Ellipse
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linked exercisesEquations of an ellipse ?, Equivalence between geometrical and analytical definitions of an ellipseEquations of an ellipse ?, Equivalence between geometrical and analytical definitions of an ellipse, Equation d’une ellipse ?
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8/6/2026, 4:22:28 PM · Sequoia
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##### 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
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:::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
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##### 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,$$15
Where $\alpha,\beta$ and $R$ are positive real numbers and $(x_0,y_0)$ is the center of the ellipse.15
where $a,b$ are positive real numbers called respectively "semi major-axis" and "semi minor-axis". Also, $(x_0,y_0)$ is the center of the ellipse.16
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##### Examples18
- Let $C$ be the (double) [[cone|cone]] in $\R^3$ with vertex at $0$ obtained by rotation the line generated by $(1,0,1)$ around the $z$-axis. Then the plane section of a cone is an ellipse, when the plane doesn't pass through $0$ and has the angle with the horizontal plane less than $\frac{\pi}{4}$.Revision 977
8/6/2026, 4:10:00 PM · Sequoia
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1
##### Geometrical definition2
Given two points $F_{1}$ and $F_{2}$ of the plane, and a real number $2a$ greater than the distance $F_{1}F_{2}$, the ellipse with foci $F_{1},F_{2}$ and major axis $2a$ is the set of points $P$ of the plane such that:3
$$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
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##### 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
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8/5/2026, 8:59:25 AM · Ancient Tree
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8/5/2026, 8:58:35 AM · Sequoia
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8/5/2026, 8:42:00 AM · Ancient Tree
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## 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
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## 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
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8/4/2026, 4:09:48 PM · Ancient Tree
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## 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
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## Formal definition6
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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
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## 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
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8/4/2026, 3:39:14 PM · Paulownia
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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 $\frac{\pi}{4}$.