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For an ellipse, two diameters are conjugate if and only if the tangent line to the ellipse at an endpoint of one diameter is parallel to the other diameter. Each pair of conjugate diameters of an ellipse has a corresponding tangent parallelogram, sometimes called a bounding parallelogram (skewed compared to a bounding rectangle).
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.
In polar coordinates, the polar tangential angle is defined as the angle between the tangent line to the curve at the given point and ray from the origin to the point. [6] If ψ denotes the polar tangential angle, then ψ = φ − θ, where φ is as above and θ is, as usual, the polar angle.
If a tangent contains the point (x 0, y 0), off the parabola, then the equation = + = holds, which has two solutions m 1 and m 2 corresponding to the two tangents passing (x 0, y 0). The free term of a reduced quadratic equation is always the product of its solutions.
Plot of the degenerate Jacobi curve (x 2 + y 2 /b 2 = 1, b = ∞) and the twelve Jacobi Elliptic functions pq(u,1) for a particular value of angle φ. The solid curve is the degenerate ellipse (x 2 = 1) with m = 1 and u = F(φ,1) where F(⋅,⋅) is the elliptic integral of the first kind. The dotted curve is the unit circle.
For example, one sphere that is described in Cartesian coordinates with the equation x 2 + y 2 + z 2 = c 2 can be described in spherical coordinates by the simple equation r = c. (In this system— shown here in the mathematics convention —the sphere is adapted as a unit sphere , where the radius is set to unity and then can generally be ...
If L intersects Q in only one point, then it is a tangent line and p is the point of tangency. Finally, if L does not intersect Q then p has no tangent lines passing through it and it is called an interior or inner point. [17] The equation of the tangent line (in homogeneous coordinates) at a point p on the non-degenerate conic Q is given by,
Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system (,,) that generalizes the two-dimensional elliptic coordinate system. Unlike most three-dimensional orthogonal coordinate systems that feature quadratic coordinate surfaces , the ellipsoidal coordinate system is based on confocal quadrics .