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Unit Normal of a Space Curve

Unit Normal of a Space Curve
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2014-08-26 06:03
VectorCoordinate GeometryCalculus
This example demonstrates the unit normal of a space curve. Key mathematical concepts include vectors and the geometric properties of curves. Using GeoGebra commands such as Vector, ClosestPoint, and UnitVector, the unit normal can be dynamically shown. Suitable for high school and above students learning space geometry and vector knowledge.

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. Circle(Cen, H) → d
  18. Segment(Cen, H) → r
  19. Angle(t, n) → α
  20. Line(H, n) → N
  21. Segment(C, D) → hnd2
  22. Segment(A, B) → hnd1

AI Drawing Prompt Reference

Draw a space curve and its unit normal vector. Start by creating points A and D, and use vector AD to create vector W. Then, create points A and B, and use vector AB to create vector P. Next, use a curve equation to draw the space curve β. Then, draw 20 points on curve β and connect them to form polyline b, and select point G from it. Find the point H on curve β that is closest to point G, and calculate the parameter a of H. Next, calculate the unit tangent vector t and unit normal vector n of curve β. Draw the unit vectors X and Y for the x-axis and y-axis. Finally, connect points C and D, points A and B, and points C and D to form auxiliary line segments.
The prompt above was compiled from the original drawing commands and is for reference only.