Let’s start with what splines are-
Splines are piecewise polynomial functions used to create smooth curves through a set of points. They are particularly useful in situations where a simple polynomial might not provide a good fit or where you need more control over the curve’s shape.
Common Uses of Splines:
- Computer Graphics and Animation: Splines are used to design smooth curves and surfaces, such as character animation paths or 3D modeling. For example in the image above the curves are created using the spline tool in the software !
- Data Interpolation: Splines are employed to create smooth interpolations between data points in various fields, like engineering and scientific computing.
- Robotics and Path Planning: Splines are used to plan smooth trajectories for robots to follow, ensuring that movement is efficient and accurate.
- Signal Processing: In signal processing, splines can be used to reconstruct signals or smooth out noise.
Different types of splines -
1.Linear Splines:
•Piecewise linear functions connecting data points with straight lines.
•Use Case: Simple interpolation when high smoothness is not required.
2.Quadratic Splines:
•Piecewise quadratic polynomials that provide a smoother curve than linear splines.
•Use Case: Moderate smoothness in applications where cubic splines may be overkill.
3.Cubic Splines:
•Piecewise cubic polynomials that are twice continuously differentiable. They provide smooth curves with continuity in the first and second derivatives.
•Use Case: Commonly used in data interpolation and curve fitting.
4.Bézier Splines:
•Defined by control points, Bézier splines are widely used in computer graphics and animation.
•Use Case: Designing curves and shapes in vector graphics software.
5.B-Splines (Basis Splines):
•Generalization of Bézier splines, providing greater control over the shape of the curve by using basis functions.
•Use Case: CAD/CAM systems, computer graphics, and modeling.
6.NURBS (Non-Uniform Rational B-Splines) (Not Discussed much in this)
•An extension of B-splines that includes weights, allowing for the representation of more complex shapes like circles and ellipses.
•Use Case: Advanced computer graphics, 3D modeling, and animation.
As all the topics have a huge scope of discussion in themselves here’s a few reference I used to understand the topics in detail and clarity -
There’s this amazing Yt video by Freya Holmér on The continuity of Splines She has done a wonderful work in explaining the concept of splines, what they are how they work and visualizing the mathematical equations, highly recommended !!