Normal Curves of Ellipses
$begingroup$
Consider a graph of a lot of different ellipses with foci $(-1,0)$ and $(1,0)$
These ellipses take on the form
$$sqrt{(x+1)^2 + y^2} + sqrt{(x-1)^2+y^2}= K $$
where $2 < K < infty$ (the case of $K = 2$ is a degenerate case).
You can graph these for several K such as $K=2.4, 2.3, 2.2, 2.1, 2.05, ...$ on a graph calculator of your choice and basically the ellipses nest nicely into each other (see here).
Now I want to generalize the following image. Which is basically depicts how lines from the origin form "normal" curves w.r.t circles (i.e. they always intersect at a right angle from the tangent of the circle at the point of intersect around the origin)
What are the "normal curves" of the collection of ellipses i've listed?
Work:
One route is to try to describe the ellipses in standard form (which looks somewhat algebraically messy) and then try to characterize their derivatives and proceed but this feels like it would quickly get out of hand in terms of sheer amount of work so i'm hoping someone here with better intuition can highlight what the general form of the curves are.
geometry conic-sections
asked Dec 27 '18 at 2:11
frogeyedpeasfrogeyedpeas
7,58972053
$endgroup$
add a comment |
$begingroup$
Consider a graph of a lot of different ellipses with foci $(-1,0)$ and $(1,0)$
These ellipses take on the form
$$sqrt{(x+1)^2 + y^2} + sqrt{(x-1)^2+y^2}= K $$
where $2 < K < infty$ (the case of $K = 2$ is a degenerate case).
You can graph these for several K such as $K=2.4, 2.3, 2.2, 2.1, 2.05, ...$ on a graph calculator of your choice and basically the ellipses nest nicely into each other (see here).
Now I want to generalize the following image. Which is basically depicts how lines from the origin form "normal" curves w.r.t circles (i.e. they always intersect at a right angle from the tangent of the circle at the point of intersect around the origin)
What are the "normal curves" of the collection of ellipses i've listed?
Work:
One route is to try to describe the ellipses in standard form (which looks somewhat algebraically messy) and then try to characterize their derivatives and proceed but this feels like it would quickly get out of hand in terms of sheer amount of work so i'm hoping someone here with better intuition can highlight what the general form of the curves are.
geometry conic-sections
asked Dec 27 '18 at 2:11
frogeyedpeasfrogeyedpeas
7,58972053
$endgroup$
3
$begingroup$
i am suspicious that the answer will be the set of hyperbolas with the same foci. Some intuition tells me those curves fit very well
$endgroup$
– frogeyedpeas
Dec 27 '18 at 3:07
1
$begingroup$
The answer here: en.wikipedia.org/wiki/…
$endgroup$
– Aretino
Dec 28 '18 at 16:09
$begingroup$
en.wikipedia.org/wiki/Confocal_conic_sections
$endgroup$
– Yves Daoust
Dec 31 '18 at 11:40
add a comment |
$begingroup$
Consider a graph of a lot of different ellipses with foci $(-1,0)$ and $(1,0)$
These ellipses take on the form
$$sqrt{(x+1)^2 + y^2} + sqrt{(x-1)^2+y^2}= K $$
where $2 < K < infty$ (the case of $K = 2$ is a degenerate case).
You can graph these for several K such as $K=2.4, 2.3, 2.2, 2.1, 2.05, ...$ on a graph calculator of your choice and basically the ellipses nest nicely into each other (see here).
Now I want to generalize the following image. Which is basically depicts how lines from the origin form "normal" curves w.r.t circles (i.e. they always intersect at a right angle from the tangent of the circle at the point of intersect around the origin)
What are the "normal curves" of the collection of ellipses i've listed?
Work:
One route is to try to describe the ellipses in standard form (which looks somewhat algebraically messy) and then try to characterize their derivatives and proceed but this feels like it would quickly get out of hand in terms of sheer amount of work so i'm hoping someone here with better intuition can highlight what the general form of the curves are.
geometry conic-sections
asked Dec 27 '18 at 2:11
frogeyedpeasfrogeyedpeas
7,58972053
$endgroup$
Consider a graph of a lot of different ellipses with foci $(-1,0)$ and $(1,0)$
These ellipses take on the form
$$sqrt{(x+1)^2 + y^2} + sqrt{(x-1)^2+y^2}= K $$
where $2 < K < infty$ (the case of $K = 2$ is a degenerate case).
You can graph these for several K such as $K=2.4, 2.3, 2.2, 2.1, 2.05, ...$ on a graph calculator of your choice and basically the ellipses nest nicely into each other (see here).
Now I want to generalize the following image. Which is basically depicts how lines from the origin form "normal" curves w.r.t circles (i.e. they always intersect at a right angle from the tangent of the circle at the point of intersect around the origin)
What are the "normal curves" of the collection of ellipses i've listed?
Work:
One route is to try to describe the ellipses in standard form (which looks somewhat algebraically messy) and then try to characterize their derivatives and proceed but this feels like it would quickly get out of hand in terms of sheer amount of work so i'm hoping someone here with better intuition can highlight what the general form of the curves are.
geometry conic-sections
geometry conic-sections
asked Dec 27 '18 at 2:11
frogeyedpeasfrogeyedpeas
7,58972053
asked Dec 27 '18 at 2:11
frogeyedpeasfrogeyedpeas
7,58972053
asked Dec 27 '18 at 2:11
frogeyedpeasfrogeyedpeas
7,58972053
asked Dec 27 '18 at 2:11
frogeyedpeasfrogeyedpeas
7,58972053
asked Dec 27 '18 at 2:11
frogeyedpeasfrogeyedpeas
7,58972053
7,58972053
3
$begingroup$
i am suspicious that the answer will be the set of hyperbolas with the same foci. Some intuition tells me those curves fit very well
$endgroup$
– frogeyedpeas
Dec 27 '18 at 3:07
1
$begingroup$
The answer here: en.wikipedia.org/wiki/…
$endgroup$
– Aretino
Dec 28 '18 at 16:09
$begingroup$
en.wikipedia.org/wiki/Confocal_conic_sections
$endgroup$
– Yves Daoust
Dec 31 '18 at 11:40
add a comment |
3
$begingroup$
i am suspicious that the answer will be the set of hyperbolas with the same foci. Some intuition tells me those curves fit very well
$endgroup$
– frogeyedpeas
Dec 27 '18 at 3:07
1
$begingroup$
The answer here: en.wikipedia.org/wiki/…
$endgroup$
– Aretino
Dec 28 '18 at 16:09
$begingroup$
en.wikipedia.org/wiki/Confocal_conic_sections
$endgroup$
– Yves Daoust
Dec 31 '18 at 11:40
3
3
$begingroup$
i am suspicious that the answer will be the set of hyperbolas with the same foci. Some intuition tells me those curves fit very well
$endgroup$
– frogeyedpeas
Dec 27 '18 at 3:07
$begingroup$
i am suspicious that the answer will be the set of hyperbolas with the same foci. Some intuition tells me those curves fit very well
$endgroup$
– frogeyedpeas
Dec 27 '18 at 3:07
1
1
$begingroup$
The answer here: en.wikipedia.org/wiki/…
$endgroup$
– Aretino
Dec 28 '18 at 16:09
$begingroup$
The answer here: en.wikipedia.org/wiki/…
$endgroup$
– Aretino
Dec 28 '18 at 16:09
$begingroup$
en.wikipedia.org/wiki/Confocal_conic_sections
$endgroup$
– Yves Daoust
Dec 31 '18 at 11:40
$begingroup$
en.wikipedia.org/wiki/Confocal_conic_sections
$endgroup$
– Yves Daoust
Dec 31 '18 at 11:40
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
It may not be mathematically rigorous, but look at the reflection laws of conic sections.
In your ellipses a light beam from one focus is reflected into the other focus. Now imagine turning the reflector 90 degrees, then the reflected beam is turned 180 degrees so now it goes directly away from the second focus. Which is ... the reflection law for a hyperbola.
We see, then, that the orthogonal curves to confocal ellipses are confocal hyperbolas with the same pair of foci.
edited Dec 31 '18 at 11:32
user376343
3,9584829
answered Dec 27 '18 at 3:11
Oscar LanziOscar Lanzi
13.2k12136
$endgroup$
add a comment |
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$begingroup$
It may not be mathematically rigorous, but look at the reflection laws of conic sections.
In your ellipses a light beam from one focus is reflected into the other focus. Now imagine turning the reflector 90 degrees, then the reflected beam is turned 180 degrees so now it goes directly away from the second focus. Which is ... the reflection law for a hyperbola.
We see, then, that the orthogonal curves to confocal ellipses are confocal hyperbolas with the same pair of foci.
edited Dec 31 '18 at 11:32
user376343
3,9584829
answered Dec 27 '18 at 3:11
Oscar LanziOscar Lanzi
13.2k12136
$endgroup$
add a comment |
$begingroup$
It may not be mathematically rigorous, but look at the reflection laws of conic sections.
In your ellipses a light beam from one focus is reflected into the other focus. Now imagine turning the reflector 90 degrees, then the reflected beam is turned 180 degrees so now it goes directly away from the second focus. Which is ... the reflection law for a hyperbola.
We see, then, that the orthogonal curves to confocal ellipses are confocal hyperbolas with the same pair of foci.
edited Dec 31 '18 at 11:32
user376343
3,9584829
answered Dec 27 '18 at 3:11
Oscar LanziOscar Lanzi
13.2k12136
$endgroup$
add a comment |
$begingroup$
It may not be mathematically rigorous, but look at the reflection laws of conic sections.
In your ellipses a light beam from one focus is reflected into the other focus. Now imagine turning the reflector 90 degrees, then the reflected beam is turned 180 degrees so now it goes directly away from the second focus. Which is ... the reflection law for a hyperbola.
We see, then, that the orthogonal curves to confocal ellipses are confocal hyperbolas with the same pair of foci.
edited Dec 31 '18 at 11:32
user376343
3,9584829
answered Dec 27 '18 at 3:11
Oscar LanziOscar Lanzi
13.2k12136
$endgroup$
It may not be mathematically rigorous, but look at the reflection laws of conic sections.
In your ellipses a light beam from one focus is reflected into the other focus. Now imagine turning the reflector 90 degrees, then the reflected beam is turned 180 degrees so now it goes directly away from the second focus. Which is ... the reflection law for a hyperbola.
We see, then, that the orthogonal curves to confocal ellipses are confocal hyperbolas with the same pair of foci.
edited Dec 31 '18 at 11:32
user376343
3,9584829
answered Dec 27 '18 at 3:11
Oscar LanziOscar Lanzi
13.2k12136
edited Dec 31 '18 at 11:32
user376343
3,9584829
edited Dec 31 '18 at 11:32
user376343
3,9584829
edited Dec 31 '18 at 11:32
user376343
3,9584829
3,9584829
answered Dec 27 '18 at 3:11
Oscar LanziOscar Lanzi
13.2k12136
answered Dec 27 '18 at 3:11
Oscar LanziOscar Lanzi
13.2k12136
answered Dec 27 '18 at 3:11
Oscar LanziOscar Lanzi
13.2k12136
13.2k12136
add a comment |
add a comment |
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3
$begingroup$
i am suspicious that the answer will be the set of hyperbolas with the same foci. Some intuition tells me those curves fit very well
$endgroup$
– frogeyedpeas
Dec 27 '18 at 3:07
1
$begingroup$
The answer here: en.wikipedia.org/wiki/…
$endgroup$
– Aretino
Dec 28 '18 at 16:09
$begingroup$
en.wikipedia.org/wiki/Confocal_conic_sections
$endgroup$
– Yves Daoust
Dec 31 '18 at 11:40