Given integers a & b > 0, how does one calculate the foci of the hyperbola?
What is the name of the point that lies midway on the line/ray formed between the two foci?
What is the name of the point where both nappes of a cone meet? The origin?
Can one determine how many points in the hyperbola will have co-ordinates such that x and y are whole numbers?
Is it possible to infer the complete 3-dimensional conic shape from just the hyperbola?
5 1/2. Obligatory shout-out to Opal.
Given a conic and a plane intersecting both nappes and thus defining a hyperbola (like this), is there a name for the intersecting plane? Is there a special name for the plane that contains the “origin” of the cone and forms a hyperbola consisting of two identical parabolas?
What’s the difference between a conic section and an elliptic curve?
And finally, why are hyperbolas and parabolas not considered part of Euclidean geometry?
Any input, reference sites (I think I’ve hit the big ones), or general comments are greatly appreciated.
Of course not. All possible hyperbolæ are slices of the same cone, just with different angles and distances from the point.
If the plane contains the cone point, the conic is called “degenerate”, and consists not of two parabolæ, but of two intersecting lines.
A conic section is just what you linked to: a slice of the cone by a plane. An elliptic curve is an algebraic curve (variety of dimension 1) with genus 1. That is, it is the set of solutions of equations generally looking like
y[sup]2[/sup] = x[sup]3[/sup] + ax + b
with 4a[sup]3[/sup]+27b[sup]2[/sup] nonzero. If everything in sight is a complex number, this set of solutions forms a torus (doughnut-shaped surface) in four-real-dimensional space.
To answer your follow-up right away: it’s called “elliptic” because the integrals used to calculate the lengths of ellipses involve terms of the form
sqrt(x[sup]3[/sup] + ax + b)
sometimes in the numerator and often in the denominator.
Hyperbolæ and parabolæ are not part of Euclidean geometry because Euclid didn’t write about them.
This statement suggests to me that there’s only one possible cone, which doesn’t make sense. Are you saying that any given hyperbola could be a section of any given cone depending on, for lack of a better word, perspective?
I’m probably not being clear. I picture a Cartesian co-ordinate system, on which five distinct points are plotted such that they define a hyperbola, right? Now, if you extend the co-ordinate system to include a z axis, so that you have this hyperbola now in three-dimensional space, you’re saying that one cannot then determine the shape (plot/set of points contained in) the cone of which that hyperbola is a section?
Why is the cone considered degerate? I mean, the plane’s just a plane, right? You slice the cone with another plane, one that doesn’t contain the point of the cone, and you have a different conic section but it’s still the same cone, right? Are you talking about a specific plane, like the one that contains both x and y axes?
Also:
Can you give me the long answer for determining the points of a hyperbola whose co-ordinates are both whole numbers? Or point me somewhere if the long answer is too involved for me to understand?
Would it be true to say all conic sections are elliptic curves while all elliptic curves are not conic sections?
Euclid wrote about circles, circles are conic sections, so what makes the other conic sections so different? My basic question is: when we talk about non-Euclidean geometry, is it an intellectual distinction that we’re making? You can have a circle in a Cartesian co-ordinate system and you can have a hyperbola in one as well. So we’re not adding dimensions or making some huge conceptual leap when we talk about non-Euclidean geometry, right? We’re simply talking about more complicated shapes that occur in the same system?
“Non-Euclidean geometry” doesn’t refer to geometry that simply goes beyond what’s covered in Euclid. It refers to geometry that rejects Euclids fifth, “parallel postulate,” in favor of an alternative postulate, and thus is “incompatible” with, or an alternative to, Euclidean geometry.
You can talk about hyperbolas and other conic sections while remaining firmly within the realm of Euclidean geometry.
The center? (That’s what you’d call it for an ellipse, at least, and it sorta makes sense to keep the same word in the case of a hyperbola.)
Bacially, yes. The only difference between two cones is the scaling in the z direction, and changing that just changes which plane gives which hyperbola. The cone is the set of solutions to the equation
x[sup]2[/sup] + y[sup]2[/sup] - z[sup]2[/sup] = 0
On the one hand, as I explained above, there’s only one cone to be determined in the first place. On the other hand, changing the hyperbola just means rotating that single cone by a bit to take the old intersecting plane to the new one.
[QUOTE]
Can you give me the long answer for determining the points of a hyperbola whose co-ordinates are both whole numbers? Or point me somewhere if the long answer is too involved for me to understand?
No, the two have almost nothing to do with each other.
Elliptic curve
x[sup]3[/sup] + ax + b - y[sup]2[/sup]
Euclid would definitely have seen them as different. More to the point, the study of conic sections is undertaken not by the techniques of Euclin, but those of Descartes.
As an aside, the conics are all the same thing over the complex numbers. There, they all look like spheres, though rotated in odd ways and some having points at infinity. This again shows that elliptic curves and conics are different things, since elliptic curves are shaved like tori over the complex numbers.
We’re not talking about what’s often called “non-Euclidean” geometry. We’re talking about Cartesian or analytic geometry, which is beyond the subject matter of Euclid himself and thus is not Euclidean.
OK, I think I’m starting to understand. Like putting an equation in “general form.” The form of cone vs. the actual cone itself. The shape is always the same; only the “parameters” are different. Those would be the angle of rotation, the co-ordinates of the point of the cone, etc.
So there’s only one cone in terms of the laws according to which the shape is constructed (and thus any given section of it revealed can be revealed by a specific plane). But there can be two distinct cones in the sense that the point of cone a is located at (0, 1, -1.5) and the point of cone b is located at (1, 0, 63) in a fixed xyz system, right? So what I’m asking is, if we have only the two hyperbolas revealed by the plane where z is always 0 (assuming the cones are such that that is the case), can we reconstruct the cones from just those sections?
That’s just crazy talk!! What do you mean when you say “over” the complex numbers? For them to look like a sphere, I take it the third dimension is the “imaginary one”, right? But how can they look like spheres? And doesn’t a sphere “look” the same from any perspective no matter how you rotate it? Wouldn’t a “point at infinity” have to show up as some kind of protrusion from the sphere?
Yes, see, I’m getting stuck on this whole idea of different “geometries.” I think of geometry as being singular (even though I know there are different ones), particularly within the scope of our familiar three dimensions:
Now from what Thudlow said, I inferred that Euclid’s fifth postulate was somehow wrong or incomplete, and that non-Euclidean geometry is simply an expansion by means of correcting the faulty postuate. But the above suggests to me that three “classes” of geometries are all equally valid, simply divergent on the fifth postulate. But then, that’s only “constant curvature” geometries. WTF? How many geometries are there?
If we let x and y take complex values, that’s four real variables to consider – the real and imaginary parts of x and y. The set of solutions inside this four-dimensional space in each case is a surface.
For elliptic curves (“curve”, since the two real parameters on the surface are one complex parameter), the solution set for any given (a,b) pair is shaped like a torus, though embedded in 4-d space differently for each pair. For conic sections, the solution set is shaped like a sphere. The reason that conic sections look different to you is that what you see is the slice of this sphere by the plane of real values of x and y, and that can intersect the sphere in various different ways.
In three dimensions, we’re still not sure. Perelman thinks he’s got the answer, though. In four and higher we can prove that it’s immpossibly to classify them.
Whoa. I didn’t consider that you’d need two extra dimensions to track the complex parts. But I don’t know what anything looks like in 4-d. I’ve been taught to think of 4-d images in terms of what their 3-d “shadows” would look like.
Apollonius of Perga is typically credited with first naming and analyzing the Conic sections. As he flourished some 75 years after Euclid, this would explain why Euclid himself didn’t discuss the conics, but just because Apollonius got the credit doesn’t mean previous Greeks (e.g. the Pythagoreans) didn’t discuss them.
Euclid did (in the Phenomina) prove an ellipse (loci of points the sum of whose distant from two focal points is a constant) can be obtained by slicing a cylinder with a plane at an angle. He did this by inscribing two spheres inside the cylinder and tangent to the slicing plane from above and below. Note then that the line along the cylinder from a point A on the edge of the ellipse directly up (or down) to the equator of the tangent sphere is equal to the distance from A to the point on the interior of the ellipse where the sphere touches (hard to visualize without a drawing; sorry ). From here it’s easy to see the sum of the distance from A to the two tangent points is equal to the distance between the equators of the spheres that goes thru A, and this is the same for any A.
Using the more standard form for a pair of hyperbolae:
x^2/a^2 - y^2/b^2 = 1
It is easy to see x and y will be non-zero whole numbers for at most only one possible value: The one where bx, ab, and ay form a Pythagorean triple. If a and b are coprime integers, this implies b|y and a|x. Let x=ja and y=kb, and we get j^2 - k^2 =1. This can only be solved by j=1, k=0, so the only place where such a hyperbola has x and y whole numbers is at x=0.
I suspect the same is true if a and b are not coprime, but it’s late and I’ve got to get some sleep…
Grrrr…I can’t get any sleep with an unsolved math problem lingering. Regarding the question of the points of a hyperbola whose co-ordinates are both whole numbers, in my last post I used the standard form:
x^2/a^2 - y^2/b^2 = 1
and showed that when a and b are coprime the only solution in whole numbers is the trivial x=a, y=0. Obviously there are whole-number solutions when a and b share a common factor: when a=b=3, x=5, y=4 is a solution.
If a and b share a greatest common factor of, say f, the equation can then be written:
x^2/(Af)^2 - y^2/(Bf)^2 = 1
where a=Af, b=Bf, and A and B are again coprime. Following the skeleton of the coprime argument, this leads to a pythagorean triple of (ABf, Ay, Bx). Again, y must have B as a factor and x must have A as a factor, so let x=jA and y=kB, and we are led to j^2 - k^2 = f^2. This always has non-trivial solutions for f>1, so we can say non-zero whole number solutions for x and y exist when a and b have a common factor.
What is interesting to note is that, for fixed GCF f, there are only a finite number of solutions to the equation j^2 - k^2 = f^2. This is because if j is large enough it is not possible to find another square within f^2 units of j^2 (e.g. 2j-1 > f^2). Thus, no matter the hyperbola, there are only a finitie number of points whose coordinates are each whole numbers, and these all lie within some defined radius of the origin.
Now, for cases where a and b are not integers (or at least not rational), I’ll leave that for someone else to investigate. Really, that’s it:-)
You have violated the Opal commandment, which says that the third item in every list shall be “Hi Opal”. Your soul is in great danger. Get down on your knees and beg forgiveness from Cecil.
Yes, that is correct: they are all equally valid, in the sense of being internally consistent. You can neither prove nor disprove the fifth postulate from the other four (though people tried to for centuries).
Think of it this way. Up to rigid motions (translation and rotation), two values define a hyperbola (for example, the angle between the asymptotes and the focal distance; or the denominators a and b in standard form). But there are three parameters defining the cone-plane geometry (for example, the cone’s vertex angle, the angle between the cone’s axis and the plane, and the distance from the cone’s vertex to the plane). You can’t expect to solve for three parameters given only two values (assuming that the values are some reasonably-continuous functions of the parameters). If you were very lucky the equations might have worked out so that you could solve for one or two of the parameters, but that doesn’t happen in this case.
Depens on what you mean by geometry. With the standard modern usage of the term, Mathochist’s earlier remarks apply.
But there’s another, reasonable view in which there are infinite geometries and the number of dimensions doesn’t matter. The details of that get interesting–for those in the know, the magic words are “model-theoretic” and “theory with equality”–but I thought it might be interesting to mention.
One problem with this solution is that the most common techniques to get the hyperbola in standard form (rigid rotation and completing the square) do not necessarily map integral solutions to other integral solutions; so this solution does not immediately translate into the OP’s form
Here’s an easy proof, for this equation, that there are a finite number of integral solutions when a and b are integral: Write
y = (a+x[sup]2[/sup])/(b-x) = -(x + b) + (a + b[sup]2[/sup])/(b-x).
For integer x, the inequalities 0 < |(a + b[sup]2[/sup])/(b-x)| < 1 hold for all sufficiently large |x| (specifically, whenever x > b + a + b[sup]2[/sup] or x < b - a - b[sup]2[/sup]), so for large |x| the quantity (a + b[sup]2[/sup])/(b-x) is clearly not integral, and so y is also nonintegral. So there are at most finitely many integer solutions; this argument immediately bounds the number of solutions to 2(a + b[sup]2[/sup]).
But better bounds are possible: For convenience, make the coordinate transformation from (x,y) to (u=b-x,v=y+x+b). Clearly u,v are integers iff x,y are integers (write x=b-u,y=v+u-2b). But we can rewrite the original equation (for x!=b) as
uv = a + b[sup]2[/sup] = c
and clearly there is one integral solution (u,v) (and hence one integral solution (x,y)) for every pair of integers whose product is c. The number of solutions is thus precisely the number of integral divisors of c.
). From here it’s easy to see the sum of the distance from A to the two tangent points is equal to the distance between the equators of the spheres that goes thru A, and this is the same for any A.