• Lines and curves are typically fit to data points.
- a complex curve that fits at the ends
- a complex curve (spline) fit exactly to all points
- a complex curve (spline) fit exactly at the ends only
• A spline based curve is typically represented with a polynomial.
• Lines are typically represented in a parametric form. As we vary a parameter we move along the line. (Note: typical parameter variables are s,t,u,v)
• Note: lines can also be represented in explicit form, although this is not as useful for mathematical modeling.
• A spline curve can interpolate or approximate a curve,
• A bumpy curve will require a more complicated function (a polynomial with more degrees of freedom).
• A cubic spline typically gives the best data fit. The example below is a parametric fit to four data points.
• As we deal with different splines, the coefficient matrix changes,
• A bezier curve is approximated. The two endpoints are clearly defined, but the two inside points determine the internal shape by setting slopes at the ends,
• the basic relationships are,
• Consider the shape of these curves are fixed to go through the endpoints, but are only guided by the two internal points
• Instead of the matrix form for representing splines, we can also use blending functions (the results are the same for a 4 point set). And, this allows us to generalize to a larger number of degrees of freedom.
• When using blending functions we can easily use more than three points (n=3) to define the spline.
• Bezier curves are distinctive in that changing an endpoint or control point will change the shape of the entire curve.
• These curves are defined by endpoints and slopes at the endpoints. Note - here we need to specify derivatives.
• the basic relationships are,
• These curves are of the greatest use when the endpoints of the line, and the slopes at those points are of more interest than the internal shape of the curve.
• These curves go through every point, but we have a tightness variable ‘c’.
• As the value of ‘c’ goes to zero, the connecting lines become straight.
• The basic form of the matrix is,
***************** REVISE TO CLARIFY **************************
• B-Splines are typically defined using blending functions, and they will often have more than four points in their definition.
• The basic form for the nonuniform B-Spline is shown below in the Cox-deBoor recursive function.
• If the intervals are all equal we call this uniform. When the intervals are different lengths we call this nonuniform. A nonuniform curve can take on much more complicated shapes.
• We can also weight the points to increase their effects on the final curve shape. This creates a rational spline.
• Non-Rational Uniform B-Spline (NURBS) curves are popular in CAD systems as they are the most general form of the spline curves. They combine both the rational weighting of points, and non-uniform knot spacing.
• Find a point at 140 degrees for NURB curve that models a sine wave from 0 to 180 degrees.
• We can sometimes double up (or more) points to increase the effect of a weighting. This is more valuable when the function is not rational. If we do this in the middle of a curve it tends to pull the curve tight. When done at the ends it makes the ends more regular.
• Quite often we will attach two spline segments in series. The connection between the segments will be said to have continuity. A zero order continuity means they touch, 1st order means they are smooth, etc. It is easier to ensure continuity with splines such as the Ferguson.
• A closed spline curve fully connects back to itself. (i.e., forms a loop)