Search Results for ""

The bifoliate is the quartic curve given by the Cartesian equation x^4+y^4=2axy^2 (1) and the polar equation r=(8costhetasin^2theta)/(3+cos(4theta))a (2) for theta in [0,pi]. ...

The bifolium is a folium with b=0. The bifolium is a quartic curve and is given by the implicit equation is (x^2+y^2)^2=4axy^2 (1) and the polar equation ...

(x^2)/(a^2-lambda)+(y^2)/(b^2-lambda)=z-lambda (1) (x^2)/(a^2-mu)+(y^2)/(b^2-mu)=z-mu (2) (x^2)/(a^2-nu)+(y^2)/(b^2-nu)=z-nu, (3) where lambda in (-infty,b^2), mu in ...

The pedal curve of an epicycloid x = (a+b)cost-b[((a+b)t)/b] (1) y = (a+b)sint-bsin[((a+b)t)/b] (2) with pedal point at the origin is x_p = 1/2(a+2b){cost-cos[((a+b)t)/b]} ...

Implicit differentiation is the procedure of differentiating an implicit equation with respect to the desired variable x while treating the other variables as unspecified ...

Given a plane ax+by+cz+d=0 (1) and a point x_0=(x_0,y_0,z_0), the normal vector to the plane is given by v=[a; b; c], (2) and a vector from the plane to the point is given by ...

The quadrifolium is the 4-petalled rose curve having n=2. It has polar equation r=asin(2theta) (1) and Cartesian equation (x^2+y^2)^3=4a^2x^2y^2. (2) The area of the ...

The sigmoid function, also called the sigmoidal curve (von Seggern 2007, p. 148) or logistic function, is the function y=1/(1+e^(-x)). (1) It has derivative (dy)/(dx) = ...

The Narayana triangle is the number triangle obtained from the Narayana numbers N(n,k), namely 1 ; 1 1 ; 1 3 1 ; 1 6 6 1 ; 1 10 20 10 1 ; 1 15 50 50 15 1 (OEIS A001263).

For the cardioid given parametrically as x = a(1+cost)cost (1) y = a(1+cost)sint, (2) the negative pedal curve with respect to the pedal point (x_0,y_0)=(0,0) is the circle ...