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
A contour line or lines of one potential (equipotential lines in electrostatics) for a function of two variables is a curve at each point of which the measured quantity retains the same value. They are defined by implicit multifocal plane curves.
These applets construct multifocal curves, the geometric location of whose points is determined relative to |x - xᵢ| - the distances of some selected points - foci Aᵢ (i=1..N, Nmax=9), having given properties of preserving some selected value. By analogy with electrostatics, "qi - charges" are added to the formulas as real values - weight factors to the formulas.
0. Σqi*|x - xi|2=const is the algebraic sum of the squares of the distances multiplied by the "charges" to the foci (const:=φ). In this case, the locus is a circle (regardless of the number and location of N points) with the center at the "center of mass" of this system of "charged" points and the radius calculated using the given formula. See applet.
1. Σqi/|x - xi|=const -algebraic sum of inverse distances to the foci (const:=φ). Example -equipotential lines in electrostatics. Images are shown in the applet.
2. Σqi*|x - xi|=const is the algebraic sum of the distances multiplied by the "charges" to the foci (const:=φ). Example - ellipse, hyperbola, Multifocal N-oval curves. Images are shown in the applet.
3. Πqi*|x - xi|=const is the product of the distances to the foci: (const:=φ)multiplied by the "charges". Example -N-Cassini lines. Images are shown in the applet.
4. Σqi*|x - xi|exponent=const is the algebraic sum of the distances to some power (exponent) multiplied by the "charges" to the foci (const:=φ).
Building formulas
tx1 = If(bue1, "/", "*")
tx2 = If(bue1 ∨ bue2 ∨ bue2a, "Sum", "Product")
tx3 = If(bue2a, "^"+exponent, " ")
Execute[{"h(x,y)="+tx2+"(Sequence(qi(i)"+tx1+" sqrt((x - x(l0(i)))² + (y - y(l0(i)))²)"+tx3+" , i, 1,Length(l0)) )" }]