Ken Fabian

Ellipse definition confusion

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Posted (edited)
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In mathematics , an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.

Wikipedia starts with this definition - but that looks to me like it would define an oval shape, ie one that is symmetrical. Elsewhere I have encountered the same kind of definition - taking two pegs with a length of string longer than the distance between and scribing a line with a third peg whilst pulling the string taut makes an ellipse shape. To me this looks like it describes only very specific sort of ellipse and is not a universal description of all ellipses.

Rather than a cross section through a cone shape, this looks to me like it describes a cross section through a round cylinder. Am I missing something obvious here?

Edited by Ken Fabian

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This description does work universally; it takes a bit of algebra to demonstrate it, but it does work. 

Ellipses can be seen as cross sections through both cones and cylinders.. 

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11 hours ago, Ken Fabian said:
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In mathematics , an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.

Wikipedia starts with this definition - but that looks to me like it would define an oval shape, ie one that is symmetrical. Elsewhere I have encountered the same kind of definition - taking two pegs with a length of string longer than the distance between and scribing a line with a third peg whilst pulling the string taut makes an ellipse shape. To me this looks like it describes only very specific sort of ellipse and is not a universal description of all ellipses.

Rather than a cross section through a cone shape, this looks to me like it describes a cross section through a round cylinder. Am I missing something obvious here?

There are other equivalent definitions such as

A regular conic is the locus of a point that moves so that the ratio of its undirected distance form a fixed point F to the distance from a fixed line L is equal to a constanr, e.

If e < 1 the conic is an ellipse

If e = 1 then the conic is a parabola

if e > 1 the conic is a hyperbola

This definition has the same characteristic as yours, that it is not dependent upon coordinate geometry, although a coordnate equation for an ellipse can be derived from it.

 

However a couple of points about terminology.

an ellipse is a plane curve.

That is it exists totally in a single plane.
So it is not necessary to to invoke anything to do with a third dimension (cones, cylinders etc) to say everything there is to say about the ellipse itself.
The third dimension only changes its situation.

An oval is not a single curve  but a compound of several curves, although it too is a plane figure.

Fairly generally,

There are two parts of two curves (which may be different or the same) whoes ends are linked or joined by parts of two more curves or lines which are tangent to the first two curves.

So the standard oval table has two identical part circular curves linked by straight lines.

An egg has two different parabolic curves linked by smooth transition curves.

An ellipse could be said to have two identical half elliptical ends linked by zero length straights.

and so on.

 

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Posted (edited)

The closed curve described by the definition given for an ellipse - sum of distances to two fixed focal points is constant - is symmetrical in two directions; the left mirrors the right and the top mirrors the bottom. However an angled section through a cone is symmetrical in only one direction - it has a large curve at the 'bottom' and tighter curve on the top; the left mirrors the right but the top doesn't mirror the bottom. The sum of the distances to two fixed focal points is NOT constant for every point on such a curve.Therefore the conic section shape being called an ellipse is not an ellipse by that definition.  It must be some other kind of shape. What shape is it?

Edited by Ken Fabian
clarity

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12 minutes ago, Ken Fabian said:

The closed curve described by the definition given for an ellipse - sum of distances to two fixed focal points is constant - is symmetrical in two directions; the left mirrors the right and the top mirrors the bottom. However an angled section through a cone is symmetrical in only one direction - it has a large curve at the 'bottom' and tighter curve on the top; the left mirrors the right but the top doesn't mirror the bottom. The sum of the distances to two fixed focal points is NOT constant for every point on such a curve.Therefore the conic section shape being called an ellipse is not an ellipse by that definition.  It must be some other kind of shape. What shape is it?

 

You will need to offer some very serious proof for this unsubstantiated claim.

With the origin placed at the centre of the ellipse,

The ellipse is symmetrical about the origin, and it is symmetrical about two conjugate diameters.
The conjugate diameters may form rectangular axes or may be skew.

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Meanwhile this conventional information about ellipses may be of interest / use.

It is important that the angle between the ellipse cutting plane and the cone axis is greater than the semi vertex angle or you will not get an ellipse.

ellipse1.jpg.11ca8110a6d8cc31d1cc75919a9fce8e.jpg

 

Here is an extract from an old tech drawing book showing two ways to extractt he true shape of the figure from the cone and the cutting plane.

(Obviously a saw and a piece of cone shaped wood is another. You have this with th coal scuttle picture in the attachment.)

ellipse2.thumb.jpg.b498fd341ccebe1da16ebf3cd8c16c90.jpg

 

Finally for a bit of fun you can create an ellips on a piece of tracing /greaseproof paper by drawing a circle on the paper.
Then marking a single point anywhere inside the circle.
Then folding the paper so over that the point lies on the circle circumference and creasing the fold.
Repeat several times and the creases will build up the envelope of an ellipse.

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On 3/6/2019 at 1:19 AM, studiot said:

There are other equivalent definitions such as

A regular conic is the locus of a point that moves so that the ratio of its undirected distance form a fixed point F to the distance from a fixed line L is equal to a constanr, e.

If e < 1 the conic is an ellipse

If e = 1 then the conic is a parabola

if e > 1 the conic is a hyperbola

This definition has the same characteristic as yours, that it is not dependent upon coordinate geometry, although a coordnate equation for an ellipse can be derived from it.

 

However a couple of points about terminology.

an ellipse is a plane curve.

That is it exists totally in a single plane.
So it is not necessary to to invoke anything to do with a third dimension (cones, cylinders etc) to say everything there is to say about the ellipse itself.
The third dimension only changes its situation.

An oval is not a single curve  but a compound of several curves, although it too is a plane figure. 

Fairly generally,

There are two parts of two curves (which may be different or the same) whoes ends are linked or joined by parts of two more curves or lines which are tangent to the first two curves.

So the standard oval table has two identical part circular curves linked by straight lines.

An egg has two different parabolic curves linked by smooth transition curves.

An ellipse could be said to have two identical half elliptical ends linked by zero length straights.

and so on.

 

My confusion is not diminishing. My command of correct terminology is poor, but yes, I do understand that there will be a range of angles within which a cross section through a cone that make a 'closed' shape called an ellipse - that at other angles it will make hyperbola and parabola, or at 90 degrees, a circle. (Although it makes my head hurt thinking about the precise angle, parallel to the side of the cone, where it stops being 'closed'.)

'Oval' can loosely describe a variety of shapes but I was thinking specifically of the closed curve you get with an angled cross section through a round cylinder - which I say is what the "sum of distances" definition of an ellipse describes - a curve that is symmetrical both ways. A cylinder may be seen as a special case of a cone - one with a 'point' of zero degrees - but a slice through a cone with other than zero degrees does not produce a curve that is symmetrical both ways. If you like, when I think of an ellipse I think of a shape where the semi-major axes are not equal - only the semi-minor axes are equal. I would call this eccentricity (like in an eccentric orbit) except the term looks already taken, for describing the difference between major axis and minor axis. I don't see how that shape can be made with the "sum of distances is constant" definition - seems like an additional parameter is required.

Quote

"An egg has two different parabolic curves linked by smooth transition curves."

What I think of as an ellipse resembles the cross section of an egg - but I would think an ellipse is it's very own, unique curve, not a combination of parabolic and transitionary curves; that will only be an approximation. Are you saying the conic section version of an 'ellipse' does not actually have a fatter 'bottom' than top? ('Top' being the part of the curve nearest the point of the cone)  ie it will be mirrored top to bottom as well as left to right, and my belief that it resembles the egg cross section is just wrong?

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12 hours ago, Ken Fabian said:

The closed curve described by the definition given for an ellipse - sum of distances to two fixed focal points is constant - is symmetrical in two directions; the left mirrors the right and the top mirrors the bottom. However an angled section through a cone is symmetrical in only one direction - it has a large curve at the 'bottom' and tighter curve on the top; the left mirrors the right but the top doesn't mirror the bottom. The sum of the distances to two fixed focal points is NOT constant for every point on such a curve.Therefore the conic section shape being called an ellipse is not an ellipse by that definition.  It must be some other kind of shape. What shape is it?

While you are correct that there is only one obvious symmetry, it turns out that the equations of the cone and plane result in another symmetry. The conic section (whenever the plane is at a shallower angle than the cone itself) is an ellipse.

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12 minutes ago, uncool said:

While you are correct that there is only one obvious symmetry, it turns out that the equations of the cone and plane result in another symmetry. The conic section (whenever the plane is at a shallower angle than the cone itself) is an ellipse.

No point trying showing me the equations - my algebra crashed and sank very early, on the rocks of factorisation.

Is the shape made by an angled cross section of a round cylinder the same shape as the angled cross section of a cone? I have to say they look different to me - and my confusion about what is and what is not an ellipse is not being relieved so far.

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Posted (edited)

Any shape made by an angled cross section of a circular cylinder can also be made by an angled cross section of a cone, with the exception of the degenerate case (a pair of parallel lines). It will be an ellipse. 

Edited by uncool

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On 3/5/2019 at 12:51 PM, Ken Fabian said:

Wikipedia starts with this definition - but that looks to me like it would define an oval shape, ie one that is symmetrical. Elsewhere I have encountered the same kind of definition - taking two pegs with a length of string longer than the distance between and scribing a line with a third peg whilst pulling the string taut makes an ellipse shape. To me this looks like it describes only very specific sort of ellipse and is not a universal description of all ellipses.

Rather than a cross section through a cone shape, this looks to me like it describes a cross section through a round cylinder. Am I missing something obvious here?

https://courses.lumenlearning.com/waymakercollegealgebra/chapter/equations-of-ellipses/

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There is also some overlap of terminology between oval and ellipse.

https://en.wikipedia.org/wiki/Cassini_oval

 

Quote

Wiki

A Cassini oval is a quartic plane curve defined as the set (or locus) of points in the plane such that the product of the distances to two fixed points is constant. This may be contrasted with an ellipse, for which the sum of the distances is constant, rather than the product.

But

 

http://mathworld.wolfram.com/CassiniOvals.html

 

Quote

Wolfram

The Cassini ovals are a family of quartic curves, also called Cassini ellipses,

 

I prefer to use the oval as a more general term see the shapes available on Wofram / Wikipedia,

And reserve the term ellipse for the true conic only.

 

Note again, both the oval and the ellipse are plane curves.

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Sounds like my mistake is thinking that the slice through a cone must be asymmetric top to bottom - that the part of the curve closest to the point of the cone would be different (tighter curve) than the part of the curve closest to the 'base' - and that it must be a different shape than what you get when you slice through a round cylinder. It seems counter-intuitive to me, that a slice through a cone makes the same shape as a slice through a cylinder.

I had always thought an ellipse was like this -

img029.jpg.460456d1601950c8d2701c1ff91e0754.jpg

And if this is not the shape of a slice through a cone and is not an ellipse - what shape is it? (besides 'egg' shape)

 

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46 minutes ago, Ken Fabian said:

And if this is not the shape of a slice through a cone and is not an ellipse - what shape is it? (besides 'egg' shape)

 

Quote

Oxford English Dictionary

Oval

Derivation : Latin Ovalis 

Having the outline of an egg as projected on a surface.

Egg shaped, a plane figure resembling the longitudinal section of an egg.

Loosely : elliptical.

Australian : a ground for Australian rules football.

 

 

 

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