eccentricity

Examples of eccentricity in the following topics:

• Eccentricity

  • The eccentricity of a circle is zero.
  • These are the distances used to find the eccentricity.
  • Therefore, by definition, the eccentricity of a parabola must be $1$.
  • For an ellipse, the eccentricity is less than $1$.
  • Conversely, the eccentricity of a hyperbola is greater than $1$.

• Parts of an Ellipse

  • All conic sections have an eccentricity value, denoted $e$.
  • All ellipses have eccentricities in the range $0 \leq e < 1$.
  • An eccentricity of zero is the special case where the ellipse becomes a circle.
  • An eccentricity of $1$ is a parabola, not an ellipse.
  • The orbits of comets around the sun can be much more eccentric.

• Types of Conic Sections

  • All parabolas possess an eccentricity value $e=1$.
  • All circles have an eccentricity $e=0$.
  • Ellipses can have a range of eccentricity values: $0 \leq e < 1$.
  • Since there is a range of eccentricity values, not all ellipses are similar.
  • This graph shows an ellipse in red, with an example eccentricity value of $0.5$, a parabola in green with the required eccentricity of $1$, and a hyperbola in blue with an example eccentricity of $2$.

• Conics in Polar Coordinates

  • With this definition, we may now define a conic in terms of the directrix: $x=±p$, the eccentricity $e$, and the angle $\theta$.
  • For a conic with a focus at the origin, if the directrix is $x=±p$, where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation:
  • For a conic with a focus at the origin, if the directrix is $y=±p$, where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation:

• Nonlinear Systems of Equations and Problem-Solving

  • The type of a conic corresponds to its eccentricity.
  • Conics with eccentricity less than $1$ are ellipses, conics with eccentricity equal to $1$ are parabolas, and conics with eccentricity greater than $1$ are hyperbolas.
  • In the focus-directrix definition of a conic, the circle is a limiting case of the ellipse with an eccentricity of $0$.

• Applications of Hyperbolas

  • The parameters of the traced hyperbola, such as its asymptotes and its eccentricity, are related to the specific physical conditions that produced it, namely the angle between the sunlight and the ground, and the latitude at which the sundial exists.
  • Depending on the orbital properties, including size and shape (eccentricity), this orbit can be any of the four conic sections.
  • In particular, if the eccentricity e of the orbit is greater than $1$, the path of such a particle is a hyperbola.
  • A diagram of the various forms of the Kepler Orbit and their eccentricities.

• Ellipses

  • The eccentricity of an ellipse, usually denoted by ε or e, is the ratio of the distance between the two foci, to the length of the major axis or
  • Use formulas to determine the area and eccentricity of an ellipse

• Standard Equations of Hyperbolas

  • The eccentricity e equals the ratio of the distances from a point P on the hyperbola to one focus and its corresponding directrix line (shown in green).