B-splines Assignment Help
The main properties of B-splines
- composed of (n-k+2) Bezier curves of k-order joined C k-2 continuously at knot values (t0 , t1 , ... , tn+k )
- each point affected by k control points
- each control point affected k segments
- inside convex hull
- affine invariance
- uniform B-splines don't interpolate deBoor control points (P0 , P1 , ... , Pn )
B-splines are commonly used as basis functions to fit smoothing curves to large data sets. Any B-spline whose knot vector is neither uniform nor open uniform is non-uniform. Use of non-uniform knot vectors. As regards the interactive interface, the user is shown a figure window with axes in which to choose control points of a uniform B-spline. As points are placed in the axes, the B-spline of specified order is drawn progressively. The user may terminate adding control points by pressing ENTER or ESC, or may place the last control point with a right mouse button click.
Once done, control points may be adjusted with drag-and-drop. Hold down the left mouse button over any control point and drag it to another location. Control point adjustment works in 3D; use the rotation tool to set a different camera position. It is also possible to explicitly set the x, y and z coordinates as well as the weight of a control point: click on the point, enter new values and hit ENTER. As regards the non-interactive interface, functions include calculating and drawing basis functions, computing points of a (weighted or unweighted) B-spline curve with de Boor's algorithm, and estimating B-spline control points given noisy data, either with or without parameter values associated with the observed data points. From a programmers' perspective, this example illustrates how to use nested functions to extend variable scope, implement drag-and-drop operations, combine normalized and pixel units for control docking and register multiple callbacks for a single event in an interactive user interface.
The simplest way to get started is to run "bspline_gui", which activates the figure window to place B-spline control points interactively. Examples are bundled to illustrate various B-spline curve computation and approximation methods. The term "B-spline" was coined by Isaac Jacob Schoenberg and is short for basis spline. A spline function is a piecewise polynomial function of degree <k in a variable x. The places where the pieces meet are known as knots. The number of internal knots must be equal to, or greater than k-1. Thus the spline function has limited support. The key property of spline functions is that they are continuous at the knots. Some derivatives of the spline function may also be continuous, depending on whether the knots are distinct or not. A fundamental theorem states that every spline function of a given degree, smoothness, and domain partition, can be uniquely represented as a linear combination of B-splines of that same degree and smoothness, and over that same partition.