History of definitions for an ellipse?
Recently I've been learning about ellipses.
It seems as though there are four (from what I've learned of so far) different ways to define ellipses, all which seem to be connected in kind of obscure ways:
- It's a stretched circle. We get the formula for a unit circle, $X^2 + Y^2 = 1$, and stretch it by dividing the terms like so: $(frac{X}{a})^2 + (frac{Y}{b})^2 = 1$. In order to satisfy the same equation, for every Y we previously had, X must get stretched by a factor of a, and for every X we previously had, Y must be stretched (multiplied) by a factor of b.
- An ellipse is the set of all points whose sum of the distances from two points, the foci, is a constant. We can represent this with the equation $sqrt{(x+f)^2 + y^2} + sqrt{(x-f)^2 + y^2} = c^2$,
where $c = 2a$ from the previous equation, $f$ is the distance from a focus to the origin, and $x$ and $y$ are the variables.
- An ellipse is a slice of a cone at an angle. I don't know any of the equations for this definition, if someone could edit them in that would be great.
- An ellipse is a locus of points whose distance from the focus at every $(x,y)$ is proportional to the horizontal distance from a vertical line, the directrix, where the ratio is <1.
I'd really like to know more about the history of the ellipse.
Did they come up with them separately, or was it all around the same time? Which came first, and from who? Did the same people that came up with one definition somehow come up with others? And how did mathematicians see the connections between them, and realize they were looking at the same family of curves?
The connections between, for example, the "squashed circle" definition and the "constant sum of distances" definition are pretty hard to notice...who noticed that these were the same family of shapes? I mean, without being told that the foci DO exist, I'm not sure how I would be able to figure out, only from the squashed circle definition, that they indeed exist...(I asked this in another question, but in this one I'm more interested about the history. I'll link the other question below)
Thank You!
Other Non-History Question:
How do we deduce that an ellipse, when defined as a "stretched circle", has foci?
conic-sections math-history
asked 19 hours ago
Joshua Ronis
1355
add a comment |
Recently I've been learning about ellipses.
It seems as though there are four (from what I've learned of so far) different ways to define ellipses, all which seem to be connected in kind of obscure ways:
- It's a stretched circle. We get the formula for a unit circle, $X^2 + Y^2 = 1$, and stretch it by dividing the terms like so: $(frac{X}{a})^2 + (frac{Y}{b})^2 = 1$. In order to satisfy the same equation, for every Y we previously had, X must get stretched by a factor of a, and for every X we previously had, Y must be stretched (multiplied) by a factor of b.
- An ellipse is the set of all points whose sum of the distances from two points, the foci, is a constant. We can represent this with the equation $sqrt{(x+f)^2 + y^2} + sqrt{(x-f)^2 + y^2} = c^2$,
where $c = 2a$ from the previous equation, $f$ is the distance from a focus to the origin, and $x$ and $y$ are the variables.
- An ellipse is a slice of a cone at an angle. I don't know any of the equations for this definition, if someone could edit them in that would be great.
- An ellipse is a locus of points whose distance from the focus at every $(x,y)$ is proportional to the horizontal distance from a vertical line, the directrix, where the ratio is <1.
I'd really like to know more about the history of the ellipse.
Did they come up with them separately, or was it all around the same time? Which came first, and from who? Did the same people that came up with one definition somehow come up with others? And how did mathematicians see the connections between them, and realize they were looking at the same family of curves?
The connections between, for example, the "squashed circle" definition and the "constant sum of distances" definition are pretty hard to notice...who noticed that these were the same family of shapes? I mean, without being told that the foci DO exist, I'm not sure how I would be able to figure out, only from the squashed circle definition, that they indeed exist...(I asked this in another question, but in this one I'm more interested about the history. I'll link the other question below)
Thank You!
Other Non-History Question:
How do we deduce that an ellipse, when defined as a "stretched circle", has foci?
conic-sections math-history
asked 19 hours ago
Joshua Ronis
1355
1
We do have a sister side dedicated specifically to the history of math and science. I think your question is better suited there.
– Arthur
19 hours ago
1
Regarding “appearing at the same time”: The conic section (i.e. slice of a cone) definition is from ancient Greece or earlier, which was a couple of thousand years before anybody even thought about drawing coordinate systems and writing equations for the ellipse.
– Hans Lundmark
18 hours ago
sites.math.rutgers.edu/~cherlin/History/Papers1999/…
– Hans Lundmark
17 hours ago
This might be of help: math.stackexchange.com/questions/2221890/…
– Aretino
11 hours ago
add a comment |
Recently I've been learning about ellipses.
It seems as though there are four (from what I've learned of so far) different ways to define ellipses, all which seem to be connected in kind of obscure ways:
- It's a stretched circle. We get the formula for a unit circle, $X^2 + Y^2 = 1$, and stretch it by dividing the terms like so: $(frac{X}{a})^2 + (frac{Y}{b})^2 = 1$. In order to satisfy the same equation, for every Y we previously had, X must get stretched by a factor of a, and for every X we previously had, Y must be stretched (multiplied) by a factor of b.
- An ellipse is the set of all points whose sum of the distances from two points, the foci, is a constant. We can represent this with the equation $sqrt{(x+f)^2 + y^2} + sqrt{(x-f)^2 + y^2} = c^2$,
where $c = 2a$ from the previous equation, $f$ is the distance from a focus to the origin, and $x$ and $y$ are the variables.
- An ellipse is a slice of a cone at an angle. I don't know any of the equations for this definition, if someone could edit them in that would be great.
- An ellipse is a locus of points whose distance from the focus at every $(x,y)$ is proportional to the horizontal distance from a vertical line, the directrix, where the ratio is <1.
I'd really like to know more about the history of the ellipse.
Did they come up with them separately, or was it all around the same time? Which came first, and from who? Did the same people that came up with one definition somehow come up with others? And how did mathematicians see the connections between them, and realize they were looking at the same family of curves?
The connections between, for example, the "squashed circle" definition and the "constant sum of distances" definition are pretty hard to notice...who noticed that these were the same family of shapes? I mean, without being told that the foci DO exist, I'm not sure how I would be able to figure out, only from the squashed circle definition, that they indeed exist...(I asked this in another question, but in this one I'm more interested about the history. I'll link the other question below)
Thank You!
Other Non-History Question:
How do we deduce that an ellipse, when defined as a "stretched circle", has foci?
conic-sections math-history
asked 19 hours ago
Joshua Ronis
1355
Recently I've been learning about ellipses.
It seems as though there are four (from what I've learned of so far) different ways to define ellipses, all which seem to be connected in kind of obscure ways:
- It's a stretched circle. We get the formula for a unit circle, $X^2 + Y^2 = 1$, and stretch it by dividing the terms like so: $(frac{X}{a})^2 + (frac{Y}{b})^2 = 1$. In order to satisfy the same equation, for every Y we previously had, X must get stretched by a factor of a, and for every X we previously had, Y must be stretched (multiplied) by a factor of b.
- An ellipse is the set of all points whose sum of the distances from two points, the foci, is a constant. We can represent this with the equation $sqrt{(x+f)^2 + y^2} + sqrt{(x-f)^2 + y^2} = c^2$,
where $c = 2a$ from the previous equation, $f$ is the distance from a focus to the origin, and $x$ and $y$ are the variables.
- An ellipse is a slice of a cone at an angle. I don't know any of the equations for this definition, if someone could edit them in that would be great.
- An ellipse is a locus of points whose distance from the focus at every $(x,y)$ is proportional to the horizontal distance from a vertical line, the directrix, where the ratio is <1.
I'd really like to know more about the history of the ellipse.
Did they come up with them separately, or was it all around the same time? Which came first, and from who? Did the same people that came up with one definition somehow come up with others? And how did mathematicians see the connections between them, and realize they were looking at the same family of curves?
The connections between, for example, the "squashed circle" definition and the "constant sum of distances" definition are pretty hard to notice...who noticed that these were the same family of shapes? I mean, without being told that the foci DO exist, I'm not sure how I would be able to figure out, only from the squashed circle definition, that they indeed exist...(I asked this in another question, but in this one I'm more interested about the history. I'll link the other question below)
Thank You!
Other Non-History Question:
How do we deduce that an ellipse, when defined as a "stretched circle", has foci?
conic-sections math-history
conic-sections math-history
asked 19 hours ago
Joshua Ronis
1355
asked 19 hours ago
Joshua Ronis
1355
asked 19 hours ago
Joshua Ronis
1355
asked 19 hours ago
Joshua Ronis
1355
asked 19 hours ago
Joshua Ronis
1355
1355
1
We do have a sister side dedicated specifically to the history of math and science. I think your question is better suited there.
– Arthur
19 hours ago
1
Regarding “appearing at the same time”: The conic section (i.e. slice of a cone) definition is from ancient Greece or earlier, which was a couple of thousand years before anybody even thought about drawing coordinate systems and writing equations for the ellipse.
– Hans Lundmark
18 hours ago
sites.math.rutgers.edu/~cherlin/History/Papers1999/…
– Hans Lundmark
17 hours ago
This might be of help: math.stackexchange.com/questions/2221890/…
– Aretino
11 hours ago
add a comment |
1
We do have a sister side dedicated specifically to the history of math and science. I think your question is better suited there.
– Arthur
19 hours ago
1
Regarding “appearing at the same time”: The conic section (i.e. slice of a cone) definition is from ancient Greece or earlier, which was a couple of thousand years before anybody even thought about drawing coordinate systems and writing equations for the ellipse.
– Hans Lundmark
18 hours ago
sites.math.rutgers.edu/~cherlin/History/Papers1999/…
– Hans Lundmark
17 hours ago
This might be of help: math.stackexchange.com/questions/2221890/…
– Aretino
11 hours ago
1
1
We do have a sister side dedicated specifically to the history of math and science. I think your question is better suited there.
– Arthur
19 hours ago
We do have a sister side dedicated specifically to the history of math and science. I think your question is better suited there.
– Arthur
19 hours ago
1
1
Regarding “appearing at the same time”: The conic section (i.e. slice of a cone) definition is from ancient Greece or earlier, which was a couple of thousand years before anybody even thought about drawing coordinate systems and writing equations for the ellipse.
– Hans Lundmark
18 hours ago
Regarding “appearing at the same time”: The conic section (i.e. slice of a cone) definition is from ancient Greece or earlier, which was a couple of thousand years before anybody even thought about drawing coordinate systems and writing equations for the ellipse.
– Hans Lundmark
18 hours ago
sites.math.rutgers.edu/~cherlin/History/Papers1999/…
– Hans Lundmark
17 hours ago
sites.math.rutgers.edu/~cherlin/History/Papers1999/…
– Hans Lundmark
17 hours ago
This might be of help: math.stackexchange.com/questions/2221890/…
– Aretino
11 hours ago
This might be of help: math.stackexchange.com/questions/2221890/…
– Aretino
11 hours ago
add a comment |
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We do have a sister side dedicated specifically to the history of math and science. I think your question is better suited there.
– Arthur
19 hours ago
1
Regarding “appearing at the same time”: The conic section (i.e. slice of a cone) definition is from ancient Greece or earlier, which was a couple of thousand years before anybody even thought about drawing coordinate systems and writing equations for the ellipse.
– Hans Lundmark
18 hours ago
sites.math.rutgers.edu/~cherlin/History/Papers1999/…
– Hans Lundmark
17 hours ago
This might be of help: math.stackexchange.com/questions/2221890/…
– Aretino
11 hours ago