B-Spline
A B-Spline, or Basis Spline, is a function defined piecewise by polynomial functions. B-Splines are used in various fields such as computer graphics, computer-aided geometric design (CAGD), and numerical analysis due to their flexibility and computational efficiency.
History and Development
The concept of B-Splines was developed by Isaac Jacob Schoenberg in 1946 as an extension of his earlier work on spline functions. Schoenberg, a Romanian-American mathematician, introduced B-Splines to generalize the Cardinal Splines which were limited to uniform knot spacing. His work laid the foundation for many practical applications in computer-aided design and data fitting.
Key Properties
- Local Control: Changes in the control points affect only a local segment of the curve, making them ideal for interactive design.
- Continuity: B-Splines can achieve various levels of continuity, typically C2 (second derivative continuous), but can be adjusted to have different continuity at knot points.
- Convex Hull Property: The B-Spline curve lies within the convex hull of its control points, which simplifies geometric computation and analysis.
- Parameterization: B-Splines use a knot vector, which can be uniform or non-uniform, allowing for variable spacing between control points.
Mathematical Formulation
The mathematical definition of a B-Spline curve involves:
- A set of control points Pi
- A knot vector t which defines where the polynomial pieces connect
- A degree d for the polynomial pieces
The curve is defined as:
P(t) = Σ Bi,d(t) * Pi
where Bi,d(t) are the B-Spline basis functions of degree d defined recursively as:
Bi,0(t) = 1 if ti ≤ t < ti+1, else 0
Bi,d(t) = (t - ti) / (ti+d - ti) * Bi,d-1(t) + (ti+d+1 - t) / (ti+d+1 - ti+1) * Bi+1,d-1(t)
Applications
- Computer Graphics: Used for modeling smooth curves and surfaces.
- Animation: Provides smooth interpolation for keyframe animations.
- Engineering Design: Allows for precise control over shape design in CAD systems.
- Data Fitting: Used for fitting curves to scattered data points.
External Resources
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