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The involute of the evolute of a parabola is a curve parallel to the parabola

The involute of the evolute of a parabola is a curve parallel to the parabola
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2017-12-26 22:22
CalculusParametric EquationGeometric Construction
This example demonstrates that the involute of the evolute of a parabola is a curve parallel to the original parabola. It involves core mathematical concepts such as calculus and parametric equations. Using GeoGebra commands like CurveCartesian, OrthogonalLine, and Intersect, one can construct this parallel curve. Suitable for high school mathematics education and learning calculus concepts.

GeoGebra Commands & Parameters

  1. CurveCartesian((((2 * a) * sgn((a * s))) * ((abs(s / ((2 * a))) + 1)^(2 / 3) - 1)^(3 / 2)), ((3 * a) * (abs(s / ((2 * a))) + 1)^(2 / 3)) - a, s, -120, 120) → E_s
  2. CurveCartesian(Exs(sp) - ((sgn((a * sp)) * s) * 1 / sqrt(1 + mtgE(sp)^(2))), Eys(sp) - ((sgn((a * sp)) * s) * mtgE(sp) / sqrt(1 + mtgE(sp)^(2))), s, 0, 60) → rtEpa
  3. OrthogonalLine(P_{old}, d_1) → g
  4. Intersect(g, d_1) → H
  5. Intersect(xAxis, yAxis) → O
  6. CurveCartesian(((-2 * a) * t^(3)), ((3 * a) * t^(2)) + (2 * a), t, -120, 120) → E_t
  7. Segment(PE, Q) → q
  8. Circle(PE, le0) → c1
  9. Segment(P, Q) → k
  10. If(sp < 0, (-spmax), sp) → es1
  11. If(sp < 0, sp, spmax) → es2
  12. CurveCartesian((((2 * a) * sgn((a * s))) * ((abs(s / ((2 * a))) + 1)^(2 / 3) - 1)^(3 / 2)), ((3 * a) * (abs(s / ((2 * a))) + 1)^(2 / 3)) - a, s, es1, es2) → Estring_s
  13. CurveCartesian(Exs(sp) + ((sgn(a) * s) * mtgE(sp) / sqrt(1 + mtgE(sp)^(2))), Eys(sp) - ((sgn(a) * s) * 1 / sqrt(1 + mtgE(sp)^(2))), s, 0, 60) → rnEpa
  14. Circle(CE2, PE) → c2
  15. CurveCartesian(((2 * a) * u^(3)) + (((6 * a) * u) * (u^(2) + 1)), ((3 * a) * u^(2)) + (2 * a) - (((6 * a) * u^(2)) * (u^(2) + 1)), u, -120, 120) → E2_t
  16. Segment(PE, CE2) → f
  17. Corner(5) → scrdim
  18. CurveCartesian(((2 * a) * t) - (d * t / sqrt(1 + t^(2))), (a * t^(2)) + (d * 1 / sqrt(1 + t^(2))), t, -15, 15) → pp

AI Drawing Prompt Reference

Draw the evolute of a parabola y = 2ax^2, then find its involute, and finally draw the parallel curve of this involute to the original parabola.
The prompt above was compiled from the original drawing commands and is for reference only.