Minimal number of points to define an elliptic arc?
What is the minimal number of points to uniquely define an elliptic arc (portion of an ellipse) ?
The points are not restricted to be part of the path of the ellipse, each of them can have a different meaning/semantics (i.e. they can be "the center", "a focus", or any other reference).
So far my idea is:
- Point #1: one endpoint of the elliptic arc
- Point #2: the other endpoint of the elliptic arc
- Point #3: the center of the ellipse
But, how many concentric ellipses that pass through the same two points are there? Only 2? ... or many?
geometry conic-sections
asked Nov 22 at 6:25
Abraham Sanchez
111
add a comment |
What is the minimal number of points to uniquely define an elliptic arc (portion of an ellipse) ?
The points are not restricted to be part of the path of the ellipse, each of them can have a different meaning/semantics (i.e. they can be "the center", "a focus", or any other reference).
So far my idea is:
- Point #1: one endpoint of the elliptic arc
- Point #2: the other endpoint of the elliptic arc
- Point #3: the center of the ellipse
But, how many concentric ellipses that pass through the same two points are there? Only 2? ... or many?
geometry conic-sections
asked Nov 22 at 6:25
Abraham Sanchez
111
According to the numbers of unknown - write the general formula and based on it derive the number of points.
– Moti
Nov 22 at 6:35
1
One can also know the two foci and one point on the ellipse, since then one can get the length sum.
– coffeemath
Nov 22 at 6:42
add a comment |
What is the minimal number of points to uniquely define an elliptic arc (portion of an ellipse) ?
The points are not restricted to be part of the path of the ellipse, each of them can have a different meaning/semantics (i.e. they can be "the center", "a focus", or any other reference).
So far my idea is:
- Point #1: one endpoint of the elliptic arc
- Point #2: the other endpoint of the elliptic arc
- Point #3: the center of the ellipse
But, how many concentric ellipses that pass through the same two points are there? Only 2? ... or many?
geometry conic-sections
asked Nov 22 at 6:25
Abraham Sanchez
111
What is the minimal number of points to uniquely define an elliptic arc (portion of an ellipse) ?
The points are not restricted to be part of the path of the ellipse, each of them can have a different meaning/semantics (i.e. they can be "the center", "a focus", or any other reference).
So far my idea is:
- Point #1: one endpoint of the elliptic arc
- Point #2: the other endpoint of the elliptic arc
- Point #3: the center of the ellipse
But, how many concentric ellipses that pass through the same two points are there? Only 2? ... or many?
geometry conic-sections
geometry conic-sections
asked Nov 22 at 6:25
Abraham Sanchez
111
asked Nov 22 at 6:25
Abraham Sanchez
111
asked Nov 22 at 6:25
Abraham Sanchez
111
asked Nov 22 at 6:25
Abraham Sanchez
111
asked Nov 22 at 6:25
Abraham Sanchez
111
111
According to the numbers of unknown - write the general formula and based on it derive the number of points.
– Moti
Nov 22 at 6:35
1
One can also know the two foci and one point on the ellipse, since then one can get the length sum.
– coffeemath
Nov 22 at 6:42
add a comment |
According to the numbers of unknown - write the general formula and based on it derive the number of points.
– Moti
Nov 22 at 6:35
1
One can also know the two foci and one point on the ellipse, since then one can get the length sum.
– coffeemath
Nov 22 at 6:42
According to the numbers of unknown - write the general formula and based on it derive the number of points.
– Moti
Nov 22 at 6:35
According to the numbers of unknown - write the general formula and based on it derive the number of points.
– Moti
Nov 22 at 6:35
1
1
One can also know the two foci and one point on the ellipse, since then one can get the length sum.
– coffeemath
Nov 22 at 6:42
One can also know the two foci and one point on the ellipse, since then one can get the length sum.
– coffeemath
Nov 22 at 6:42
add a comment |
3 Answers
3
active
oldest
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For an ellipse, you need five independent pieces of information, so for an arc, which has two more degrees of freedom, you would need seven. Note that this isn't always enough to uniquely determine the arc, but it is enough to ensure that only finitely many possible arcs exist.
A point on the arc is a single piece. An end point of the arc is two pieces, and the length of the arc is one piece. A focal point of the ellipse is two pieces, as is the center. The line through the focal points is two pieces, although if you already know the center or one of the focal points, it is only a single piece of information.
answered Nov 22 at 6:43
Arthur
110k7105186
add a comment |
In general something else is needed. If the two endpoints of the arc happen to be on the line through the center, and the ellipse is not rotated (so has its major/minor axes parallel to the $x,y$ axes), then there will be an infinite family of such elliptic arcs, not to mention the choice of whether to use the top or bottom or left or right arc.
answered Nov 22 at 6:38
coffeemath
2,3181413
add a comment |
Three points and a real are enough.
You can describe the ellipse by a focus, excentricity and directrix.
- focus: one point or two reals
- directrix: an arbitrary line not passing through the focus, so two reals (say, distance from focus and orientation). That can be a point too: the point of the directrix nearest to the ellipse.
- eccentricity: one real
You then need two reals to describe the two endpoints (say, distance from the point nearest to the directrix, clockwise).
That's 7 reals, which you can "encode" in 3 points and a real.
Of course, you could theoretically encode the seven reals within only one real, since there is a bijection between $Bbb R^7$ and $Bbb R$, but that's cheating.
answered Nov 22 at 6:58
Jean-Claude Arbaut
14.7k63363
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
For an ellipse, you need five independent pieces of information, so for an arc, which has two more degrees of freedom, you would need seven. Note that this isn't always enough to uniquely determine the arc, but it is enough to ensure that only finitely many possible arcs exist.
A point on the arc is a single piece. An end point of the arc is two pieces, and the length of the arc is one piece. A focal point of the ellipse is two pieces, as is the center. The line through the focal points is two pieces, although if you already know the center or one of the focal points, it is only a single piece of information.
answered Nov 22 at 6:43
Arthur
110k7105186
add a comment |
For an ellipse, you need five independent pieces of information, so for an arc, which has two more degrees of freedom, you would need seven. Note that this isn't always enough to uniquely determine the arc, but it is enough to ensure that only finitely many possible arcs exist.
A point on the arc is a single piece. An end point of the arc is two pieces, and the length of the arc is one piece. A focal point of the ellipse is two pieces, as is the center. The line through the focal points is two pieces, although if you already know the center or one of the focal points, it is only a single piece of information.
answered Nov 22 at 6:43
Arthur
110k7105186
add a comment |
For an ellipse, you need five independent pieces of information, so for an arc, which has two more degrees of freedom, you would need seven. Note that this isn't always enough to uniquely determine the arc, but it is enough to ensure that only finitely many possible arcs exist.
A point on the arc is a single piece. An end point of the arc is two pieces, and the length of the arc is one piece. A focal point of the ellipse is two pieces, as is the center. The line through the focal points is two pieces, although if you already know the center or one of the focal points, it is only a single piece of information.
answered Nov 22 at 6:43
Arthur
110k7105186
For an ellipse, you need five independent pieces of information, so for an arc, which has two more degrees of freedom, you would need seven. Note that this isn't always enough to uniquely determine the arc, but it is enough to ensure that only finitely many possible arcs exist.
A point on the arc is a single piece. An end point of the arc is two pieces, and the length of the arc is one piece. A focal point of the ellipse is two pieces, as is the center. The line through the focal points is two pieces, although if you already know the center or one of the focal points, it is only a single piece of information.
answered Nov 22 at 6:43
Arthur
110k7105186
answered Nov 22 at 6:43
Arthur
110k7105186
answered Nov 22 at 6:43
Arthur
110k7105186
answered Nov 22 at 6:43
Arthur
110k7105186
110k7105186
add a comment |
add a comment |
In general something else is needed. If the two endpoints of the arc happen to be on the line through the center, and the ellipse is not rotated (so has its major/minor axes parallel to the $x,y$ axes), then there will be an infinite family of such elliptic arcs, not to mention the choice of whether to use the top or bottom or left or right arc.
answered Nov 22 at 6:38
coffeemath
2,3181413
add a comment |
In general something else is needed. If the two endpoints of the arc happen to be on the line through the center, and the ellipse is not rotated (so has its major/minor axes parallel to the $x,y$ axes), then there will be an infinite family of such elliptic arcs, not to mention the choice of whether to use the top or bottom or left or right arc.
answered Nov 22 at 6:38
coffeemath
2,3181413
add a comment |
In general something else is needed. If the two endpoints of the arc happen to be on the line through the center, and the ellipse is not rotated (so has its major/minor axes parallel to the $x,y$ axes), then there will be an infinite family of such elliptic arcs, not to mention the choice of whether to use the top or bottom or left or right arc.
answered Nov 22 at 6:38
coffeemath
2,3181413
In general something else is needed. If the two endpoints of the arc happen to be on the line through the center, and the ellipse is not rotated (so has its major/minor axes parallel to the $x,y$ axes), then there will be an infinite family of such elliptic arcs, not to mention the choice of whether to use the top or bottom or left or right arc.
answered Nov 22 at 6:38
coffeemath
2,3181413
answered Nov 22 at 6:38
coffeemath
2,3181413
answered Nov 22 at 6:38
coffeemath
2,3181413
answered Nov 22 at 6:38
coffeemath
2,3181413
2,3181413
add a comment |
add a comment |
Three points and a real are enough.
You can describe the ellipse by a focus, excentricity and directrix.
- focus: one point or two reals
- directrix: an arbitrary line not passing through the focus, so two reals (say, distance from focus and orientation). That can be a point too: the point of the directrix nearest to the ellipse.
- eccentricity: one real
You then need two reals to describe the two endpoints (say, distance from the point nearest to the directrix, clockwise).
That's 7 reals, which you can "encode" in 3 points and a real.
Of course, you could theoretically encode the seven reals within only one real, since there is a bijection between $Bbb R^7$ and $Bbb R$, but that's cheating.
answered Nov 22 at 6:58
Jean-Claude Arbaut
14.7k63363
add a comment |
Three points and a real are enough.
You can describe the ellipse by a focus, excentricity and directrix.
- focus: one point or two reals
- directrix: an arbitrary line not passing through the focus, so two reals (say, distance from focus and orientation). That can be a point too: the point of the directrix nearest to the ellipse.
- eccentricity: one real
You then need two reals to describe the two endpoints (say, distance from the point nearest to the directrix, clockwise).
That's 7 reals, which you can "encode" in 3 points and a real.
Of course, you could theoretically encode the seven reals within only one real, since there is a bijection between $Bbb R^7$ and $Bbb R$, but that's cheating.
answered Nov 22 at 6:58
Jean-Claude Arbaut
14.7k63363
add a comment |
Three points and a real are enough.
You can describe the ellipse by a focus, excentricity and directrix.
- focus: one point or two reals
- directrix: an arbitrary line not passing through the focus, so two reals (say, distance from focus and orientation). That can be a point too: the point of the directrix nearest to the ellipse.
- eccentricity: one real
You then need two reals to describe the two endpoints (say, distance from the point nearest to the directrix, clockwise).
That's 7 reals, which you can "encode" in 3 points and a real.
Of course, you could theoretically encode the seven reals within only one real, since there is a bijection between $Bbb R^7$ and $Bbb R$, but that's cheating.
answered Nov 22 at 6:58
Jean-Claude Arbaut
14.7k63363
Three points and a real are enough.
You can describe the ellipse by a focus, excentricity and directrix.
- focus: one point or two reals
- directrix: an arbitrary line not passing through the focus, so two reals (say, distance from focus and orientation). That can be a point too: the point of the directrix nearest to the ellipse.
- eccentricity: one real
You then need two reals to describe the two endpoints (say, distance from the point nearest to the directrix, clockwise).
That's 7 reals, which you can "encode" in 3 points and a real.
Of course, you could theoretically encode the seven reals within only one real, since there is a bijection between $Bbb R^7$ and $Bbb R$, but that's cheating.
answered Nov 22 at 6:58
Jean-Claude Arbaut
14.7k63363
edited Nov 22 at 7:04
answered Nov 22 at 6:58
Jean-Claude Arbaut
14.7k63363
answered Nov 22 at 6:58
Jean-Claude Arbaut
14.7k63363
answered Nov 22 at 6:58
Jean-Claude Arbaut
14.7k63363
14.7k63363
add a comment |
add a comment |
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According to the numbers of unknown - write the general formula and based on it derive the number of points.
– Moti
Nov 22 at 6:35
1
One can also know the two foci and one point on the ellipse, since then one can get the length sum.
– coffeemath
Nov 22 at 6:42