Outline
Equations of branches of implicitly defined curves
Statement of the problem:
The curve is given in implicit form as g(x, y) = 0, for example a circle as y² + x² = 1.
Find:
the explicit form of the equations y=f(x) for each of the k branches of the curve, i.e. {fᵢ(x)}, where i=1..k. For a circle, for example, f₁(x) = sqrt(1-x²) for y > 0, and f₂(x) = − sqrt(1-x²) for y < 0. If f(x, y) is a quadratic function with respect to variable y, GeoGebra can easily find the roots in symbolic form, y₁(x) and y₂(x).
For polynomials of the 3rd and 4th degree, knowing the existing rigorous https://www.geogebra.org/m/v4fvf8nx of their equations in symbolic form, one can find the equations of the branches {fᵢ(x)} of the corresponding plane curves using complex functions.
Table of Contents
Multifocal plane curves
- Ellipses and Hyperbolas defined geometrically as locus of points
- Cassini ovals and their orthogonal trajectories (hyperbolas)
- What is the Locus of a point that moves in such a way that it's sum of the squares of the distance (with weight factors) to N fixed points remains constant?
- Construction of multifocus curves, whose locus is relative to |x - xᵢ| - distances of some selected points - foci, having the given conservation properties of some selected value
- 1. Images of the construction of multifocal curves corresponding to the potential lines of the electrostatic field
- 2. Images of the construction of multifocal curves corresponding to "potential lines"-Contour lines for various "charge" configurations of n-ellipse/hyperbola
- 3. Images of the construction of multifocal curves corresponding to "potential lines"-Contour lines for various "charge" configurations of n-Lemniscate
Biquadratic equation
- Finding explicit expressions of four real functions for an implicitly defined plane curve (Trifolium curve) whose equation is biquadratic in the variable y
- Branches of an implicitly defined biquadratic curve found using complex functions: Trifolium curve
- Branches of an implicitly defined biquadratic curve found using complex functions: Cartesian oval
- Images: Branches of an implicitly defined biquadratic curve found using complex functions: Cartesian oval
Examples of finding the equations of branches of implicitly defined cubic curves
Examples of an implicit equation of a plane curve of the quartic equations
- An example of finding explicit equations for curves using exact roots as complex functions that make up an implicitly defined quartic plane curve whose equation has 15 coefficients
- Images. An example of finding explicit equations for curves using exact roots as complex functions that make up an implicitly defined quartic plane curve whose equation has 15 coefficients