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Continuity for Bezier Curves

Continuity for Bezier Curves
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2017-12-14 00:45
VectorFunctionParametric EquationSlider AnimationLocus
In collections Mathematics of Computer Graphics
This interactive GeoGebra construction explores the continuity (C0, C1, etc.) of Bezier curves at their joining points. It demonstrates how the positions of control points and tangent vectors affect the smoothness of curve connections. Core concepts include parametric curves, control polygons, tangent vector continuity, and conditions for geometric smoothness. The construction uses Locus to trace curve paths, Segment and Vector to build control polygons and tangents, If for conditional continuity checks, and multiple InputBoxes and sliders (t1, a) to dynamically adjust control points. Suitable for high school to university courses in mathematics or computer graphics, this example helps visualize the geometric conditions for smooth Bezier curve splicing.

GeoGebra Commands & Parameters

  1. Locus(P, t1) → loc1
  2. Textfield(C_1) → InputBox1
  3. Textfield(C_2) → InputBox2
  4. Textfield(C_3) → InputBox3
  5. Textfield(C_4) → InputBox4
  6. Segment(C_1, C_2) → f
  7. Segment(C_2, C_3) → g
  8. Segment(C_3, C_4) → h
  9. Locus(Q, s) → loc2
  10. Segment(D_1, D_2) → i
  11. Segment(D_2, D_3) → j
  12. Segment(D_3, D_4) → k
  13. Textfield(D_1) → InputBox5
  14. Textfield(D_2) → InputBox6
  15. Textfield(D_3) → InputBox7
  16. Textfield(D_4) → InputBox8
  17. Vector(D_3, D5) → u
  18. Vector(C_2, C5) → v
  19. Vector(C_3, C_4) → C3C4
  20. Vector(D_1, D_2) → D1D2
  21. If(t ≤ 1, ((1 - t)^(3) * C_1) + (((3 * t) * (1 - t)^(2)) * C_2) + (((3 * t^(2)) * (1 - t)) * C_3) + (t^(3) * C_4), ((2 - t)^(3) * D_1) + (((3 * (t - 1)) * (2 - t)^(2)) * D_2) + (((3 * (t - 1)^(2)) * (2 - t)) * D_3) + ((t - 1)^(3) * D_4)) → R

AI Drawing Prompt Reference

Draw the continuity of Bezier curves with four control points C1, C2, C3, C4 and four auxiliary points D1, D2, D3, D4. Create four line segments connecting the control points and auxiliary points, then use the Locus command to trace the curve paths. Adjust the parameters t1 and s in the input boxes to observe how the continuity of the curve changes at the connection points. Pay attention to how the positions of control points and auxiliary points, as well as the tangent vectors, affect the smoothness of the curve.
The prompt above was compiled from the original drawing commands and is for reference only.