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Pedal and Co-pedal Curves of Logarithmic Spiral

Pedal and Co-pedal Curves of Logarithmic Spiral
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2014-10-01 01:02
FunctionCoordinate GeometryParametric EquationGeometric ConstructionLocus
In collections Logarithmic Spiral and Fibonacci Numbers
This example demonstrates that the pedal and co-pedal curves of the logarithmic spiral are congruent to the original spiral. It involves core mathematical concepts such as parametric equations, coordinate geometry, and congruent figures. The construction utilizes GeoGebra commands like CurveCartesian, Point, Tangent, OrthogonalLine, Intersect, Locus, Segment, and Polygon. Suitable for high school students and educators exploring the geometric properties and applications of parametric equations.

GeoGebra Commands & Parameters

  1. CurveCartesian(((a * exp((b * t))) * cos(t)), ((a * exp((b * t))) * sin(t)), t, left, right) → c
  2. Point(c) → R
  3. Tangent(R, c) → d
  4. OrthogonalLine(P, d) → e
  5. Intersect(e, d) → A
  6. Locus(A, R) → loc1
  7. OrthogonalLine(P, e) → f
  8. OrthogonalLine(R, d) → g
  9. Intersect(f, g) → C
  10. Segment(R, C) → h
  11. Segment(C, P) → i
  12. Segment(P, A) → j
  13. Segment(A, R) → k
  14. Locus(C, R) → loc2
  15. Point(d) → D
  16. Point(d) → E
  17. Segment(D, E) → t
  18. CurveCartesian(((a_1 * exp((b_1 * t))) * cos(t + p)), ((a_1 * exp((b_1 * t))) * sin(t + p)), t, left, right) → m
  19. Polygon(Corner[1], (x(Corner[1]) + (0.3 * Distance[Corner[1], Corner[2]]), y(Corner[1])), (x(Corner[4]) + (0.3 * Distance[Corner[1], Corner[2]]), y(Corner[4])), Corner[4]) → Left, l, q, r, s
  20. Point(d) → B

AI Drawing Prompt Reference

Draw a logarithmic spiral and show its pedal and co-pedal curves, demonstrating their congruence with the original spiral. Start by plotting the spiral, then find the tangent line at an arbitrary point on the spiral. Draw the perpendicular line to this tangent, and the intersection point is the pedal point. Trace the path of this pedal point as the point moves along the spiral to form the pedal curve. Similarly, find the co-pedal point by drawing a perpendicular to the pedal line, and trace its path to form the co-pedal curve. Finally, compare all three curves to verify their congruence.
The prompt above was compiled from the original drawing commands and is for reference only.