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Trammel of Archimedes: A Focus-Driven Construction of the Locus

Trammel of Archimedes: A Focus-Driven Construction of the Locus
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2023-11-27 12:41
CircleVectorGeometric ConstructionLocus
In collections Conic Section Collection
This GeoGebra example demonstrates how the Trammel of Archimedes is generated as a geometric locus: the curve is traced by the intersection points produced under circular/elliptical constraints, yielding a characteristic closed curve. The main mathematical ideas include the definition and geometric behavior of the trammel curve, the role of foci and constraints in determining a locus, and the use of circle/ellipse geometry. Construction highlights drawn from the provided build commands: Vector and UnitVector to set directions for moving points, Circle/Ellipse/Line/Ray/Segment to build the underlying geometric entities, Point and Midpoint for key reference points, DynamicCoordinates for interactive or parametric motion, Distance to enforce length-based constraints, and CurveCartesian together with DynamicCoordinates to render the traced locus. The presence of foci and rotation-related objects (e.g., rot1) supports a focus-oriented parametrization/transform view. Suitable for high-school enrichment, early university geometry, and math visualization activities focusing on locus, conic sections, and geometric construction methods.

GeoGebra Commands & Parameters

  1. Vector((l1 / 2, h1 / 2)) → p1'
  2. Vector((214, 0)) → q1'
  3. UnitVector(xAxis) → X
  4. Circle(O, r) → d
  5. Point(d) → R'
  6. Vector(O, R') → u
  7. UnitVector(u) → u'
  8. Vector((prp * u)) → v
  9. UnitVector(v) → v'
  10. Line(O, u') → i
  11. Ray(O, Vector[UnitVector[u]]) → h
  12. Segment(O, S') → g
  13. Point(h) → S
  14. Segment(O - u, O + u) → ax1
  15. Segment(O - v, O + v) → ax2
  16. Segment(A, B) → AB
  17. Midpoint(A, B) → M
  18. Circle(M, A) → k_1
  19. UnitVector(Vector[O, N]) → n'
  20. DynamicCoordinates(N, x(W), y(W)) → C
  21. Distance(C, N) → CN
  22. Vector(O, W) → z
  23. UnitVector(Vector[B, W]) → v2'
  24. Vector((62.5, 38.5)) → p3'
  25. Segment(dartA, dartB) → j
  26. Point(j) → E
  27. Segment(dartA, E) → drtLen
  28. Ellipse(F_1, F_2, P) → e
  29. CurveCartesian(x(O + (k * ((cos(t) * u) + (sin(t) * v)) / sqrt(1 + (ζ * (sin(t))^(2))))), y(O + (k * ((cos(t) * u) + (sin(t) * v)) / sqrt(1 + (ζ * (sin(t))^(2))))), t, tmin, tmax) → b

AI Drawing Prompt Reference

Draw the trammel of Archimedes as a locus curve. Start with a central point O and define two key unit direction vectors u and v in the plane. Use a circle with center O and a chosen radius to mark a point R on the circle, then take the direction from O to R to obtain u; construct another unit direction v to complete the two-direction parametrization. Create supporting lines and segments through O along these directions and use a midpoint-to-circle construction for visual reference. Then introduce two foci F_1 and F_2 together with a moving point P, and parameterize a locus using a variable t that combines the directions u and v and incorporates the focus/constraint relationship (including the dependence on sin(t)). Finally, render the resulting closed curve for t within a chosen interval.
The prompt above was compiled from the original drawing commands and is for reference only.