Spline Toolbox

rpmak, rsmak

Put together a rational spline

Syntax

• rp = rpmak(breaks,coefs)
  rp = rpmak(breaks,coefs,d)
  rs = rsmak(knots,coefs)
  rs = rsmak(shape,parameters)
  

Description

Both rpmak and rsmak put together a rational spline from minimal information. rsmak is also equipped to provide rational splines that describe standard geometric shapes.

rpmak(breaks,coefs) has the same effect as the command ppmak(breaks, coefs), -- except that the resulting ppform is tagged as a rational spline, i.e., as a rpform.

To describe what this means, let be the piecewise-polynomial put together by the command ppmak(breaks,coefs), and let be the rational spline put together by the command rpmak(breaks,coefs). If v is the value of at , then v(1:end-1)/v(end) is the value of at . In other words, . Correspondingly, the dimension of the target of is one less than the dimension of the target of . In particular, the dimension (of the target) of must be at least 2, i.e., the coefficients specified by coefs must be d-vectors with d > 1. See ppmak for how the input arrays breaks and coefs are being interpreted, hence how they are to be specified in order to produce a particular piecewise-polynomial.

rpmak(breaks,coefs,d) has the same effect as ppmak(breaks,coefs,d+1), except that the resulting ppform is tagged as being a rpform. Note that the desire to have that optional third argument specify the dimension of the target requires different values for it in rpmak and ppmak for the same coefficient array coefs.

rsmak(knots,coefs) is similarly related to spmak(knots,coefs). In particular, rsmak(knots,coefs) puts together a rational spline in B-form, i.e., it provides a rBform. See spmak for how the input arrays knots and coefs are being interpreted, hence how they are to be specified in order to produce a particular piecewise-polynomial.

rsmak(shape,parameters) provides a rational spline in rBform that describes the shape being specified by the string shape and the optional additional parameters. Specific choices are:

• rsmak('circle',radius,center)
  rsmak('cone',radius,halfheight)
  rsmak('cylinder',radius,height)
  rsmak('southcap',radius,center)
  

From these, one may generate related shapes by affine transformations, with the help of fnbrk.

All fn... commands except fnint, fnder, fndir can handle rational splines.

Examples

The commands

• runges = rsmak([-5 -5 -5 5 5 5],[1 1 1; 26 -24 26]); 
  rungep = rpmak([-5 5],[0 0 1; 1 -10 26],1);
  

both provide a description of the rational polynomial on the interval [-5 .. 5]. However, outside the interval [-5 .. 5], the function given by runges is zero, while the rational spline given by rungep agrees with for every .

The figure of a rotated cone is generated by the commands

• fnplt(fncmb(rsmak('cone',1,2),[0 0 -1;0 1 0;1 0 0]))
  axis equal, axis off, shading interp
  
  
  

Figure 3-4: A Rotated Cone Given By a Rational Quadratic Spline

For further, illustrated examples, seeNURBS and Other Rational Splines.

See Also

fnbrk, ppmak, spmak

ppmak

slvblk