AN EXTENSION TO THE BEZIER SUB-DIVISION METHOD

TO COMPLETELY APPROXIMATE CURVES AND SURFACES

Andreas Savva, Vasso Stylianou and George Portides

Department of Computer Science, School of Sciences and Engineering, University of Nicosia

46 Makedonitissas Avenue, P.O. Box 24005, 1700 Nicosia, Cyprus

Keywords: Splines, Geometric modelling, Curves, Surfaces.

Abstract: Sub-division splines generate a number of new control points calculated fron the old control points. Both

control polygons/grids define the same curve/surface. At each iteration the resulting new points are much

greater in number than the old points and lie nearer to the actual curves. After a number of iterations, the

generated points lie on the actual curve, very close to each other, and by displaying them on a computer

screen the result is a smooth curve/surface. This paper describes a method, which is an extension to the

Bezier sub-division method, where the resulting curve is an approximation curve which interpolates only

the first and the last control points. The method is also derived for surfaces.

1 INTRODUCTION

The most popular methods for curve-fitting are

based on approximation – that is, the generated

curve passes near the original points approximating

their shape – like Bézier splines (Bezier, 1970,

1974, 1977), B-splines (Bartels et al, 1987), Beta-

splines (Barsky, 1981, 1986), ν-Spline (N

ielson,

1986), NURBS (Bartels et al, 1987; Piegl, 1990),

Kochanek spline (

Kochanek, 1984), Catmull-Rom

spline (Catmull and Rom, 1974) and many others.

Subdivision methods have also attracted a large

amount of research due to their ability to generated

complex surfaces defined by an arbitrary topology

of points, which are not based on a regular

rectangular mesh and even still today the most

important methods are those of Catmull and Clark

(Catmull and Clark, 1978) and Doo and Sabin (Doo

and Sabin, 1978).

This paper describes a subdivision method that

subdivides a Bézier curve and then generalizes it to

produce an approximated curve that interpolates

only the first and the last original points. Section 2

describes the Bezier sub-division method, section 3

derives its extension, and section 4 concludes.

2 THE BÉZIER SUBDIVISION

METHOD

The cubic Bézier curve defined by the control points

, V

and V

is given by Eq.1 for

01≤≤u

)1(3)1(3)1()( uVuuVuuVuVuQ +−+−+−=

(1)

The same curve can also be generated by another

two Bezier curves,

(

)

and

()

Q +

, where the

first is defined by the control points S

, S

, and the second by T

, T

where S

, as illustrated in Fig.1. The control points S

, S

= T

, T

are called new points

and they can be calculated by the old points V

, V

and V

by Eq.1 and Eq.2 (Bartels et al, 1987).

The new points are twice as much as the original

points, (in-fact double minus one, since S

and T

the same point) and they lie nearer to the actual

curve than the original control polygon V

, V

. Then from the new control points another set of

points (double minus one in size than the previous

points) can be calculated that are even nearer to the

actual curve. After a number of iterations the new

points lie on the actual curve, very close to each

other, and by displaying them on a computer screen

the result is a smooth curve.

143

Savva A., Stylianou V. and Portides G. (2008).

AN EXTENSION TO THE BEZIER SUB-DIVISION METHOD TO COMPLETELY APPROXIMATE CURVES AND SURFACES.

In Proceedings of the Third International Conference on Computer Graphics Theory and Applications, pages 143-146

DOI: 10.5220/0001099001430146

 SciTePress

()

()()

()

101

01 12

2012

01 23

30123

()

22 2

(+) (+)

()

33 +2 +

42 2 2

SVV

VV VV

SVVV

VV VV

SVVVV

⎫

⎪

⎬

++⎛⎞

⎪

=++= +

⎜⎟

⎪

⎝⎠

⎪

⎛⎞

=+++=

⎪

⎜⎟

⎝⎠

⎪

⎭

(2)

()

()()

01 23

00123

12 23

1123

223

(+) (+)

()

33 +2 +

42 2 2

22 2

()

VV VV

TVVVV

VV VV

TVVV

TVV

⎫

⎛⎞

=+++=

⎪

⎜⎟

⎝⎠

⎪

++⎛⎞

⎪

=++= +

⎜⎟

⎪

⎝⎠⎪

⎬

⎪

⎭

(3)

= S

= VT

= T

Figure1: The old and new points.

Consecutive segments in a composite Bézier

curve are C

continuous if the penultimate control

vertex of the first curve, the shared endpoint and the

second vertex of the next curve are collinear and

equally spaced. In Fig.2 the unprimed vertices

define one curve segment and the primed vertices

define another. Because V

, V

′

and

′

are

collinear and | V

- V

|=|

′

| the composite

curve will be C

continuous.

Figure2: Two Bézier segments.

From Eq.1 and Eq.2 it yields

()()()()

01 23

01 12 01 12

(+) (+)

1()

+2 +

42 2 2

22 2 22 2

(4)

VV VV

VV VV VV VV

⎛⎞

⎜⎟

⎝⎠

++ ++

⎛⎞⎛⎞

++ +

⎜⎟⎜⎟

⎝⎠⎝⎠

which means that S

(which is the same point as T

)

is the midpoint of S

and T

and therefore the

required condition for continuity between successive

Bézier segments at each step is satisfied. Fig.3

illustrates a Bezier curve after 1, 2 and 6

subdivisions. It can be noted that the actual curve

interpolates the first and fourth control point of each

segment.

Figure 3: A recursive Bézier curve after 1, 2 and 6

subdivisions.

GRAPP 2008 - International Conference on Computer Graphics Theory and Applications

144

3 GENERATING AN

APPROXIMATION CURVE

The new points can be divided into four different

categories (Fig.4). The V-points which correspond to

an old point at the two ends of the segment and are

equal to the old point, the V’-points which also

correspond to an old edge but not at the ends of the

segment, the E-points that correspond to an old edge

that one of the old points sharing the edge is at the

ends of the segment, and the E’-points that

corresponds to an old edge that none of the two old

vertices sharing the edge is at the ends of the

segment. Additionally, it can be emphasized that the

E’-points lie on the actual curve.

V' V'

= V V

V =

Figure 4: The new control polygon resulted from the

original one after one iteration.

It has been derived in (Savva and Clapworthy,

1998) that if we use V’-points and E’-points

everywhere on a curve except at the corner points

then the resulting curve becomes an approximation

curve that interpolates only the first and the last

control points. Also the penultimate control vertex

of the each previous segment, the shared endpoint

and the second vertex of the next segment does not

need to be collinear and equally spaced for

continuity, and actually the resulting curve is has C

continuity everywhere. This is illustrated in Fig.5.

Eq.4 shows that

. This condition is

satisfied after the first iteration. But if we add the

midpoints of all the edges in the initial control

polygon then the resulting curve is similar as above

(Fig.5) but it gives a better approximation to the

control points as shown in Fig.6.

The same method is also derived for surfaces.

The resulting surface is an approximation to the

control grid but interpolates the corner points of the

surface.

Figure 5: An approximated curve that interpolates the first

and last control points after 1, 2 and 6 sub-divisions.

4 CONCLUSIONS

A Bezier curve interpolates the first and last control

point for each segment and in order to achieve C

continuity between successive segments the

penultimate control vertex of the first segment, the

shared endpoint and the second vertex of the next

curve must be collinear and equally spaced. Despite

the fact that there is only C

continuity at these

points, having to make the three control points

collinear makes it difficult to be used in modelling

applications.

AN EXTENSION TO THE BEZIER SUB-DIVISION METHOD TO COMPLETELY APPROXIMATE CURVES AND

SURFACES

145

Figure 6: Adding the edge midpoints after 1, 2 and 6

subdivisions.

This paper describes an extension to the Bezier

sub-division scheme. The resulting curve is an

approximation curve that interpolates only the first

and the last control points and the curve has C2

continuity everywhere.

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