Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

Put together rational spline

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. A rational spline must be scalar- or vector-valued.

`rp = 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 *R* be the
piecewise-polynomial put together by the command `ppmak(breaks,coefs)`

,
and let *r*(*x*) = *s*(*x*)/*w*(*x*)
be the rational spline put together by the command `rpmak(breaks,coefs)`

.
If `v`

is the value of *R* at *x*,
then `v(1:end-1)/v(end)`

is the value of *r* at *x*.
In other words, *R*(*x*) = [*s*(*x*);*w*(*x*)].
Correspondingly, the dimension of the target of *r* is
one less than the dimension of the target of *R*.
In particular, the dimension (of the target) of *R* 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.

`rp = 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`

.

`rpmak(breaks,coefs,sizec) `

has the same effect as `ppmak(breaks,coefs,sizec)`

except
that the resulting ppform is tagged as being a rpform, and the target
dimension is taken to be `sizec(1)-1`

.

is similarly related to `rs = rsmak(knots,coefs) `

`spmak(knots,coefs)`

, and `rsmak(knots,coefs,sizec)`

to `spmak(knots,coefs,sizec)`

.
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.

provides a rational
spline in rBform that describes the shape being specified by the character
vector ```
rs
= rsmak(shape,parameters)
```

`shape`

and the optional additional `parameters`

.
Specific choices are:

rsmak('arc',radius,center,[alpha,beta]) rsmak('circle',radius,center) rsmak('cone',radius,halfheight) rsmak('cylinder',radius,height) rsmak('southcap',radius,center) rsmak('torus',radius,ratio)

with `1`

the default value for `radius`

, `halfheight`

and `height`

,
and the origin the default for `center`

, and the
arc running through all the angles from `alpha`

to `beta`

(default
is `[-pi/2,pi/2]`

), and the cone, cylinder, and torus
centered at the origin with their major circle in the (x,y)-plane,
and the minor circle of the torus having radius `radius*ratio`

,
the default for `ratio`

being `1/3`

.

From these, one may generate related shapes by affine transformations,
with the help of `fncmb`

`(rs,transformation)`

.

All `fn...`

commands except `fnint`

, `fnder`

, `fndir`

can
handle rational splines.

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 *r*(*x*)
= 1/(*x*^{2} + 1) 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 1/(*x*^{2} +
1) for every *x*.

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

**A Rotated Cone Given by a Rational Quadratic
Spline**

A Helix, showing a helix with several windings, is generated by the commands

arc = rsmak('arc',2,[1;-1],[0 7.3*pi]); [knots,c] = fnbrk(arc,'k','c'); helix = rsmak(knots, [c(1:2,:);aveknt(knots,3).*c(3,:); c(3,:)]); fnplt(helix)

**A Helix**

For further illustrated examples, see NURBS and Other Rational Splines

Was this topic helpful?