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An ellipse (red) obtained as the intersection of a cone with an inclined plane. Ellipse: notations Ellipses: examples with increasing eccentricity. In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant.
Area#Area formulas – Size of a two-dimensional surface; Perimeter#Formulas – Path that surrounds an area; List of second moments of area; List of surface-area-to-volume ratios – Surface area per unit volume; List of surface area formulas – Measure of a two-dimensional surface; List of trigonometric identities
An ellipse has two axes and two foci. Unlike most other elementary shapes, such as the circle and square, there is no algebraic equation to determine the perimeter of an ellipse. Throughout history, a large number of equations for approximations and estimates have been made for the perimeter of an ellipse.
Since the area of the rectangle is ab, the area of the ellipse is π ab/4. We can also consider analogous measurements in higher dimensions. For example, we may wish to find the volume inside a sphere. When we have a formula for the surface area, we can use the same kind of "onion" approach we used for the disk.
More formulas of this nature can be given, as explained by Ramanujan's theory of elliptic functions to alternative bases. Perhaps the most notable hypergeometric inversions are the following two examples, involving the Ramanujan tau function τ {\displaystyle \tau } and the Fourier coefficients j {\displaystyle \mathrm {j} } of the J-invariant ...
An elliptic equation can mean: The equation of an ellipse; An elliptic curve, describing the relationships between invariants of an ellipse; A differential equation with an elliptic operator; An elliptic partial differential equation
The equation is for an ellipse, since both eigenvalues are positive. (Otherwise, if one were positive and the other negative, it would be a hyperbola.) The principal axes are the lines spanned by the eigenvectors. The minimum and maximum distances to the origin can be read off the equation in diagonal form.
The classic applications of elliptic coordinates are in solving partial differential equations, e.g., Laplace's equation or the Helmholtz equation, for which elliptic coordinates are a natural description of a system thus allowing a separation of variables in the partial differential equations. Some traditional examples are solving systems such ...
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