How Do Knots Influence a B-Spline Curve?

[Danh860]

Hi,

I am working on a a project that involves the use of B-splines. Using the information from this informative http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/B-spline/bspline-curve-prop.html" I have managed to write up a simple script in VBA that allows me to construct the basis functions for a given sequence of knots and then apply this to a set of control points.

Whilst the code works and I can generate splines, I still don't have a full understanding as to how the knots influence the curve - Ultimately, I need the curve to pass through a specific control point (e.g P[k]) and be influenced by a couple of control points before and after P[k]. Additionally, the curve needs to be at least C[1] continuous at this particular control point.

Can anyone suggest the knot sequence and degree of curve that I should be using?

Many thanks,
Dan

 

[mathman]

In most applications cubic polynomials are used.

 

[AlephZero]

You would be better using a sequence of spline segments, not one high-order spline.

The end points of each segment lie on the curve. The second and second-to-last control points of each segment define the tangent direction at the end of the curve, so you can get continuity slope by constraining those control points to lie on a straight line (i.e. the tangent line to the curve), and you can control the continuity of curvature by constraining the distance of those control points from the end points.

For many applications cubic spline segments (constraining the slopes but not worrying about the curvature) is a practical method, but you can use higher order segments, and/or different orders in different segments, if you want to.

 

[chiro]

Like mathman said, you're better of using cubic polynomials.

What you can do is use cubic polynomials and maintain the correct continuity for joining splines. From memory you automatically have C0 continuity, but you want to maintain higher continuity between splines: to do this you want the tangent between points 2 and 3 (point 0 being the first point) and 3 and 4 have the same value. That is tangent(2,3) = tangent(3,4).

Its been a while since I've looked at NURBS though, so I'm not confident I can help you there.

 

[CellCoree]

I would first like to commend you on your progress in understanding and implementing B-splines in your project. B-splines are a powerful mathematical tool that has a wide range of applications in various fields, including computer graphics, engineering, and data analysis.

To address your question about the influence of knots on the curve, it is important to understand that knots are essentially breakpoints that divide the curve into different segments. The number and placement of knots determine the degree of flexibility and control over the curve. A higher degree of flexibility is achieved with a larger number of knots, but this also increases the complexity of the curve. On the other hand, a smaller number of knots may result in a simpler curve but with less control over its shape.

In terms of selecting a knot sequence and degree of curve for your specific project, it would depend on the specific requirements and constraints of your application. Generally, a knot sequence is chosen based on the desired shape and smoothness of the curve. For example, if you want the curve to pass through a specific control point and maintain C[1] continuity, you may need to adjust the knot sequence to achieve this. Additionally, the degree of the curve can also be adjusted to control the smoothness of the curve.

I would suggest experimenting with different knot sequences and degrees of curve to see which combination best meets your requirements. Additionally, there are various resources available online and in textbooks that provide guidelines and algorithms for selecting knot sequences and degrees of curve for different applications.

I hope this helps in your understanding of B-splines and their application in your project. Keep up the good work!