Ellipse
Concept history
A revision trail for this concept page.
Revision 985
8/6/2026, 4:22:28 PM · Sequoia
Concept saved without content changes
No content or metadata changes were recorded.
Revision 978
8/6/2026, 4:12:26 PM · Sequoia
Updated text
Compare with revision 9774 changed lines
1
##### Geometrical definition2
Given two points $F_{1}$ and $F_{2}$ of the plane, and a real number $2a$ greater than the distance $F_{1}F_{2}$, the ellipse with foci $F_{1},F_{2}$ and major axis $2a$ is the set of points $P$ of the plane such that:3
$$PF_{1}+PF_{2}=2a.$$4
5
6
7
:::fold Explanation8
This definition is called the gardener's method: plant two stakes at $F_{1}$ and $F_{2}$, loop a string around them, and trace the curve keeping the string taut. This will produce an ellipse. This construction generalizes the one for the [[Circle|circle]], where a single stake and a taut string of fixed length give the set of points at constant distance from a single center.9
:::10
11
##### Cartesian equation definition12
An ellipse is the set of points $M:=(x,y)\in\R^2$ satisfying an equation 13
$$14
\left(\frac{x-x_{0}}{\alpha}\right)^{2} + \left(\frac{y-y_{0}}{\beta}\right)^{2} = R^2$$14
\left(\frac{x-x_{0}}{a}\right)^{2} + \left(\frac{y-y_{0}}{b}\right)^{2} = 1,$$15
Where $\alpha,\beta$ and $R$ are positive real numbers and $(x_0,y_0)$ is the center of the ellipse.15
where $a,b$ are positive real numbers called respectively "semi major-axis" and "semi minor-axis". Also, $(x_0,y_0)$ is the center of the ellipse.16
17
##### Examples18
- Let $C$ be the (double) [[cone|cone]] in $\R^3$ with vertex at $0$ obtained by rotation the line generated by $(1,0,1)$ around the $z$-axis. Then the plane section of a cone is an ellipse, when the plane doesn't pass through $0$ and has the angle with the horizontal plane less than $\frac{\pi}{4}$.Revision 977
8/6/2026, 4:10:00 PM · Sequoia
Updated text
Compare with revision 94110 changed lines
1
##### Geometrical definition2
Given two points $F_{1}$ and $F_{2}$ of the plane, and a real number $2a$ greater than the distance $F_{1}F_{2}$, the ellipse with foci $F_{1},F_{2}$ and major axis $2a$ is the set of points $P$ of the plane such that:3
$$PF_{1}+PF_{2}=2a.$$4
5
6
7
:::fold Explanation8
This definition is called the gardener's method: plant two stakes at $F_{1}$ and $F_{2}$, loop a string around them, and trace the curve keeping the string taut. This will produce an ellipse. This construction generalizes the one for the [[Circle|circle]], where a single stake and a taut string of fixed length give the set of points at constant distance from a single center.9
:::10
11
##### Implicit equation definition11
##### Cartesian equation definition12
An ellipse is the set of points $p$ in $\R^{2}$ satisfying an equation 12
An ellipse is the set of points $M:=(x,y)\in\R^2$ satisfying an equation 13
\[ \alpha (x-x_{0})^{2 } + \beta (y-y_{0})^{2} = \gamma \]13
$$14
with $\alpha, \beta,\gamma$ positive real numbers and $x,y$ coordinates of $p$ in some orthonormal basis of $\R^{2}.$14
\left(\frac{x-x_{0}}{\alpha}\right)^{2} + \left(\frac{y-y_{0}}{\beta}\right)^{2} = R^2$$15
Where $\alpha,\beta$ and $R$ are positive real numbers and $(x_0,y_0)$ is the center of the ellipse.15
16
##### Examples17
18
- Let $C$ be the (double) [[cone|cone]] in $\R^3$ with vertex at $0$ obtained by rotation the line generated by $(1,0,1)$ around the $z$-axis. Then the plane section of a cone is an ellipse, when the plane doesn't pass through $0$ and has the angle with the horizontal plane less than $\frac{\pi}{4}$.Revision 941
8/5/2026, 9:15:19 AM · Ancient Tree
Concept edited
This older revision predates detailed metadata tracking.
Revision 940
8/5/2026, 8:59:25 AM · Ancient Tree
Concept edited
This older revision predates detailed metadata tracking.
Revision 939
8/5/2026, 8:58:35 AM · Sequoia
Concept edited
This older revision predates detailed metadata tracking.
Revision 937
8/5/2026, 8:42:00 AM · Ancient Tree
Concept edited
This older revision predates detailed metadata tracking.
Revision 933
8/5/2026, 8:20:29 AM · Ancient Tree
Concept marked usable
This older revision predates detailed metadata tracking.
Revision 932
8/5/2026, 8:16:31 AM · Ancient Tree
Concept edited
Compare with revision 92916 changed lines
1
## Intuitive definition1
##### Geometrical definition2
Given two points $F_{1}$ and $F_{2}$ of the plane, and a real number $2a$ greater than the distance $F_{1}F_{2}$, the ellipse with foci $F_{1},F_{2}$ and major axis $2a$ is the set of points $P$ of the plane such that:3
$$PF_{1}+PF_{2}=2a.$$2
3
An ellipse is a shape obtained by stretching a [[Circle|circle]] in any direction.5
4
5
6
## Formal definition7
:::fold Explanation8
This definition is called the gardener's method: plant two stakes at $F_{1}$ and $F_{2}$, loop a string around them, and trace the curve keeping the string taut. This will produce an ellipse. This construction generalizes the one for the [[Circle|circle]], where a single stake and a taut string of fixed length give the set of points at constant distance from a single center.9
:::7
11
##### Implicit equation definition8
An ellipse is the set of points $p$ in $\R^{2}$ satisfying an equation 9
\[ \alpha (x-x_{0})^{2 } + \beta (y-y_{0})^{2} = \gamma \]10
with $\alpha, \beta,\gamma$ positive real numbers and $x,y$ coordinates of $p$ in some orthonormal basis of $\R^{2}.$11
12
## Examples16
##### Examples13
14
- Let $C$ be the (double) cone in $\R^3$ with vertex at $0$ obtained by rotation the line generated by $(1,0,1)$ around the $z$-axis. Then the plane section of a cone is an ellipse, when the plane doesn't pass through $0$ and has the angle with the horizontal plane less than $\frac{\pi}{4}$.18
- Let $C$ be the (double) [[cone|cone]] in $\R^3$ with vertex at $0$ obtained by rotation the line generated by $(1,0,1)$ around the $z$-axis. Then the plane section of a cone is an ellipse, when the plane doesn't pass through $0$ and has the angle with the horizontal plane less than $\frac{\pi}{4}$.Revision 929
8/5/2026, 7:21:08 AM · Sequoia
Concept edited
This older revision predates detailed metadata tracking.
Revision 906
8/4/2026, 4:09:48 PM · Ancient Tree
Concept edited
Compare with revision 9044 changed lines
1
## Intuitive definition2
3
An ellipse is a shape obtained by stretching a circle in any direction.3
An ellipse is a shape obtained by stretching a [[Circle|circle]] in any direction.4
4
5
## Formal definition6
7
An ellipse is the set of points $p$ in $\R^{2}$ satisfying an equation 8
\[ \alpha (x-x_{0})^{2 } + \beta (y-y_{0})^{2} = \gamma \]9
with $\alpha, \beta,\gamma$ positive real numbers and $x,y$ coordinates of $p$ in some orthonormal basis of $\R^{2}.$11
10
## Examples11
12
- Let $C$ be the (double) cone in $\R^3$ with vertex at $0$ obtained by rotation the line generated by $(1,0,1)$ around the $z$-axis. Then the plane section of a cone is an ellipse, when the plane doesn't pass through $0$ and has the angle with the horizontal plane less than $\frac{\pi}{4}$.Revision 904
8/4/2026, 4:02:04 PM · Ancient Tree
Concept marked usable
This older revision predates detailed metadata tracking.
Revision 903
8/4/2026, 3:39:14 PM · Paulownia
Concept created
Recorded titleEllipse
Recorded typeDefinition
## Intuitive definition
An ellipse is a shape obtained by stretching a circle in any direction.
## Formal definition
An ellipse is the set of points $p$ in $\R^{2}$ satisfying an equation
\[ \alpha (x-x_{0})^{2 } + \beta (y-y_{0})^{2} = \gamma \]
with $\alpha, \beta,\gamma$ positive real numbers and $x,y$ coordinates of $p$ in some orthonormal basis of $\R^{2}.$
## Examples
- Let $C$ be the (double) cone in $\R^3$ with vertex at $0$ obtained by rotation the line generated by $(1,0,1)$ around the $z$-axis. Then the plane section of a cone is an ellipse, when the plane doesn't pass through $0$ and has the angle with the horizontal plane less than $\frac{\pi}{4}$.