Curve and surface interpolation and approximation

Introduction
B-Spline basis functions
B-Spline curves and surfaces
Global interpolation
Local interpolation
Global approximation of curves and surfaces
Approximation to within a speciﬁed accuracy

Curve and surface interpolation and approximation by piecewise
polynomial functions

Alejandro Cosin Ayerbe

June 2012

Curve and surface interpolation and approximation by piecewise polynomial

Introduction
Local interpolation

Introduction

This presentation is an abstract of my ﬁnal project in bachelor’s degree in Mathematics,
the goals of the project were the following:

Study and develop powerful methods of curve and surface interpolation and
approximation.


Introduction
Local interpolation

Introduction


approximation.
Cover the two main current approaches: global and local.


Introduction
Local interpolation

Introduction


approximation.
Matlab is used as the programming tool, developing methods so that the translation
to C++ is straightforward.


Introduction
Local interpolation

Introduction


approximation.
Matlab is used as the programming tool, developing methods so that the translation
to C++ is straightforward.
Automate the approximation process for surfaces and curves in order to generate a
solution that meets a preset maximum error.


Introduction
Local interpolation

B-Spline basis functions can be used to build curves and surfaces, they are highly
versatile and have important mathematical properties.

Given the knot vector U = {0, 0, 0, 1, 2, 3, 3, 4, 4, 5, 5, 5}, the B-Spline functions of degree
0, 1 and 2 are as follows:


Introduction
Local interpolation


Introduction
Local interpolation

B-Spline curves and surfaces they are linear combination of B-Spline basis functions, so
they have also good properties:

Strong convex hull:


Introduction
Local interpolation

Local modiﬁcation scheme:


Introduction
Local interpolation

Coincident control points:

A well known example of such curves and surfaces are the NURBS (Non-Uniform
Rational B-Splines). In this case, the curves and surfaces generated will be Non-Uniform
Non-Rational B-Splines, very similar.

Introduction
Local interpolation

Once functions to obtain the values of the B-Spline basis functions have been
programmed, interpolation conditions can be imposed to obtain the control points that
deﬁne a curve or surface.

If the degree is given and the knot and parameter vectors are estimated, a global
approach to the problem results in a linear system, easy to solve: given the set of points
Q=[[1,1]’,[3,3]’, [6,0]’,[8,2]’,[11,5]’], a second degree B-Spline curve
interpolating these points is shown below:


Introduction
Local interpolation

It is possible to add derivative constraints to the interpolation problem, and interpolate
the derivative vector at the beginning and the end of the curve: given a degree, 2, the set
of points Q=[[1,1]’,[3,3]’,[6,0]’,[8,2]’,[11,5]’], and the vectors
D=[[-3,-1]’,[0,3]’] and D=[[6,-4]’,[-3,4]’], the B-Spline curves interpolating
this data are shown below:

Derivative constraints can be added to all the points of the curve.


Introduction
Local interpolation

The process of interpolating a set of points arranged in grid using global techniques is
much easier than it looks. If the degree (p, q) of the surface is given, and knot vectors U
and V and parameter vectors are estimated, the interpolating surface can be obtained
through a small number of curve interpolations, because B-Spline sufaces are tensor
product surfaces. This avoids to solve large linear systems.


Introduction
Local interpolation

Example: given the set of points
Q=[[4,0,1]’,[3,0,1]’,[2,0,1]’,[1,0,1]’,[0,0,1]’,[4,1,1]’,[3,1,1]’,...
[2,1,1]’,[1,1,1]’,[0,1,1]’,[4,2,1]’,[3,2,1]’,[2,2,2]’,[1,2,1]’,...
[0,2,1]’,[4,3,1]’,[3,3,1]’,[2,3,1]’,[1,3,1]’,[0,3,1]’,[4,4,1]’,...
[3,4,1]’,[2,4,1]’,[1,4,1]’,[0,4,1]’];
the B-Spline interpolating surface of degree (2, 2) for this set is shown below:


Introduction
Local interpolation

Local interpolation

A local interpolation scheme consists in generating segments of curve or surface wich join
with a pre established level of continuity, given by the method of interpolation used.

In the case of curves, each segment is known as B´zier segment, and in the case of
e
surfaces, each segment is known as B´zier patch.
e

There is a local interpolation method, due to Renner, which performs local interpolation
of a set of points, generating a cubic B-Spline curve.


Introduction
Local interpolation

Local interpolation example for curves: given the set of points
Q=[[1,2]’,[2,4]’,[3,1]’,[5,3]’,[6,1]’,[7,4]’], the cubic interpolating curve is
shown below (in blue):

Note that each control point is in the (estimated) tangent of each point to be
interpolated.


Introduction
Local interpolation

For the case of surfaces, given a set of points arranged in grid, bicubic (degree (3, 3))
B´zier patches are build. The construction of each B´zier patch is the key of this
e e
interpolation method. The inner control points of each patch are obtained with the help
of estimates of mixed partial derivatives.
The next image shows a scheme of the simplest case of local interpolation, when there
are only four points in the grid (in the corners). The outer control points are obtained
with the local interpolation method for curves seen before, and the inner ones with
estimates of mixed partial derivatives (control points are denoted Pi,j ):


Introduction
Local interpolation

Once the control points of each B´zier patch are obtained, the control points of the
e
interpolating surface are obtained by eliminating B´zier points along inner rows and
e
columns. An example is shown below (with control points in blue, and interpolated
points in red), for the set of points:
Q=[[2,0,2]’,[1,0,2]’,[0,0,3]’,[2,1,2]’,[1,1,2]’,[0,1,3]’,[2,2,1]’,
[1,2,1]’,[0,2,2]’,[2,3,1]’,[1,3,1]’,[0,3,2]’]


Introduction
Local interpolation


The approximation of a set of points with a curve can be achieved in various ways. One
of them is the least squares approximation scheme, a global scheme in which the
resulting curve minimizes the error in the least squares sense, i.e., the sum of the squared
distances between each point and the resulting curve is minimum with respect to the
unknowns (the control points in this case).

For being a global scheme, the degree of the curve must be given, as well as the knot and
parameter vectors.

Given a set of m + 1 points, the curve can be build with up to m control points, because
the case of m + 1 control points is the interpolation case. The endpoints of the curve are
interpolated, while the inner points are approximated in the least squares sense.


Introduction
Local interpolation

Example: given the set of points:
Q=[[0,0]’,[3,1]’,[2,4]’,[-1,5]’,[-1,6]’,[2,7]’,[5,10]’,...
[-3,12]’,[0,14]’,[3,16]’,[-5,17]’,[2,19]’];
the next ﬁgure shows two cubic curves (in blue) approximating these points, with six and
nine control points respectively (the more control points, the better the approximation):


Introduction
Local interpolation

For the case of surfaces, the approximation process is analogous to the interpolation
process: using the preceding method for curves, only a few approximations are required
to obtain the least squares surface. The next two ﬁgures show this process:


Introduction
Local interpolation

The resulting approximation, and an interpolation of the same set of points are shown
below (control points in blue color, initial set of point in red):


Introduction
Local interpolation


The preceding techniques of curve and surface approximation can be used in iterative
methods of approximating data to within some speciﬁed error bound.

Iterative methods proceed in two ways: adding control points (starting with only a few of
them), or removing control points (starting with many or enough control points).

Given a set of points to be approximated, the degree and an error bound E , a technique
based in adding control points proceeds as follows:
1 Start with the minimum number of control points.


Introduction
Local interpolation




2 Using a global method, approximate a curve (or surface) to the data.


Introduction
Local interpolation




3 Check the deviation of the curve (or surface) from the data.


Introduction
Local interpolation




3 Check the deviation of the curve (or surface) from the data.
4 If the deviation is greater than E at any point, return to step 2, else end the process.


Introduction
Local interpolation

To check the error after each approximation, is necessary to obtain the closest point of a
curve or surface to a given point. This is an inverse function problem, which can be
solved through the Newton method or similar. The following image shows examples for
curves and surfaces.


Introduction
Local interpolation

The following is a curve approximation example with an error bound E = 0.45. Initially
the the number of control points is three (the degree plus two). Each curve generated
passes closer to the points to be approximated.
E = 0.45;
Q=[[0,0]’,[3,1]’,[2,4]’,[-1,5]’,[-1,6]’,[2,7]’,[5,10]’,...
[-3,12]’,[0,14]’,[3,16]’,[-5,17]’,[2,19]’];


Introduction
Local interpolation

For the case of surfaces, the iterative process benefits from the tensor product surface
properties: when checking the error, the isoparametric curves with more error are
considered and approximated separatedly until an error bound less than E is achieved.
This way, new knot vectors (and thus control points) are generated for the next iteration.

An example is shown in the next two slides, where the error bound is E = 0.6.

The first figure in green color is the set of points to be approximated, and an
interpolation of that set is seen in the right. The next figures correspond to the iterative
process of approximation on the left, and the error in each point of the grid in the right.


Introduction
Local interpolation

Bibliography

H. Akima.
A new method of interpolation and smooth curve ﬁtting based on local procedures.
Jour. ACM, 17:589–602, 1970.

W. Boehm, W. Farin, and J. KahMann.
A survey of cuve and surface methods in cagd.
Computer Aided Geometric Design, 1:1–60, 1984.

M. G. Cox.
The numerical evaluation of b-splines.
Journal of the Institute of Mathematics and its Applications, 10:134–149, 1972.

C. de Boor.
On calculating with b-splines.
The Journal of Approximation Theory, 6:50–62, 1972.

Carl de Boor.
A practical Guide to Splines.
Springer-Verlag, ﬁrst edition, 1978.


Introduction
Local interpolation

G. Farin, J. Hoschek, and M. S. Kim.
Handbook of Computer Aided Geometric Design.
Elsevier, ﬁrst edition, 2002.

H. Prautzsch, W. Boehm, and M. Paluszny.
B´zier and B-Spline Techniques.
e
Springer, ﬁrst edition, 2002.

L. Piegl.
Interactive data interpolation by rational b´zier curves.
e
IEEE Computer Graphics and Applications, 7:45–58, 1987.

Les Piegl and Wayne Tiller.
The NURBS Book.
Springer, second edition, 1997.

G. Renner.
A method of shape description for mechanical engineering practice.
Computers in Industry, 3:137–142, 1982.


Curve and surface interpolation and approximation

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