 |
Business | Industries | Finance | Tax |
| Index: > A B C D E F G H I J K L M N O P Q R S T U V W X Y Z |
|
|||||
| First Prev [ 1 2 ] Next Last |
In mathematics, an ellipse is a figure corresponding to a circle which has been stretched in one direction. This is an example of a conic section and can be defined as the locus of all points, in a plane, which have the same sum of distances from two given fixed points (called foci, plural of focus). According to Kepler's laws of planetary motion, the orbit of a planet is an ellipse with the system's center of mass (very near the surface of the Sun in our system) at one focus.
An ellipse can also be characterized as the intersection of a cone with a suitably situated plane; that is why the term conic section is used. For a short elementary proof of the equivalance of these two characterizations, see Dandelin spheres.
If the two foci coincide, then the ellipse becomes a circle; depending on context, a circle may or may not be considered a type of ellipse. Most properties of ellipses also apply to circles as a special case. However, one can not define a circle as the set of points with a distance to the center equal to the eccentricity times the distance to the directrix, because we get zero times infinity, which doesn't have a defined result in mathematics.
The line which passes through the foci is the major axis and also the longest line which passes through the ellipse. The line which passes through the centre (halfway between the foci), at right angles to the major axis, is the minor axis. The semimajor axis is one half the major axis; running from the center, through a focus, and to the edge of the ellipse. Likewise, the semiminor axis is one half the minor axis. The two axes are the elliptic equivalents of the diameter, while the two semiaxes are the elliptic equivalents of the radius.
The size and shape of an ellipse are determined by two constants, conventionally denoted a and b. The constant a equalsSee also the disambiguation page title equality. In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. This defines a binary predicate, equality denoted " "; x y iff x and y are equal. Equivalence in th the lengthIn general English usage, length (symbol: l is but one particular instance of distance an object's length is how long the object is but in the physical sciences and engineering, the word length is in some contexts used synonymously with " distance". Heigh of the semimajor axis; the constant b equals the lengthIn general English usage, length (symbol: l is but one particular instance of distance an object's length is how long the object is but in the physical sciences and engineering, the word length is in some contexts used synonymously with " distance". Heigh of the semiminor axis.
An ellipse centered at the origin of an x-y coordinate systemGeometry See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of scala with its major axis along the x-axis is defined by the equation
The following diagram shows an ellipse demonstrating the Pythagoras equation a˛ = b˛ + c˛ as a special case of the non-parametric equation above (x=0, y=b).
The same ellipse is also represented by the parametric equations:
which use the trigonometric functionIn mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They may be defined as ratios of two sides of a right triangle containing the angle, or, more generally, as ratios ofs sine and cosine.
If an ellipse is not centered at the origin of an x-y coordinate system, it may be specified by the equation
where (h,k) is the center.
A Gauss-mappedDifferential geometry Riemannian geometry In differential geometry, the Gauss map maps a surface in Euclidean space R 3 to the unit sphere. Namely, given a surface lying in R 3, the Gauss map is a continuous map such that is orthogonal to S at p''. The Ga form:
has normal .
| Politics | Business | Finance |
 |
 |
 |
|