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Curvature of a Space Curve

Curvature of a Space Curve
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2014-08-26 06:39
VectorCoordinate GeometryCalculusParametric EquationGeometric Construction
This example demonstrates the curvature of a space curve shown in the plane with its osculating circle. It involves core mathematical concepts like curvature, osculating circle, tangent vector, and normal vector. The construction uses GeoGebra commands such as Vector, CurveCartesian, Zip, PolyLine, Point, ClosestPoint, PathParameter, UnitVector, DynamicCoordinates, Polygon, Circle, Segment, and Angle. It is suitable for high school and above students to learn and understand the geometric properties of curves in calculus, as well as for related demonstrations.

GeoGebra Commands & Parameters

  1. Vector(A, D) → W
  2. Vector(A, B) → P
  3. CurveCartesian(x(A + ((3 * P) * t) + ((3 * Q) * t^(2)) + (R * t^(3))), y(A + ((3 * P) * t) + ((3 * Q) * t^(2)) + (R * t^(3))), t, 0, 1) → β
  4. Zip(A + ((3 * P) * k) + ((3 * Q) * k^(2)) + (R * k^(3)), k, Sequence[k / 20, k, 0, 20]) → list1
  5. PolyLine(list1) → b
  6. Point(b) → G
  7. ClosestPoint(β, G) → H
  8. PathParameter(H) → a
  9. UnitVector(χ') → T
  10. UnitVector(xAxis) → X
  11. Corner(imgWhite, 3) → crn3
  12. DynamicCoordinates(PN', x(Corner[4] + PN'' / ppu), y(Corner[4] + PN'' / ppu)) → PN
  13. Polygon(PN, p2, p3, p4) → panel, pn1, pn2, pn3, pn4
  14. Polygon(PN2, PN2 + (w, 0), PN2 + (w, h), PN2 + (0, h)) → pnl2, pn5, pn6, pn7, pn8
  15. UnitVector(yAxis) → Y
  16. Vector(C, D) → c
  17. Vector((ί * T)) → n
  18. Circle(Cen, H) → d
  19. Segment(A, B) → hnd1
  20. Segment(C, D) → hnd2
  21. Angle(T, n) → α
  22. Segment(Cen, H) → e

AI Drawing Prompt Reference

Draw a space curve projected onto the plane and show its osculating circle. First, plot the tangent and normal vectors of the curve. Then, draw the osculating circle using points on the curve. Next, discretize the curve and connect the points to form a polyline. Find the closest point between the polyline and the curve, and use it as the center to draw the osculating circle. Finally, label the tangent and normal vectors, the osculating circle, and display the angle between them.
The prompt above was compiled from the original drawing commands and is for reference only.