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Sum of Distances to Foci is Constant: Ellipse Locus Demo

Sum of Distances to Foci is Constant: Ellipse Locus Demo
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2020-03-22 07:37
Coordinate GeometryGeometric ConstructionSlider AnimationLocus
① This applet demonstrates the fundamental definition of an ellipse: the set of all points in the plane for which the sum of distances to two fixed points (foci) is constant. ② Core mathematical concepts include the first definition of an ellipse, foci, sum of distances, locus, symmetry, and eccentricity. ③ Construction highlights: based on the construction summary, GeoGebra commands such as Intersect, Ray, Point, Mirror, Line, Segment, Circle, Ellipse, Distance, If, Polygon, OrthogonalLine, Vector, and Midpoint are used to create foci F_1 and F_2, a moving point P, auxiliary segments, circles, polygons, and locus objects. Sliders and buttons enable dynamic interaction, visually confirming the constant sum of distances. ④ Suitable for high school analytic geometry or college geometry courses to illustrate the ellipse definition, and for students to explore the relationship between foci, distances, and the resulting locus.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → E
  2. Ray((0.01, 0), (1, 0)) → xx
  3. Ray((0.01, 0), (0, 1)) → yy
  4. Point(axis) → F_1
  5. Mirror(F_1, E) → F_2
  6. Line(B, xAxis) → i
  7. Point(i) → D
  8. Point(i) → C
  9. Segment(C, D) → aa
  10. Circle(F_2, a) → e
  11. Circle(F_1, a) → d
  12. Intersect(e, d) → P, G, ,
  13. Ellipse(F_2, F_1, P) → f
  14. Point(f) → R
  15. Distance(E, F_1) → c
  16. Segment(F_2, R) → b
  17. Segment(R, F_1) → g
  18. If(aa > (2 * c), true, false) → sh
  19. Polygon(Corner[1], Corner[2], (x(Corner[2]), y(Corner[2]) + (0.28 * Distance[Corner[3], Corner[2]])), (x(Corner[4]), y(Corner[1]) + (0.28 * Distance[Corner[3], Corner[2]]))) → Bottom, j, k, l, m
  20. Line(F_1, F_2) → n
  21. OrthogonalLine(E, n) → p
  22. Intersect(f, n) → A, K
  23. Intersect(f, p) → L, M
  24. Point(K, Vector[((UnitVector[p] * 1.2) * a)]) → K1
  25. Mirror(K1, p) → K1'
  26. Vector(K1, K) → u
  27. Vector(K1', A) → v
  28. Segment(K1, K1') → q
  29. Midpoint(q) → N
  30. Point(M, Vector[(-(((UnitVector[n] * 1.28) * a)))]) → M1
  31. Mirror(M1, n) → M1'
  32. Segment(M1, M1') → r
  33. Vector(M1', L) → w
  34. Vector(M1, M) → s
  35. Midpoint(r) → Q
  36. If(y(M) > 0, M, L) → PO
  37. If(y(M) < 0, M, L) → NE

AI Drawing Prompt Reference

In a Cartesian plane, place two fixed points symmetric about the origin as the foci of an ellipse. Introduce an adjustable constant representing the sum of distances from a moving point to the two foci. Draw circles centered at each focus with radius equal to half the constant; their intersections trace the ellipse. Connect the intersection points to form the ellipse, and also draw its major and minor axes, vertices, and center. Add a slider below the figure to vary the constant, dynamically showing that the sum of distances remains unchanged as the point moves along the curve.
The prompt above was compiled from the original drawing commands and is for reference only.