Apollonius Distance Ratio

Apollonius Distance Ratio

The Apollonius Distance Ratio module explores the geometric locus of a point P such that its distance to a fixed point A is a constant multiple of its distance to another fixed point B. That is, |P - A| = k|P - B| for some constant k. For points A = (-1, 0) and B = (1, 0), this locus generally forms a circle, but takes on degenerate forms like a line or a point for specific values of k. This classical problem bridges distance formulas and algebraic curves.

Worked Example

Consider the case where k = 0.5. This implies |P - A| = 0.5 * |P - B|, which can be expanded using coordinates P = (x, y):

A = (-1, 0), B = (1, 0)
(x + 1)^2 + y^2 = (0.5)^2 * ((x - 1)^2 + y^2)
x^2 + 2x + 1 + y^2 = 0.25 * (x^2 - 2x + 1 + y^2)
0.75x^2 + 2.5x + 0.75y^2 + 0.75 = 0
x^2 + 10/3 x + y^2 + 1 = 0
(x + 5/3)^2 + y^2 = 25/9 - 1 = 16/9

The locus is a circle centered at (-5/3, 0) with a radius of 4/3. When probing at y = 0, substituting y = 0 yields intersections at x = -3 and x = -1/3.

Experiment

You can interact with the Apollonius Distance Ratio module to see how the geometric locus shifts based on the chosen ratio k and how the horizontal probe y intersects the curve.

1. Set the distance ratio k to 0.5 and observe the circular locus. Note the center and radius.
2. Set the probe y to 1 to find horizontal intersections with the circle.
3. Change the distance ratio k to 1 to see the locus degenerate into the perpendicular bisector (the y-axis). Note how it has one horizontal intersection for any y.
4. Change the distance ratio k to 0 to observe the limit where the locus collapses to the point A.

Limitations

This module considers only a finite discrete domain of ratios and horizontal probes. It is not a general-purpose solver. The focus is specifically on the algebraic transitions and degeneracies in this configuration. The distance sum (ellipse) case is separate from this distance ratio locus.

References

The classical problem of finding the locus of points whose distances to two fixed points are in a constant ratio is a foundational topic in analytic geometry. The primary reference below provides a detailed exploration of this concept.

• Apollonius Circle (University of Georgia) - A comprehensive guide on the construction and properties of the Apollonius Circle.