Why b spline

The "knots" , Then the curve defined by. Specific types include the nonperiodic B-spline first knots equal 0 and last equal to 1; illustrated above and uniform B-spline internal knots are equally spaced. A curve is times differentiable at a point where duplicate knot values occur.

The knot values determine the extent of the control of the control points. Weisstein, Eric W. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.

Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own. Without moving the control points position, through assigning appropriate value to the shape parameter, C 1 continuity of combined spline curves can be realized easily. Results show that the methods are simple and suitable for the engineering application. B-spline methods are very popular in computer-aided geometric design and associated fields because of their distinct advantages.

In recent years, some other methods have been presented for representing curves and surfaces. Papers [ 1 — 7 ] presented successively C-curves, T-curves, TC-curves, and -spline in trigonometric functions space. In order to improve the flexibility of product design, researchers give further consideration to introduce shape parameters.

Through the parameters, designers can adjust flexibly the shape of curves and surfaces. Wang et al. Xiong et al. Bashir et al. In recent years, researchers also paid attention to extension of traditional B-spline methods. But they mainly concentrated on Bezier curves [ 13 ], quadratic and cubic uniform B-spline curves [ 14 — 19 ].

Uniform B-splines can represent overall continuity closed curves and surfaces. But they use equally spaced knots; the spline does not interpolate the first and last control points. Because a nonuniform B-spline uses repeated knots technology, the curves have clamped property. The designers can locate more easily the two end points of the curve and achieve smooth connection between adjacent B-spline segments.

So it has more practical significance for us to study extension of nonuniform B-spline curves and surfaces. This paper discusses mainly cubic B-spline curves with shape parameter and presents the matrix representation and analyzes the influence of shape parameter on the curve shape. The application of the shape parameter in shape design is discussed deeply. In the end, we focus on discussions about how to realize C 1 continuity between adjacent B-spline segments by only adjusting the value of the shape parameters without changing the position of the control points.

Results show that the methods given by this paper are simple and suitable for the engineering application. Given control points , let the knot vector be. Then we have a piecewise representation of the quadratic open uniform B-spline with parameter where is called shape parameter and are basis functions.

Figure 1 shows the influence of the parameter on , , and , where solid line, dash line, and dotted line correspond to , , and. Also it can be expressed in the following matrix form:. Figure 2 shows the influence of on the shape of , , , and , where solid line and dash line correspond to and.

Figure 3 shows the influence of the parameter on the shape of basis functions , , , , , and , where solid line and dash line correspond to and. Especially when , the curve is C1. Nonuniform B-spline methods have important applications in shape design.

By modifying the shape parameters, the designers get additional choice in two-dimensional design. Figures 4 — 6 illustrate the influence of the parameter on the shape of curve, where Figure 4 shows , , and corresponding, respectively, to the thick line, the thin line, and the dash line. Figure 5 shows corresponding to the solid line and the dash line. Figure 6 shows , , and corresponding, respectively, to the solid line, the dotted line, and the dash line. Figures 7 — 11 show the application of the spline cure in this paper in fractal modeling.

In the practical application, we usually construct composite spline curves that satisfy some smooth conditions. By adjusting shape parameters, designers can achieve the goal. Two spline curves are given: The first one is The knot vector.

The second one is The knot vector. From the equations above,. The adjusting methods are shown below. We only prove that and can be gotten in the same way. If , we can choose freely the value of as long as. Across the knots is C k -2 -continuous. Even more properties of B-splines are described in Rogers and Adams pp.

The knot vector The above explanation shows that the knot vector is very important. The knot vector can, by its definition, be any sequence of numbers provided that each one is greater than or equal to the preceding one.

Some types of knot vector are more useful than others. Knot vectors are generally placed into one of three categories: uniform, open uniform, and non-uniform. These are knot vectors for which. The shapes of the N i , k basis functions are determined entirely by the relative spacing between the knots.

Scaling or translating the knot vector has no effect on the shapes of the N i , k. The above gives a description of the various types of knot vector but it doesn't really give you any insight into how the knot vector determines the shape of the curve. The following subsections look at the different types of knot vector in more detail. However, the best way to get to feel for these is to derive and draw the basis functions yourself.

Things you can change about a uniform B-spline With a uniform B-spline, you obviously cannot change the basis functions they are fixed because all the knots are equispaced. However you can alter the shape of the curve by modifying a number of things: Moving control points.

Moving the control points obviously changes the shape of the curve. Multiple control points. Sticking two adjacent control points on top of one another causes the curve to pass closer to that point. Stick enough adjacent control points on top of one another and you can make the curve pass through that point.

Increasing the order k increases the continuity of the curve at the knots, increases the smoothness of the curve, and tends to move the curve farther from its defining polygon. Joining the ends. You can join the ends of the curve to make a closed loop. Say you have M points,. You want a closed B-spline defined by these points.