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B^^ = T^^xN^^ (1) = (r^'xr^(''))/(|r^'xr^('')|), (2) where the unit tangent vector T and unit "principal" normal vector N are defined by T^^ = (r^'(s))/(|r^'(s)|) (3) N^^ = ...

The most common "sine integral" is defined as Si(z)=int_0^z(sint)/tdt (1) Si(z) is the function implemented in the Wolfram Language as the function SinIntegral[z]. Si(z) is ...

The most common form of cosine integral is Ci(x) = -int_x^infty(costdt)/t (1) = gamma+lnx+int_0^x(cost-1)/tdt (2) = 1/2[Ei(ix)+Ei(-ix)] (3) = -1/2[E_1(ix)+E_1(-ix)], (4) ...

The Mercator projection is a map projection that was widely used for navigation since loxodromes are straight lines (although great circles are curved). The following ...

Given a circle C with center O and radius k, then two points P and Q are inverse with respect to C if OP·OQ=k^2. If P describes a curve C_1, then Q describes a curve C_2 ...

A spherical harmonic of the form sin(mphi)P_m^m(costheta) or cos(mphi)P_m^m(costheta).

Let a spherical triangle have sides of length a, b, and c, and semiperimeter s. Then the spherical excess E is given by

Any linear combination of real spherical harmonics A_lP_l(costheta)+sum_(m=1)^l[A_l^mcos(mphi)+B_l^msin(mphi)]P_l^m(costheta) for l fixed whose sum is not premultiplied by a ...

A helix, sometimes also called a coil, is a curve for which the tangent makes a constant angle with a fixed line. The shortest path between two points on a cylinder (one not ...

The epispiral is a plane curve with polar equation r=asec(ntheta). There are n sections if n is odd and 2n if n is even. A slightly more symmetric version considers instead ...