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Negative Pedal Curve of an Ellipse

Negative Pedal Curve of an Ellipse
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2016-04-18 09:30
Coordinate GeometryLocus
This example demonstrates the negative pedal curve of an ellipse. The negative pedal curve is formed by connecting the foot of the perpendicular from the origin to the tangent of the ellipse. GeoGebra commands such as Line, Sequence, Point, Segment, Tangent, OrthogonalLine, LineBisector, Intersect, and Locus are used to construct it. Suitable for high school mathematics education, particularly in coordinate geometry and calculus courses.

GeoGebra Commands & Parameters

  1. Line(O, xAxis) → xax
  2. Line(O, yAxis) → yax
  3. Sequence(OrthogonalLine[(i, 0), xax], i, (-a), a, 0.1) → list1
  4. Sequence(Intersect[el, OrthogonalLine[(i, 0), xax]], i, (-a), a, 0.1) → list2
  5. Sequence(Line[A, Intersect[el, OrthogonalLine[(i, 0), xax]]], i, (-a), a, 0.1) → radiant1
  6. Sequence(OrthogonalLine[Intersect[el, OrthogonalLine[(i, 0), xax]], Line[A, Intersect[el, OrthogonalLine[(i, 0), xax]]]], i, (-a), a, 0.1) → negped1
  7. Sequence(OrthogonalLine[Intersect[el, OrthogonalLine[(i, 0), xax], 2], Line[A, Intersect[el, OrthogonalLine[(i, 0), xax], 2]]], i, (-a), a, 0.1) → negped2
  8. Sequence(Line[A, Intersect[el, OrthogonalLine[(i, 0), xax], 2]], i, (-a), a, 0.1) → radiant2
  9. Point(el) → B
  10. Segment(A, B) → d
  11. Tangent(B, el) → e,
  12. OrthogonalLine(B, e) → f
  13. LineBisector(d) → g
  14. Intersect(f, g) → C
  15. Locus(C, B) → loc1
  16. Intersect(g, d) → D
  17. Locus(D, B) → loc2

AI Drawing Prompt Reference

Draw an ellipse centered at the origin with coordinate axes. Generate a series of points on the ellipse by intersecting it with vertical lines at equally spaced x-coordinates. From the origin, draw rays to each intersection point, then construct perpendiculars to these rays at the points; these perpendiculars form two families of envelope curves representing the negative pedal curves on both sides of the ellipse. Additionally, take a moving point on the ellipse, connect it to the origin, and draw the perpendicular bisector of this segment. Also draw the tangent to the ellipse at the moving point and its perpendicular. The locus of the intersection of these two lines gives the negative pedal curve, while the locus of the intersection of the bisector with the original segment gives another related curve.
The prompt above was compiled from the original drawing commands and is for reference only.