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Richmond Curvature Circle

Richmond Curvature Circle
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2023-03-15 21:27
CircleAngle
In collections geometriadinamica.es
This example demonstrates the construction of the Richmond curvature circle. Key mathematical concepts include circles, ellipses, and angles. Using GeoGebra commands such as Segment, Point, Ellipse, etc., the Richmond curvature circle can be constructed. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Segment(Q3, Q4) → p2
  2. Point(p2) → P3
  3. Segment(P3, Q4) → p4
  4. Point(p4) → P4
  5. Segment(Q1, Q2) → p1
  6. Point(p1) → P1
  7. Segment(P1, Q2) → p3
  8. Point(p3) → P2
  9. OrthogonalLine(C, yAxis) → zz1
  10. Circle(C, 0.62) → zz0
  11. Intersect(zz0, zz1, 1) → ZZ1
  12. Intersect(zz0, zz1, 2) → ZZ2
  13. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  14. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  15. Point(zElipse1) → XY
  16. Angle(ZZ6, C, XY) → zAngRot
  17. OrthogonalLine(C, xAxis) → zz2
  18. Intersect(zz0, zz2, 2) → ZZ3
  19. Intersect(zz0, zz2, 1) → ZZ4
  20. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  21. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  22. Point(zElipse2) → Z
  23. Angle(ZZ8, C, Z) → zAngInc
  24. Segment(Q5, Q6) → p5
  25. Point(p5) → P5
  26. CurveCartesian(((r / 4 * ((-3 * u) - u^(5)) / ((6 * u^(2)))) * sin(zAngRot)), (r / 4 * (((((3 * u) + u^(5)) / ((6 * u^(2))) * cos(zAngRot)) * sin(zAngInc)) + (u * cos(zAngInc)))), u, u1, u2) → curva1
  27. Segment(ZZ5, ZZ6) → zz4
  28. Segment(ZZ7, ZZ8) → zz3
  29. Intersect(zElipse2, zElipse1, 1) → zPunto1
  30. Intersect(zElipse2, zElipse1, 4) → zPunto2
  31. Segment(zPunto1, zPunto2) → zSeg2
  32. CurveCartesian((r / 4 * ((((-3 * α) - α^(5) + ((2 * α^(3)) * v^(2)) + ((3 * α) * v^(4))) / ((6 * (α^(2) + v^(2)))) * sin(zAngRot)) + (((-3 * v) - ((3 * α^(4)) * v) - ((2 * α^(2)) * v^(3)) + v^(5)) / ((6 * (α^(2) + v^(2)))) * cos(zAngRot)))), (r / 4 * (((((-((-3 * α) - α^(5) + ((2 * α^(3)) * v^(2)) + ((3 * α) * v^(4))))) / ((6 * (α^(2) + v^(2)))) * cos(zAngRot)) * sin(zAngInc)) + ((((-3 * v) - ((3 * α^(4)) * v) - ((2 * α^(2)) * v^(3)) + v^(5)) / ((6 * (α^(2) + v^(2)))) * sin(zAngRot)) * sin(zAngInc)) + (α * cos(zAngInc)))), v, v1, v2) → curva2
  33. CurveCartesian((r / 4 * ((((-3 * u) - u^(5) + ((2 * u^(3)) * β^(2)) + ((3 * u) * β^(4))) / ((6 * (u^(2) + β^(2)))) * sin(zAngRot)) + (((-3 * β) - ((3 * u^(4)) * β) - ((2 * u^(2)) * β^(3)) + β^(5)) / ((6 * (u^(2) + β^(2)))) * cos(zAngRot)))), (r / 4 * (((((-((-3 * u) - u^(5) + ((2 * u^(3)) * β^(2)) + ((3 * u) * β^(4))))) / ((6 * (u^(2) + β^(2)))) * cos(zAngRot)) * sin(zAngInc)) + ((((-3 * β) - ((3 * u^(4)) * β) - ((2 * u^(2)) * β^(3)) + β^(5)) / ((6 * (u^(2) + β^(2)))) * sin(zAngRot)) * sin(zAngInc)) + (u * cos(zAngInc)))), u, u1, u2) → a
  34. Sequence(CurveCartesian[(r / 4 * ((((-3 * u) - u^(5) + ((2 * u^(3)) * v^(2)) + ((3 * u) * v^(4))) / ((6 * (u^(2) + v^(2)))) * sin(zAngRot)) + (((-3 * v) - ((3 * u^(4)) * v) - ((2 * u^(2)) * v^(3)) + v^(5)) / ((6 * (u^(2) + v^(2)))) * cos(zAngRot)))), (r / 4 * (((((-((-3 * u) - u^(5) + ((2 * u^(3)) * v^(2)) + ((3 * u) * v^(4))))) / ((6 * (u^(2) + v^(2)))) * cos(zAngRot)) * sin(zAngInc)) + ((((-3 * v) - ((3 * u^(4)) * v) - ((2 * u^(2)) * v^(3)) + v^(5)) / ((6 * (u^(2) + v^(2)))) * sin(zAngRot)) * sin(zAngInc)) + (u * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list3
  35. Sequence(CurveCartesian[(r / 4 * ((((-3 * u) - u^(5) + ((2 * u^(3)) * v^(2)) + ((3 * u) * v^(4))) / ((6 * (u^(2) + v^(2)))) * sin(zAngRot)) + (((-3 * v) - ((3 * u^(4)) * v) - ((2 * u^(2)) * v^(3)) + v^(5)) / ((6 * (u^(2) + v^(2)))) * cos(zAngRot)))), (r / 4 * (((((-((-3 * u) - u^(5) + ((2 * u^(3)) * v^(2)) + ((3 * u) * v^(4))))) / ((6 * (u^(2) + v^(2)))) * cos(zAngRot)) * sin(zAngInc)) + ((((-3 * v) - ((3 * u^(4)) * v) - ((2 * u^(2)) * v^(3)) + v^(5)) / ((6 * (u^(2) + v^(2)))) * sin(zAngRot)) * sin(zAngInc)) + (u * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso) → list1

AI Drawing Prompt Reference

Draw a Richmond curvature circle construction. Start with several initial points on the plane, create segments between them and take points on those segments as adjustable control points. Using a point near the origin as center, draw a circle with radius about 0.62, then create horizontal and vertical lines through the center intersecting the circle at four pairs of points. Using these intersection points as foci, construct two ellipses with major axis around 0.65, and find their intersection points. Measure angles between points on the ellipses and the center to obtain rotation and inclination angles. Finally, using these angle parameters, plot a family of complex parametric curves distributed around the base circle, forming a geometric figure resembling a rotational surface that demonstrates the mathematical relationship between curvature circles and elliptical parametric curves.
The prompt above was compiled from the original drawing commands and is for reference only.