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 $1$.

• Conic Sections in Polar Coordinates

  • One of the most useful definitions, in that it involves only the plane, is that a conic consists of those points whose distances to some point—called a focus—and some line—called a directrix—are in a fixed ratio, called the eccentricity.
  • The type of a conic corresponds to its eccentricity, those with eccentricity less than 1 being ellipses, those with eccentricity equal to 1 being parabolas, and those with eccentricity greater than 1 being hyperbolas.
  • In the focus-directrix definition of a conic, the circle is a limiting case with eccentricity 0.
  • where e is the eccentricity and l is half the latus rectum.

• Conic Sections

  • There are a number of other geometric definitions possible, one of the most useful being that a conic consists of those points whose distances to some other point (called a focus) and some other line (called a directrix) are in a fixed ratio, called the eccentricity.
  • The type of a conic corresponds to its eccentricity—those with eccentricity less than 1 being ellipses, those with eccentricity equal to 1 being parabolas, and those with eccentricity greater than 1 being hyperbolas.
  • In the focus-directrix definition of a conic, the circle is a limiting case with eccentricity 0.

• Exercise-Induced Muscle Damage

  • Exercise damages muscles due to eccentric and concentric muscle loading and often results in delayed onset muscle soreness (DOMS).
  • Exercise damages muscles due to eccentric and concentric muscle loading.
  • Resistance training, and particularly high loading during eccentric contractions, results in delayed onset muscle soreness (DOMS).
  • Acute inflammation of the muscle cells, as understood in exercise physiology, can result after induced eccentric and concentric muscle training.
  • Participation in eccentric training and conditioning, including resistance training and activities that emphasize eccentric lengthening of the muscle including downhill running on a moderate to high incline can result in considerable soreness within 24 to 48 hours, even though blood lactate levels, previously thought to cause muscle soreness, were much higher with level running.

• 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:

• Kepler's First Law

  • How stretched out an ellipse is from a perfect circle is known as its eccentricity: a parameter that can take any value greater than or equal to 0 (a circle) and less than 1 (as the eccentricity tends to 1, the ellipse tends to a parabola).
  • The eccentricities of the planets known to Kepler varied from 0.007 (Venus) to 0.2 (Mercury).
  • The dwarf planet Pluto, discovered in 1929, has an eccentricity of 0.25.
  • The eccentricity $\epsilon$ is the coefficient of variation between $r_{\text{min}}$ and $r_{\text{max}}$:
  • The orbits of planets with very small eccentricities can be approximated as circles.

• Types of Muscle Contractions: Isotonic and Isometric

  • Isotonic muscle contractions can be either concentric or eccentric.
  • An eccentric contraction results in the elongation of a muscle while the muscle is still generating force; in effect, resistance is greater than force generated.
  • For example, a voluntary eccentric contraction would be the controlled lowering of the heavy weight raised during the above concentric contraction.
  • An involuntary eccentric contraction may occur when a weight is too great for a muscle to bear and so it is slowly lowered while under tension.
  • An isotonic concentric contraction results in the muscle shortening, an isotonic eccentric contraction results in the muscle lengthening.

• Planetary Motion According to Kepler and Newton

  • The eccentricity of an ellipse tells you how stretched out the ellipse is.
  • The eccentricity can be from 0 to 1.
  • If the eccentricity is equal to zero, that means it is a circle.
  • The eccentricity is what makes an ellipse different from a circle.
  • These values are important because the equation for eccentricity is: