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Instrument for Constructing a Parabola

Instrument for Constructing a Parabola
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2014-11-08 00:43
CircleCoordinate GeometryGeometric ConstructionLocus
In collections Selected Examples
①This example demonstrates an instrument for constructing a parabola using geometric tools, showing dynamically how the curve emerges from a construction process. ②The core mathematical concept is the definition of a parabola as the locus of points equidistant from a fixed point (focus) and a fixed line (directrix), along with geometric construction principles involving unit circles, perpendiculars, and angle bisectors. ③Construction details: Based on the summary, the GeoGebra commands used include Segment, Point, Polygon, Circle, LineBisector, Intersect, OrthogonalLine, Parabola, UnitVector, UnitOrthogonalVector, and Line. Objects such as focus F, directrix, auxiliary points (S, K_S, C_S, D_S, etc.), and polygon poly1 are created; intersections and perpendicular relationships determine points on the parabola. ④This is suitable for high school or college analytic geometry and geometric construction courses, as well as for teachers demonstrating parabola properties or students exploring geometric construction and locus generation.

GeoGebra Commands & Parameters

  1. Segment((0, 0), (3, 0)) → a
  2. Point(a) → F
  3. Segment(((-1 * x(F)), (-1 * sqrt(36 - (4 * x(F)^(2))))), ((-1 * x(F)), sqrt(36 - (4 * x(F)^(2))))) → j
  4. Point(j) → S
  5. Polygon(K_S, C_S, D_S, E_S, F_S, G_S, J_S, I_S) → poly1, h, i, s, t, a_1, b_1, c_1, d_1
  6. Circle(F, 3) → k
  7. Segment(S, F) → c
  8. LineBisector(c) → d
  9. Intersect(k, d) → B, C
  10. OrthogonalLine(S, yAxis) → b
  11. Intersect(d, c) → A
  12. Parabola(F, l) → p
  13. UnitVector(Vector[B, F]) → u_1
  14. UnitOrthogonalVector(Vector[B, F]) → u_2
  15. Polygon(L_1, L_2, L_3, L_4) → Vieleck1, m, n, q, r
  16. Polygon(N_1, N_2, N_3, N_4) → Vieleck2, n_1, n_2, n_3, n_4
  17. UnitVector(Vector[F, C]) → v_1
  18. UnitOrthogonalVector(Vector[F, C]) → v_2
  19. Polygon(M_1, M_2, M_3, M_4) → Vieleck3, m_1, m_2, m_3, m_4
  20. Polygon(O_1, O_2, O_3, O_4) → Vieleck4, o_1, o_2, o_3, o_4
  21. Line(S, xAxis) → e_1
  22. Intersect(d, e_1) → P
  23. Polygon((x(S), y(S) - 0.05), (x(S) + 9, y(S) - 0.05), (x(S) + 9, y(S) + 0.05), (x(S), y(S) + 0.05)) → Vieleck5, f_1, g_1, h_1, i_1
  24. Point(d) → D
  25. Point(d) → E
  26. UnitOrthogonalVector(Vector[B, P]) → u
  27. Polygon(A_d, B_d, F_d, E_d) → Vieleck6, a_d, b_d, f_d, e_d
  28. Polygon(C_d, D_d, H_d, G_d) → Vieleck7, j_1, k_1, l_1, p_1

AI Drawing Prompt Reference

Draw a geometric construction tool for a parabola. Start with a horizontal segment as the directrix and place a focus point on it. Draw a circle centered at the focus. Pick a point on the directrix, connect it to the focus, and construct the perpendicular bisector of that segment. Through the directrix point, draw a line perpendicular to the directrix; its intersection with the bisector gives a point on the parabola. Then draw the parabola defined by the focus and directrix. Additionally, construct several small squares or parallelograms around the focus and directrix point to illustrate equal distances. Mark intersections and show the dynamic construction process.
The prompt above was compiled from the original drawing commands and is for reference only.