For an ellipse, the aspect ratio denotes the ratio of the major axis to the minor axis.

For an ellipse the eccentricity is between 0 and 1 ( 0 < e < 1 ).

For an ellipse in canonical form, we have

For comparison, the other two general conic sections, the ellipse and the parabola, derive from the corresponding Greek words for " deficient " and " comparable "; these terms may refer to the eccentricity of these curves, which is greater than one ( hyperbola ), less than one ( ellipse ) and exactly one ( parabola ), respectively.

For constant u, that is on the ellipse which is the intercept with a constant z plane, v then plays the role of the eccentric anomaly for that ellipse.

For constant v on a plane through the Oz axis the parameter u plays the same role for the ellipse of intersection.

For an object in an eccentric orbit orbiting a much larger body, the length of the orbit decreases with eccentricity, and is given at ellipse.

For example, eliminating z between the two equations x < sup > 2 </ sup > + y < sup > 2 </ sup > − z < sup > 2 </ sup > = 0 and x + 2y + 3z − 1 = 0, which defines an intersection of a cone and a plane in three dimensions, we obtain the conic section 8x < sup > 2 </ sup > + 5y < sup > 2 </ sup > − 4xy + 2x + 4y − 1 = 0, which in this case is an ellipse.

For example, consider the ellipse x < sup > 2 </ sup > + xy + y < sup > 2 </ sup > = 1, where (− 1, 0 ) is a rational point.

For example, the projective parameterization of above ellipse is

( For a real-world tire, the circle is likely to be closer to an ellipse, with the y axis slightly longer than the x axis.

For example, a circle is a concept that makes sense in Euclidean geometry, but not in affine linear geometry or projective geometry, where circles cannot be distinguished from ellipses, since one may squeeze a circle to an ellipse.

For example, the pencil of curves ( 1-dimensional linear system of conics ) defined by is non-degenerate for but is degenerate for concretely, it is an ellipse for two parallel lines for and a hyperbola with < math > a < 0 </ math > – throughout, one axis has length 2 and the other has length which is infinity for

For as Kepler knew the orb to be not circular but oval, and guessed it to be elliptical, so Mr Hooke, without knowing what I have found out since his letters to me, can know no more, but that the proportion was duplicate quam proximè at great distances from the centre, and only guessed it to be so accurately, and guessed amiss in extending that proportion down to the very centre, whereas Kepler guessed right at the ellipse.

The appearance of the figure is highly sensitive to the ratio a / b. For a ratio of 1, the figure is an ellipse, with special cases including circles ( A

For example, for the ellipse

For example, let C < sub > 1 </ sub > and C < sub > 2 </ sub > both be the ellipse

For the point P orbiting around an ellipse, the eccentric anomaly is the angle E in the figure.

* For the Earth and a just little more than this is ; the quantity is the height the ellipse extends above the surface, plus the periapsis distance ( the distance the ellipse extends beyond the center of the Earth ); the latter times g is the kinetic energy of the horizontal component of the velocity.

For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola.

For the ellipse, both the focus and the center of the directrix circle have finite coordinates and the radius of the directrix circle is greater than the distance between the center of this circle and the focus ; thus, the focus is inside the directrix circle.

For a detailed discussion of the differences including a more comprehensive table ( several essential tables are given below ) of Biblical scripture for both Testaments and the intertestamental period with regard to canonical acceptance in Christendom's various major traditions, see Wikipedia's article on " Biblical canon ".

For the example, we have a canonical form available that reduces any string to one of length at most three, by decreasing the length monotonically.

For a given diagram F: J → C and functor G: C → D, if both F and GF have specified limits there is a unique canonical morphism

For nonzero Q there exists a canonical linear isomorphism between Λ ( V ) and Cℓ ( V, Q ) whenever the ground field K does not have characteristic two.

For any normal modal logic L, a Kripke model ( called the canonical model ) can be constructed, which validates precisely the theorems of L, by an adaptation of the standard technique of using maximal consistent sets as models.

For example, the phase space measure of the microcanonical ensemble ( see below ) is different from that of the canonical ensemble.

For example, one can specify the density operators describing microcanonical, canonical, and grand canonical ensembles of quantum mechanical systems, in a mathematically rigorous fashion.

For hyperbolic local geometry, many of the possible three-dimensional spaces are informally called horn topologies, so called because of the shape of the pseudosphere, a canonical model of hyperbolic geometry.

For example, he developed a theory of canonical transformations which allowed changing coordinates so that some coordinates disappeared from the Lagrangian, as above, resulting in conserved canonical momenta.

For each neighborhood U of x, the canonical morphism F ( U ) → F < sub > x </ sub > associates to a section s of F over U an element s < sub > x </ sub > of the stalk F < sub > x </ sub > called the germ of s at x.

For the purposes of consistency, the game accepts The Lord of the Rings, The Hobbit and The Silmarillion and to a lesser extent the other works of Tolkien as canonical materials.

For this purpose he decreed that partial indulgences, previously granted as the equivalent of a certain number of days, months, " quarantines " ( Lent-like forty-day periods ) or years of canonical penance, simply supplement, and to the same degree, the remission that those performing the indulgenced action already gain by the charity and contrition with which they do it.

For instance, often it is possible to choose the Hamiltonian itself H = H ( q, p ; t ) as one of the new canonical momentum coordinates.

For, one simply gets the well-known canonical commutation relations.

( For a detailed derivation of this result, see canonical ensemble.

Also, starting with any compact real form of a semisimple Lie algebra g its complexification as a real Lie algebra of twice the dimension splits into g and a certain solvable Lie algebra ( the Iwasawa decomposition ), and this provides a canonical bicrossproduct quantum group associated to g. For su ( 2 ) one obtains a quantum group deformation of the Euclidean group E ( 3 ) of motions in 3 dimensions.

For Sunni Muslims, Salat al-Eid differs from the five daily canonical prayers in that no adhan ( Call to Prayer ) or iqama ( call ) is pronounced for the two Eid prayers.

For example, the Künneth standard conjecture, which states the existence of algebraic cycles π < sup > i </ sup > ⊂ X × X inducing the canonical projectors H < sup >∗</ sup >( X ) ↠ H < sup > i </ sup >( X ) ↣ H < sup >∗</ sup >( X ) ( for any Weil cohomology H ) implies that every pure motive M decomposes in graded pieces of weight n: M = ⊕ Gr < sub > n </ sub > M.

But it turns out that ( if A is " nice " enough ) there is one canonical way of doing so, given by the right derived functors of F. For every i ≥ 1, there is a functor R < sup > i </ sup > F: A → B, and the above sequence continues like so: 0 → F ( A ) → F ( B ) → F ( C ) → R < sup > 1 </ sup > F ( A ) → R < sup > 1 </ sup > F ( B ) → R < sup > 1 </ sup > F ( C ) → R < sup > 2 </ sup > F ( A ) → R < sup > 2 </ sup > F ( B ) → ....

For the canonical values the Hénon map is chaotic.

For the canonical map, an initial point of the plane will either approach a set of points known as the Hénon strange attractor, or diverge to infinity.