Talk:Dandelin spheres

Making the article more user friendly

Hi, now that the i=1,2 is included it makes more sense to the reader. However, this statement now seems a little out of place and needs rewording: "because both are tangent to the same sphere" etc etc.

We can either remove the "i" term and just make statements for F1 and F2, or change the statement to read "beacuse both are tangents to sphere Gi

Personally I'd like to remove the i term altogether as it's unnecessary.

I would do this:

Each sphere touches the plane at a point, and let us call those two points F₁ and F₂.
Let P be a typical point on the ellipse.
Prove: The sum of distances d(F₁, P) + d(F₂, P) remain constant as the point P moves along the curve.
- A line passing through P and the vertex S of the cone intersects the two circles at points P₁ and P₂.
- As P moves along the ellipse, P₁ and P₂ move along the two circles.
- The distance from F₁ to P is the same as the distance from P₁ to P, because both are tangent to the same sphere (G1)
- Likewise, the distance from F₂ to P is the same as the distance from P₂ to P, because both are tangent to the same sphere (G2).
- Consequently, the sum of distances d(F₁, P) + d(F₂, P) must be constant as P moves along the curve because the sum of distances d(P₁, P) + d(P₂, P) also remains constant.
- This follows from the fact that P lies on the straight line from P₁ to P₂, and the distance from P₁ to P₂ remains constant.

--Milesoneill (talk) 08:00, 26 June 2009 (UTC)[reply]

Definition

"In geometry, the Dandelin spheres are one or two spheres that are tangent both to a plane and to a cone that intersects the plane."

Looks like the center of a Dandelin sphere is always on the axis of the cone. If that's true, the definition should say so-- if being tangent to the inside of the cone at one point isn't enough to be a Dandelin sphere — Preceding unsigned comment added by 205.154.244.243 (talk) 23:36, 5 February 2017 (UTC)[reply]

Improvement to the proof about constant sum of distances to foci

The following line in the proof (and the one after it, using the same argument) took me a little while to understand:

The distance from F₁ to P is the same as the distance from P₁ to P, because both are tangent to the same sphere (G1)

After some thought, I realized why this works. The tangency implies we have two right triangles: F1, P, M and P1, P, M. The right angles are at the vertices F1 and P1, respectively, and the line from M to P is the hypotenuse of both triangles. Moreover, the lengths from M to F1 and M to P1 are both radii of the sphere (G1), and hence are equal. The Pythagorean theorem therefore implies that the distances F1,P and P1,P are equal.

Since this is not an entirely trivial argument, perhaps it could be incorporated into the article?

Nilesj (talk) 16:25, 27 March 2017 (UTC)[reply]

We typically put new comments at the bottom of a talk page (unlike some other places on the web), so I repositioned yours ... I hope you don't mind.

You have correctly provided a proof that the lengths of the tangent line segments from a point outside of a sphere are equal. The question you raise is whether or not to include this argument in the article. If this was a textbook or a set of class notes, I would say, "yes, it certainly should be included." However, Wikipedia is not a textbook and serves a different function. There is an on-going debate about whether or not proofs should even be included in Wikipedia articles. Personally, I am ambivalent on this issue, but I can say that when they are included they should not be full of all the details that one would find in a text; proof outlines should be as far as we go. This particular proof outline is a good example, as it stands. When writing an outline, some material has to be left out and an easy choice of what to leave out are those statements that are intuitively obvious (granted, that is a somewhat subjective call). I would say that this statement about tangent line segments falls into that category, as would say, that the points of contact of these tangent segments form a circle on the sphere. These are easily believable due to our experiences with ice cream cones or dunce caps, and while they would have to be proved in a more rigorous setting they are readily accepted by a reader who does not wish to see all the details, but still wants to understand why something is true.--Bill Cherowitzo (talk) 17:12, 27 March 2017 (UTC)[reply]

Thanks for the thoughtful response. Until you mentioned ice cream cones and dunce caps, I didn't at all see how this statement should be obvious. (Not that I didn't spend time thinking about it -- I just wasn't successful.) Now I see what you mean, and I'm happy to leave it alone as you suggest. If there were another article which included basic geometric properties such as this one, I would suggest referencing it, but I haven't yet found one. (The list Sphere#Eleven_properties_of_the_sphere has some basic properties, but doesn't seem helpful for this point.)--Nilesj (talk) 17:47, 28 March 2017 (UTC)[reply]

Two more graphics, one for each parabola and hyperbola

The graphic is an incredible companion to the text, and a good graphic for a hyperbola and parabola showing the circle would be great, since most users will not be able to wrap their heads around the sphere and it's relationship to the rest until they actually see it.Brinerustle (talk) 06:48, 19 February 2021 (UTC)[reply]