- #1
- 19,864
- 10,851
Definition/Summary
The eccentricity e of a conic section (other than a parabola or a pair of crossed lines) is its focal length divided by its major axis: e = f/a
The eccentricity of a conic section (other than a pair of crossed lines) is the distance from any point P on the conic section to a focus F divided by the distance from P to the directrix accompanying F.
Eccentricity is a measure of circularity:
e = 0 circle
0 < e < 1 ellipse (other than a circle)
e = 1 parabola
1 < e < \infty hyperbola
e = \infty pair of crossed lines
Equations
For an ellipse or hyperbola with major axis 2a along the x-axis, and focal length 2f:
e = \frac{f}{a}
\frac{x^2}{a^2}\,+\,\frac{y^2}{a^2 - f^2}\,=\,\frac{x^2}{a^2}\,+\,\frac{y^2}{a^2(1 - e^2)}\,=\,1
distance from centre to directrix: a/e
Defining b\,=\,a\sqrt{|1- e^2|} gives:
for e < 1 (ellipse):
f^2\,=\,a^2\,-\,b^2
\frac{x^2}{a^2}\,+\,\frac{y^2}{b^2}\,=\,1 (so the minor axis is 2b)
e\,=\,\frac{f}{a}\,=\,\sqrt{1 - \left (\frac{b}{a} \right)^2}
for e > 1 (hyperbola):
f^2\,=\,a^2\,+\,b^2
\frac{x^2}{a^2}\,-\,\frac{y^2}{b^2}\,=\,1
e\,=\,\frac{f}{a}\,=\,\sqrt{1 + \left (\frac{b}{a} \right)^2}
Extended explanation
Orbital eccentricity:
For astronomical orbits or trajectories, an alternative convenient definition is:
e = \frac{r_A - r_P}{r_A + r_P}
where r_A = a(1 + e) is the apoapse distance
and r_P = a(1 - e) is the periapse distance.
For parabolic trajectories, r_A is taken to be ∞.
For hyperbolic trajectories, r_A is the closest distance if gravity were repulsive.
These formulas of course are valid for any inverse-square-law force.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
The eccentricity e of a conic section (other than a parabola or a pair of crossed lines) is its focal length divided by its major axis: e = f/a
The eccentricity of a conic section (other than a pair of crossed lines) is the distance from any point P on the conic section to a focus F divided by the distance from P to the directrix accompanying F.
Eccentricity is a measure of circularity:
e = 0 circle
0 < e < 1 ellipse (other than a circle)
e = 1 parabola
1 < e < \infty hyperbola
e = \infty pair of crossed lines
Equations
For an ellipse or hyperbola with major axis 2a along the x-axis, and focal length 2f:
e = \frac{f}{a}
\frac{x^2}{a^2}\,+\,\frac{y^2}{a^2 - f^2}\,=\,\frac{x^2}{a^2}\,+\,\frac{y^2}{a^2(1 - e^2)}\,=\,1
distance from centre to directrix: a/e
Defining b\,=\,a\sqrt{|1- e^2|} gives:
for e < 1 (ellipse):
f^2\,=\,a^2\,-\,b^2
\frac{x^2}{a^2}\,+\,\frac{y^2}{b^2}\,=\,1 (so the minor axis is 2b)
e\,=\,\frac{f}{a}\,=\,\sqrt{1 - \left (\frac{b}{a} \right)^2}
for e > 1 (hyperbola):
f^2\,=\,a^2\,+\,b^2
\frac{x^2}{a^2}\,-\,\frac{y^2}{b^2}\,=\,1
e\,=\,\frac{f}{a}\,=\,\sqrt{1 + \left (\frac{b}{a} \right)^2}
Extended explanation
Orbital eccentricity:
For astronomical orbits or trajectories, an alternative convenient definition is:
e = \frac{r_A - r_P}{r_A + r_P}
where r_A = a(1 + e) is the apoapse distance
and r_P = a(1 - e) is the periapse distance.
For parabolic trajectories, r_A is taken to be ∞.
For hyperbolic trajectories, r_A is the closest distance if gravity were repulsive.
These formulas of course are valid for any inverse-square-law force.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!