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3D Bezier Curves

3D Bezier Curves
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2015-12-29 05:58
VectorFunctionSolid GeometryParametric EquationGeometric Construction
In collections surface 3D surface 3D
This example demonstrates the generation and visualization of 3D Bezier curves. It involves core mathematical concepts like parametric equations, spatial geometry, and vector operations. Construction highlights include using commands such as Segment, CurveCartesian, and Surface, along with objects like points A, B, C, D, E, and parameters a, b, c, d. Suitable for high school students and math enthusiasts studying spatial curves and parametric surfaces.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. Segment(B, C) → b
  3. Segment(C, D) → c
  4. Segment(D, E) → d
  5. CurveCartesian(((1 - t)^(2) * A) + (((2 * (1 - t)) * t) * B) + (t^(2) * C), t, 0, 1) → c11
  6. CurveCartesian(((1 - t)^(2) * B) + (((2 * (1 - t)) * t) * C) + (t^(2) * D), t, 0, 1) → c12
  7. CurveCartesian(((1 - t)^(2) * C) + (((2 * (1 - t)) * t) * D) + (t^(2) * E), t, 0, 1) → c13
  8. CurveCartesian(((1 - t)^(3) * A) + (((3 * (1 - t)^(2)) * t) * B) + (((3 * (1 - t)) * t^(2)) * C) + (t^(3) * D), t, 0, 1) → c21
  9. CurveCartesian(((1 - t)^(3) * B) + (((3 * (1 - t)^(2)) * t) * C) + (((3 * (1 - t)) * t^(2)) * D) + (t^(3) * E), t, 0, 1) → c22
  10. CurveCartesian(((1 - t)^(4) * A) + (((4 * (1 - t)^(3)) * t) * B) + (((6 * (1 - t)^(2)) * t^(2)) * C) + (((4 * (1 - t)) * t^(3)) * D) + (t^(4) * E), t, 0, 1) → c3
  11. Surface(x((((A * (1 - p)) + (p * B)) * (1 - t)) + (t * ((B * (1 - p)) + (p * C)))), y((((A * (1 - p)) + (p * B)) * (1 - t)) + (t * ((B * (1 - p)) + (p * C)))), z((((A * (1 - p)) + (p * B)) * (1 - t)) + (t * ((B * (1 - p)) + (p * C)))), t, 0, U, p, 0, 1) → s11
  12. Surface(x((((B * (1 - p)) + (p * C)) * (1 - t)) + (t * ((C * (1 - p)) + (p * D)))), y((((B * (1 - p)) + (p * C)) * (1 - t)) + (t * ((C * (1 - p)) + (p * D)))), z((((B * (1 - p)) + (p * C)) * (1 - t)) + (t * ((C * (1 - p)) + (p * D)))), t, 0, U, p, 0, 1) → s12
  13. Surface(x((((C * (1 - p)) + (p * D)) * (1 - t)) + (t * ((D * (1 - p)) + (p * E)))), y((((C * (1 - p)) + (p * D)) * (1 - t)) + (t * ((D * (1 - p)) + (p * E)))), z((((C * (1 - p)) + (p * D)) * (1 - t)) + (t * ((D * (1 - p)) + (p * E)))), t, 0, U, p, 0, 1) → s13
  14. Surface(x((c11(u) * (1 - v)) + (v * c12(u))), y((c11(u) * (1 - v)) + (v * c12(u))), z((c11(u) * (1 - v)) + (v * c12(u))), u, 0, U, v, 0, 1) → s112
  15. Surface(x((c12(u) * (1 - v)) + (v * c13(u))), y((c12(u) * (1 - v)) + (v * c13(u))), z((c12(u) * (1 - v)) + (v * c13(u))), u, 0, U, v, 0, 1) → s123
  16. Surface(x((c21(u) * (1 - v)) + (v * c3(u))), y((c21(u) * (1 - v)) + (v * c3(u))), z((c21(u) * (1 - v)) + (v * c3(u))), u, 0, U, v, 0, 1) → s213
  17. Surface(x((c22(u) * (1 - v)) + (v * c3(u))), y((c22(u) * (1 - v)) + (v * c3(u))), z((c22(u) * (1 - v)) + (v * c3(u))), u, 0, U, v, 0, 1) → s223