Branches of an implicitly defined biquadratic curve found using complex functions: Cartesian oval
Solving an implicitly defined plane curve means splitting it into separate "curve sections" for which explicit expressions of functional dependencies can be written. These may take the form of analytical function definitions, y = f(x), or parametrically defined curves where the x-coordinate coincides with the parameter, x = t.
In case of the Cartesian oval where an implicit equation of a plane curve is given in the form of a biquadratic equation in the variable y. Using its root formulas, 4 explicit functional dependencies of the real variable x are found that make up the plane curve under consideration. However, this method is not always applicable.
Like the previous applet, this applet illustrates how to use of the method of parametrically defined complex functions to solve the same implicit biquadratic equation in y: eq: (x² + y² - 2a x)² - b² (x² + y²) - c=0. Alternatively, it can be expanded into powers of y: y4 + ( -4 a x + 2 x2 - b2) * y2 +(4 a2 x2-4 a x3+x4-x2 b2-c)=0. Using the new variable assignments: k1:=a; k2:=b; k3:=c and the function assignments b(x)=(-4 k1 x + 2 x² - k2²); c(x)= 4 k1² x² - 4 k1 x³ + x⁴ - x² k2² - k3 we can rewrite the equation as y4 + b(x) y2 + c(x) = 0. With the parameter a before y⁴, we have a*y⁴+b(x)*y²+c(x)=0.
In the complex plane the variable x→z, y→f(z). The root formulas used can be easily extended to the complex plane in a certain way. Roots: complex functions g1(z), g2(z), g3(z) and g4(z) are solutions of the corresponding complex equation.
*Images made with this applet can be seen in the applet.