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Specifying two adjacent angles A and B and the side between them c uniquely (up to geometric congruence) determines a triangle with area K=(c^2)/(2(cotA+cotB)). (1) The angle ...

Let the nth composition of a function f(x) be denoted f^((n))(x), such that f^((0))(x)=x and f^((1))(x)=f(x). Denote the composition of f and g by f degreesg(x)=f(g(x)), and ...

Brocard's conjecture states that pi(p_(n+1)^2)-pi(p_n^2)>=4 for n>=2, where pi(n) is the prime counting function and p_n is the nth prime. For n=1, 2, ..., the first few ...

A quartic curve with implicit equation (a^2)/(x^2)-(b^2)/(y^2)=1 (1) or a^2y^2-b^2x^2=x^2y^2 (2) and a,b>0. In parametric form, x = +/-acost (3) y = bcott. (4) The curvature ...

A map projection defined by x = sin^(-1)[cosphisin(lambda-lambda_0)] (1) y = tan^(-1)[(tanphi)/(cos(lambda-lambda_0))]. (2) The inverse formulas are phi = sin^(-1)(sinDcosx) ...

The conical spiral with angular frequency a on a cone of height h and radius r is a space curve given by the parametric equations x = (h-z)/hrcos(az) (1) y = (h-z)/hrsin(az) ...

The dumbbell curve is the sextic curve a^4y^2=x^4(a^2-x^2). (1) It has area A=1/4pia^2 (2) and approximate arc length s approx 5.541a. (3) For the parametrization x = at (4) ...

The W-polynomials obtained by setting p(x)=3x and q(x)=-2 in the Lucas polynomial sequence. The first few Fermat polynomials are F_1(x) = 1 (1) F_2(x) = 3x (2) F_3(x) = ...

Let r be the correlation coefficient. Then defining z^'=tanh^(-1)r (1) zeta=tanh^(-1)rho, (2) gives sigma_(z^') = (N-3)^(-1/2) (3) var(z^') = 1/n+(4-rho^2)/(2n^2)+... (4) ...

Given a semicircular hump f(x) = sqrt(L^2-(x-L)^2) (1) = sqrt((2L-x)x), (2) the Fourier coefficients are a_0 = 1/2piL (3) a_n = ((-1)^nLJ_1(npi))/n (4) b_n = 0, (5) where ...