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An alternative parametrization exists that closely follows the angular parametrization of spherical coordinates: [1] = , = , = . Here, > parametrizes the concentric ellipsoids around the origin and [,] and [,] are the usual polar and azimuthal angles of spherical coordinates, respectively.
Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as , see conventions in spherical coordinates). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range ...
The lemniscate sine (red) and lemniscate cosine (purple) applied to a real argument, in comparison with the trigonometric sine y = sin(πx/ϖ) (pale dashed red).. In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli.
Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is (x − a) 2 + (y − b) 2 = r 2 where a and b are the coordinates of the center (a, b) and r is the radius.
Some of these formulas are expressed in terms of the Cartesian expansion of the spherical harmonics into polynomials in x, y, z, and r. For purposes of this table, it is useful to express the usual spherical to Cartesian transformations that relate these Cartesian components to θ {\displaystyle \theta } and φ {\displaystyle \varphi } as
The old coordinates (x, y, z) of a point Q are related to its new coordinates (x′, y′, z′) by [14] [′ ′ ′] = [ ] []. Generalizing to any finite number of dimensions, a rotation matrix A {\displaystyle A} is an orthogonal matrix that differs from the identity matrix in at most four elements.
The arc length of one branch between x = x 1 and x = x 2 is a ln y 1 / y 2 . The area between the tractrix and its asymptote is π a 2 / 2 , which can be found using integration or Mamikon's theorem. The envelope of the normals of the tractrix (that is, the evolute of the tractrix) is the catenary (or chain curve) given by y = a ...
Then the Cartesian oval is the locus of points S satisfying d(P, S) + m d(Q, S) = a. The two ovals formed by the four equations d( P , S ) + m d( Q , S ) = ± a and d( P , S ) − m d( Q , S ) = ± a are closely related; together they form a quartic plane curve called the ovals of Descartes .