - y + 2x - y = 0 (ii) if the gradient of the polynomial has a zero along the curve, then there can be a singularity (such as an isolated point, or the crossing of two lines): 2 2 example: x + y = 0 (iii) it is possible to have an empty point set for the solution set. 2 2 example: x + y + 1 = 0. If you've established that (i) the quadratic doesn't factor, (ii) the quadratic doesn't have any solutions in common with zeroes of both its partial derivatives (iii)there is at least one solution, then you're home free, and have a real-live conic section you can call your own. by now you've quit reading this, so I'll go quietly. Dale Hall
bs@faron.UUCP (Robert D. Silverman) (01/23/86)
> In article <532@well.UUCP> rab@well.UUCP (Bob Bickford) writes: > > > > 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? > > > > The general quadratic equation in two real variables: > > 2 2 > a x + b xy + c y + d x + e y + f = 0 > > yields (possibly degenerate) conic sections as solution sets. > The way to determine which of the three types (ellipse, parabola, > hyperbola) emerges with a given set of coefficients is to turn > the equation into a homogeneous equation in three variables: > > 2 2 2 > a x + b xy + c y + d xz + e yz + f z = 0. > > This new equation represents a curve in the (real) projective plane, > essentially a completion of the affine plane which preserves limits > of curves at infinity. To be strictly honest, points are triples: > [x,y,z], in which [x,y,z] ~ [tx,ty,tz] for any non-zero number t; > the subspace of points for which z is non-zero (equivalently z = 1) > is the standard (affine) plane, and the "line at infinity" is the > set of points for which z = 0. etc. etc. You can remove the xy , yz , and xz terms via rotation without affecting any properties of the curve. Rotating the curve through an angle theta where: tan(2 theta) = B/(A-C) will remove the Bxy term. Bob Silverman
ladkin@kestrel.ARPA (01/23/86)
In article <532@well.UUCP>, rab@well.UUCP (Bob Bickford) writes: > > 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? > They are all intersections of a plane with a cone. The equation of a cone with the origin at the point (rather, a double cone intersecting at the points),with a vertical axis, is x**2 + y**2 = a*z for some constant a. This is because the cross section is a circle, whose radius varies linearly with height above the origin (the projection on the y-z plane is straight lines (two of them). The equation of a plane is the general 3-d linear equation. Intersect the two. The resulting conic depends on the slope and position of the plane only, which are determined by the coefficients of x, y, z, and the constant term, in the plane equation. If you need the conic to be elsewhere in 3-space, apply the appropriate change-of-coordinates function, which is linear+a-constant, otherwise known as affine, and doesn't change the power of any of the x,y or z occurrences. I thought a constuctive-type answer might be more appropriate than just giving an equation, even though it takes longer Peter Ladkin ladkin@kestrel.arpa
lotto@brahms.BERKELEY.EDU (Ben Lotto) (01/23/86)
The general equation that includes all conics is a*x^2 + b*x*y + c*y^2 + d*x + e*y + f = 0 where if all of a, b, and c = 0 the graph degenerates into a line. Here are some cases: b=d=e=0, a,c>0: Circle or ellipse b=d=e=0, a>0, c<0: Hyperbola b=c=0: Parabola Varying the other coefficients will transform the graph by rotation, translation, or dilation. -Ben
berry@zinfandel.UUCP (Berry Kercheval) (01/24/86)
In article <532@well.UUCP> rab@well.UUCP (Bob Bickford) writes: > > 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? Yes. My reference is the CRC Standard Mathematical Tables, 22 ed., page 373. Consider the graph of 2 2 ax + 2hxy + by + 2gx +2fy + c = 0 a h g a h let del = det h b f , J = det , g f c h b a g b f I = a + B, K = det + det . g c f c Then we can build this neat table: Case del J del/I K Conic ----------------------------------------------- 1 !=0 >0 <0 real ellipse (circle is degenerate ellipse) 2 !=0 >0 >0 imag. ellipse 3 !=0 <0 hyperbola 4 !=0 0 parabola 5 0 <0 real intersecting lines 6 0 >0 conjugate complex lines 7 0 0 <0 real distinct parallel lines 8 0 0 >0 conjugate complex parallel lines 9 0 0 0 coincident lines in cases 1, 2 and 3 the center (x0, y0) of the conic is given by the simultaneous solution of ax + hy + g = 0, hx + by + f = 0 The equations of the axes of the conic are: y - y0 = m(x - x0), y - y0 = - 1/m (x - x0), where m is the positive root of 2 hm + (a - b)m -h = 0. I do not promis that I have made no errors of transcription, for a (more or less) guaranteed correct version consult the CRC std. math. tables; it's a wondrous useful book! -- Berry Kercheval Zehntel Inc. (ihnp4!zehntel!zinfandel!berry) (415)932-6900 (kerch@lll-tis.ARPA)
markb@sdcrdcf.UUCP (Mark Biggar) (01/24/86)
In article <144@linus.UUCP> bs@linus.UUCP (Robert D. Silverman) writes: >> >> 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? >> >This is a simple high school analytic geometry problem. A general quadratic >form in two variables is: > > 2 2 > F(x,y) = Ax + Cy + Dx + Ey + F > >F(x,y) = 0 is the general equation representing a conic section. The following >conditions hold: > >If A or C is zero the conic is a parabola or in special cases two parallel >lines which may be distinct, coincident or imaginary. > >If A and C have the same sign we have an ellipse or in special cases it can >degenerate into a single point or imaginary ellipse. It is a circle when >A = C. > >If A and C have different signs we have a hyperbola or in degenerate cases >two intersecting lines. 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 Mark Biggar {allegra,burdvax,cbosgd,hplabs,ihnp4,akgua,sdcsvax}!sdcrdcf!markb
aglew@ccvaxa.UUCP (01/25/86)
There is a formula that can encapsulate any finite set of formulae Fi(x), i=1..n: SIGMA (x-i).Fi(x). It's like asking "what is the next number in the sequence N0..Ni" - SIGMA (Nj.PI k<>j (x-Nk)) + Arb(x) evaluated at x=i+1, with Arb() totally arbitrary. But I probably never learnt the proper definition of "formula", did I?
leimkuhl@uiucdcsp.CS.UIUC.EDU (01/26/86)
Boy, remind me not to go to any of your parties.
rab@well.UUCP (Bob Bickford) (01/26/86)
Many thanks to all those who answered my request for help.
Yes, to all who said so, I'm well aware that this is really
a high-school-math problem (at least, that's where most of us
were taught it). But, when it's been ten years since high-school,
it's a little difficult to derive the answer (especially at a party).
Most answers I got included the generic quadratic equation, which
isn't much help by itself, but several gave examples of coefficient
values to obtain the various conic sections, which is exactly what
I wanted. I've downloaded the whole mess to paper (paper!?) and
given it to the unbeliever...
Now, pardon me while I review my ancient and dusty algebra and
calculus texts so I don't look so dumb next time...
Robert Bickford (rab@well.uucp)
================================================
| I doubt if these are even my own opinions. |
================================================stephen@datacube.UUCP (01/27/86)
> 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? Yes. The equation is: A*x**2 + C*y**2 + 2*D*x + 2*E*y + F = 0. This is borrowed from the VNR Concise Encyclopedia of Math (Van Nostrand Reinhold, 1975). Vary A,C,D,E, and F to get any conic.
bhuber@sjuvax.UUCP (B. Huber) (01/28/86)
>>> 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? >> >> 2 2 >> F(x,y) = Ax + Cy + Dx + Ey + F >> >>F(x,y) = 0 is the general equation representing a conic section. The following >>conditions hold: > ... >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 5600 City Ave Philadelphia, PA 19131
weemba@brahms.BERKELEY.EDU (Matthew P. Wiener) (01/30/86)
In article <2731@sjuvax.UUCP> bhuber@sjuvax.UUCP (B. Huber) writes: >>>> Is there a formula which describes *all* conic sections, which >(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. > >[a fine discussion follows] 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. ucbvax!brahms!weemba Matthew P Wiener/UCB Math Dept/Berkeley CA 94720
robison@uiucdcsb.CS.UIUC.EDU (02/04/86)
> > Does anyone know (or know where to find) expressions for the coefficients of > the (oft quoted) general second order equation in terms of the radii > ($r_{major},r_{minor}$), center ($x_c,y_c$) and rotation ($theta$) of a > circle or ellipse... One simple approach is use matrix transformations. The equation 2 2 Ax + Bxy + Cy + Dx + Ey + F = 0 can be expressed with the matrix equation: t (1) V M V = 0 t where V is the column vector [x,y,1] and M is a 3x3 matrix. (Note that M has 9 coefficients, we could define M as being symmetric to remove the extra 3 degrees of freedom without losing generality). Any translation, scaling, or rotation of (x,y) to (x',y') can be expressed with a matrix multiplication (as shown in most computer graphics texts). (2) U = T V t where U is the column vector [x',y',1] We can solve (2) for V and plug back into (1), i.e. -1 V = T U -1 t -1 (T U) M T U = 0 which simplifies to the equation: t -1t -1 U (T M T ) U = 0 from which you can dig out the coefficients for the conic section equation. - Arch D. Robison University of Illinois at Urbana-Champaign