1-column matrix 1-row matrix 2D 3D almost block-diagonal collocation matrix in spcol linear system use appropriate knot sequence aptknt augknt use use use use use use use augmented knot sequence aveknt use use B in B-spline banded banded banded linear system basic interval as set in ppmak cautionary note cautionary note extension outside for the B-form for the pp-form of B-form outside of use use in fnder fnint fnjmp fnmin newknt basis basis function in stform of thin-plate spline in stcol in stmak (Overview) BBform bell-shaped best interpolant B-form conversion to ppform discussion of in spmak introduction to smoothness conditions vs ppform bias bicubic spline example bicubic spline example bivariate bkbrk in spcol boundary layer break example in ppform interior use vs knot break sequence example example in ppform breaks breaks (Glossary) breaks vs knots conversion B-representation brk2knt bspligui B-spline coefficients example example (Glossary) in CAGD in spcol in spcrv normalized of order k some sample figures support of via bspline bspline use Bézier CAGD center of a shifted power form of an stform centers centripetal chbpnt chebdem Chebyshev polynomial Chebyshev spline circle, spline approximation to clamped end condition collocation in spcol matrix constructed in spcol use in spapi use collocation matrix in stcol column-vector composing function with a matrix constructive approach to splines control point example of a spline curve of a spline function control polygon example of a spline function use conversion B-form to ppform via splpp,sprpp coordinates with respect to a basis csape csapi example use csaps cscvn in getcurve cubic B-spline example Hermite smoothing spline spline example cubic means order 4 cubic smoothing spline via csaps cubic spline via spap2 via spapi cubic spline curve via cscvn cubic spline interpolation via csapi Curry-Schoenberg Theorem curvature curve plotted via fnplt via spmak vs function data point multiplicity in sorted in spapi data site data value degree raising degrees of freedom derivative of a rational spline differential equation non-standard solved with splines differentiation discrete in the pp sense of B-form via fnder dimension discrete differentiation least-squares approximation domain of a function draftsman's spline dual functional use in fn2fm d-valued d-vector end break knot end conditions clamped complete curved Lagrange natural not-a-knot other variational equidistribute error measure (Glossary) in csaps in spaps in splinetool in tpaps error weight evaluation of tensor product spline simultaneous via fnval extension beyond basic interval B-form cautionary note ppform fn2fm fnbrk use use use fnchg fncmb use use fnder use use fndir fnint vs fnder fnjmp fnmin fnplt use use use use vs spcrv fnrfn fntlr fnval use use use use fnzeros franke Franke function picture function vs curve functional dual Gauss points getcurve good interpolation sites from chbpnt from Chebyshev spline via aveknt graphic accuracy Greville site gridded data example example smoothing Hermite cubics Hermite interpolation implicit integral definite indefinite integral equation integration interior break knot interpolant variational via csaps via spaps interpolation by thin-plate spline Hermite Hermite optimal via cscvn via spapi via spaps interpolation points, good interval notation (Glossary) jump allow for ignored in fnder in derivative knot average use via aveknt insertion used in fn2fm used in spcrv used in splpp interior multiplicity at endpoints multiplicity vs smoothness sequence appropriate improved in B-form in spcol of a spline (Glossary) optimal simple (Glossary) knots vs breaks conversion knt2brk knt2mlt Lagrange end condition least-squares approximation discrete (Glossary) via spap2 in csaps in spaps limit from the left in splpp via fnval limit from the right linear combination of functions dependence operations space local polynomial coefficients power form (Glossary) in ppform matrix banded matrix-valued maximum via fnmin mesh meshgrid minimize minimum via fnmin Mobius multiplicity in a sequence of a data point of a knot of a knot of smoothness conditions vs smoothness example in bspligui multivariable (see multivariate) multivariate example in fnval overview tensor product vs univariate m-variate naming conventions natural (Glossary) in csaps in cscvn ND-valued ND-valued nested multiplication newknt use Newton's method example in optknt noise noisy nonlinear system example in optknt normalized B-spline not-a-knot end condition in csape NURBS optimal interpolation knot sequence optknt order of a polynomial of a pp of a spline of ppform osculatory parabolic curve example spline parametric bicubic spline cubic spline curve parametrization chord-length parametrized perfect spline periodic PGS piecewise cubic example piecewise-polynomial (Glossary) in ppform placeholder notation plotting polygon polynomial part of stform polyval power form pp ppform conversion from B-form from fncmb from spline introduction to of a B-spline (Overview) via ppmak vs B-form ppmak pp-representation QR factorization in spap2 in spapi use in slvblk quadratic convergence quartic radial basis function radial basis function range of a function rational spline from rpmak,rsmak (Glossary) RBF rBform recovery scheme recurrence relation for B-splines use in fnval use in spcol Remez algorithm restriction to an interval roughness measure (Glossary) in csaps in spaps in splinetool in tpaps roughness weight row-vector rpform rpmak rsmak scalar-valued vs vector-valued scaling of a function scattered data in tpaps translates Schoenberg Schoenberg Schoenberg-Whitney conditions (Glossary) in optknot in spap2 in spapi theorem (Glossary) (Overview) secant method shifted power form side conditions simple knot site slvblk in spap2 in spapi smoothing parameter (Glossary) in csaps in spaps in tpaps (Overview) spline smoothness across breaks across knot condition enforced guaranteed in B-form multiplicity of sort sorted spap2 use use use use use spapi use use use spaps sparse sparse matrix spcol in spap2 in spapi use use use use use use spcrv vs fnplt sphere via csape via rsmak spline approximation to a circle curve example via cscvn via spapi via spcrv draftsman's naming of specified splinetool splpp spmak use use use use use use sprpp staircase shape stcol stform from stmak stmak subdivision support of a B-spline surface target of a function Taylor polynomial example via fntlr series tensor product tensor product example (Glossary) polynomial thin-plate spline (Glossary) tpaps trivariate truncated tspdem uniform knot sequence mesh unimodal unique spline uniqueness of B-form unit circle univariate vs multivariate value outside basic interval variational approach to splines interpolant via csaps via spaps vector curve in this toolbox is always a column matrix -valued example in spmak splines vs scalar-valued weight