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Parabolic Reflector Light Path Simulation 3D

Parabolic Reflector Light Path Simulation 3D
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2015-09-17 08:40
FunctionCoordinate GeometrySolid GeometryParametric EquationGeometric Construction
In collections Properties of the Parabola
This example demonstrates the light reflection principle of a parabolic reflector, commonly used in car headlights and flashlights. Core mathematical concepts include parabola equations, focal point properties, law of light reflection, and solid geometry. Construction highlights: CurveCartesian command draws parametric curves of the parabolic surface; Point command creates focal points and incident points; If conditions dynamically display light reflection paths; Sequence command generates multiple light rays for demonstration. Suitable for high school and above, ideal for mathematics extension courses, physics optics teaching, or as an intuitive tool for demonstrating parabola properties.

GeoGebra Commands & Parameters

  1. CurveCartesian((p * t^(2)), t, t, (-y(P)), y(P)) → a
  2. Point(a) → A
  3. Point(a) → B
  4. If(n ≥ 2, (y(B) - y(A)) / (n - 1), 1) → dd
  5. Sequence((((y(A) + ((i - 1) * dd))^(2) * p), y(A) + ((i - 1) * dd)), i, 1, n) → H
  6. Sequence(If[y(Element[H, i]) ≠ 0, atan(1 / (((2 * p) * y(Element[H, i])))), 1.5707963267948966], i, 1, n) → H''
  7. Sequence((y(S) - y(Element[H, i])) / (x(S) - x(Element[H, i])), i, 1, n) → MM'
  8. Sequence(If[y(S) ≟ y(Element[H, i]), 1.5707963267948966, atan(Element[MM', i])], i, 1, n) → MM''
  9. Sequence((2 * Element[H'', i]) - Element[MM'', i] + 3.14159, i, 1, n) → FI
  10. Sequence(tan(Element[FI, i]), i, 1, n) → KH
  11. Sequence(((1 - (((2 * Element[KH, i]) * p) * y(Element[H, i])) + (((Element[KH, i] * p) * y(Element[H, i])))^(2)) / ((Element[KH, i]^(2) * p)), (1 - ((Element[KH, i] * p) * y(Element[H, i]))) / ((Element[KH, i] * p))), i, 1, n) → P_1
  12. Sequence(If[x(Element[P_1, i]) ≤ 1000, Element[P_1, i], (1000, y(Element[H, i]) + (Element[KH, i] * (1000 - x(Element[H, i]))))], i, 1, n) → PP
  13. Sequence(Segment[S, Element[H, i]], i, 1, n) → MM
  14. Sequence(Segment[Element[PP, i], Element[H, i]], i, 1, n) → ZZ

AI Drawing Prompt Reference

Create a 3D parabolic reflector showing light reflection paths. Draw a parabolic surface generated by rotating a parabola around its axis. Place multiple incident points on the surface and mark the focal point. From the focus, draw incident rays to each point, then construct reflected rays according to the law of reflection (angle of incidence equals angle of reflection), making them parallel to the axis. Use several rays at different positions to demonstrate the reflection principle, highlighting the focal property and symmetry of light paths in a solid geometric view.
The prompt above was compiled from the original drawing commands and is for reference only.