2

The Wikipedia Article on Hyperbola gives the following explanation of geometrical similarity of two hyperbolas, under the topic Eccentricity:

Two hyperbolas are geometrically similar to each other – meaning that they have the same shape, so that one can be transformed into the other by rigid left and right movements, rotation, taking a mirror image, and scaling (magnification) – if and only if they have the same eccentricity.

I imagine hyperbolas having the same eccentricity are obtained by slicing planes (cutting planes) which are parallel to each other but slice (cut) both the cones of the double-napped cone. On a two dimensional coordinate plane, the two sections of the hyperbolas either come towards the centre of move away from it at the same rate. Other qualities like shape, size remain the same during this process (moving the slicing plane parallel to itself or simply translating it).

By which process, could we transform one hyperbola to the other one which is similar to the first one - rigid left and right movements, rotation, taking a mirror image, or scaling (magnification)? I think both - movement (of right and left parts of a hyperbola separately towards or away from the centre) as well as scaling could be done. But, it would be great if you could clarify this.

Thank you in advance.

Vishnu
  • 1,816
  • 1
    Take the cone with two parallel cuts and scale it so that the radius at the bottom of the cut closer to the apex becomes the same as the one as at the bottom of the cut further from it. Since the cone is self-similar this will superimpose the hyperbolas. – Conifold Nov 10 '19 at 08:43
  • @Conifold, Thank you. So, two hyperbolas which are similar (obtained by two parallel slicing planes) may have different sizes or different latus rectum, unlike the one suggested by me - they have the same latus rectum. Am I right now? – Vishnu Nov 10 '19 at 08:48
  • 1
    Latus rectum is a measure of size, like the radius of a circle. Just like circles with different radii are similar so are hyperbolas with different latus recta (but the same eccentricity). We can see it for the circles by scaling the same cone, just slicing it perpendicularly, and it also works for ellipses and parabolas. – Conifold Nov 10 '19 at 09:03
  • @Conifold, Thank you again. So can we conclude this - "on translating the slicing plane, we obtain similar conics but of different sizes" - or are there exceptions to this? – Vishnu Nov 10 '19 at 09:06
  • 1
    I like to think of the cone as a beam of light coming from the point source at the apex. If you place a transparent sheet with a shape drawn on it, and then a white sheet parallel to it further out, you'll see the same shape enlarged on the white sheet (the principle of a projector). It won't work when your plane cuts through the apex parallel to the axis, you just get a pair of lines. – Conifold Nov 10 '19 at 09:16
  • 1
    "I imagine hyperbolas having the same eccentricity are obtained by slicing planes (cutting planes) which are parallel to each other ..." Indeed, as described in this answer, the eccentricity of a conic is $\sin P/\sin C$, where $P$ and $C$ are the angles made with the "horizontal" by, respectively, the cutting plane and the cone surface (rather, a generator). ... Now, note that the Dandelin sphere configurations with parallel cutting planes must be similar; thus also, the conics they determine. – Blue Nov 10 '19 at 09:33
  • Provided that neither plane intersects the vertex of the cone, there is a simple dilation that takes the figure including the cone and an intersecting plane to the same cone and a parallel intersecting plane. So there is a geometric similarity between the hyperbolas formed by the intersections of the cone and those planes. – David K Nov 10 '19 at 15:10

0 Answers0