is defined as the sum of two functions 
and 
:Computer Graphics & Geometry
Y. Deniskin
Moscow State Aviation Institute
Department of Applied Geometry
125871 Moscow, Volokolamskoye shosse, 4
Tel. 007 095 158 4022
E-mail: denis@mai.ru
Contents
Abstract: The problems of geometric modeling of closed composite
parametric Bezier curves are discussed in this paper. The main attention is given to
control of the constructed curve’s shape without usage of Bezier points which are
already computed according to continuity conditions. The local modification of C2-continuous
parametric curves with preserving smoothness conditions is possible due to a method of
compound functions, which was for the first time offered by Coons S.A. Some problems of
designing of so-called “surfaces of dependent cross-sections” also are considered in
this paper. Some examples of the plots and pictures are resulted. The discussed techniques
are used at training of the students in the Moscow State Aviation Institute.
Key words: Bezier curves and surfaces, geometric modeling, parametric approximation, cross-section, blending function, continuity conditions.
In geometric modeling the effective methods of design composite curves and surfaces play an important role. Thus one of the main requirements is the obtaining required shape of geometrical object with usage of minimum number of parameters. It is desirable also, that the designer had a capability to set these parameters in a graphic presentation. Selected for designing of objects the class of curves or surfaces should be described rather simply (better in a parametric form). The chosen curves and the surfaces should have tangent continuity, curvature continuity and torsion continuity (for space composite curve). In methods the ”simple” algorithms of global and local modification of a geometric shape should be used. The composite curve must approximate to enough large set of points (oscillation should not exceed pre-set values, the singular points should easily be determined). In most cases the parametric Bezier curves are used for the mathematical description of these composite curves.
In geometric modeling the different methods are used for control of the shape of composite curves formed from Bezier segments. For instance, degree elevation of Bernstein polynomials and, therefore, increasing of number of Bezier points. Besides, any segment of composite curve can be defined as a particular case of a rational Bezier curve, and its local modification will be made by selection of new weighting coefficients which are not equal to one. This technique is called NURBS (Non-Uniform Rational B-Splines) and is the powerful tool for geometric modeling of curvilinear surfaces.
At the same time there is a method which is equal or better then mentioned above ones on capabilities and practical results, though computationally much more simple. This method of compound parametric functions was for the first time offered by Coons S.A. [1].
The purpose of this paper is to present a method of designing and control of the shape
of closed composite curvature continuous parametric curves, compounded of Bezier segments.
The techniques for design of surfaces of dependent cross-sections with closed generatrix
are present also.
The main idea of this method is in following. Some piecewise parametric function is defined as the sum of two functions and :
, 
, (1)
where 
- arbitrary constant, 
,
- arbitrary parametric vector function,
- vector function is given by
.
Function has remarkable properties. The
first and second derivatives (vectors) vanish (or are null vectors) at 
and 
. As the polynomial 
increases
in monotonous way from 0 up to 1 in an interval , the function describes a
straight line segment passing through two points 
and 
.
When 
, 
, and when 
, 
, with other shapes in between. Fig. 1 shows a
typical situation for some choice of constant 
.
Fig. 1. , , 
relationship
The first and second derivatives of the curves 
and also coincide on a
direction, but differ in length 
(see an
equation (1)). Therefore, if composite curve of the given order of a smoothness was formed
from several curves , these properties will
be saved at local modification of its segments with usage of one variable constant 
.
Let's take a case, when the variable constant 
vary as some function of the independent piecewise variably 
. The only requirement, so as not to disturb the 
continuity of the compound curve 
at the joints, is that the quantity be at least 
continuous
at this joints [1].
3. Design of closed curvature continuous curve
Let's construct closed C2-continuous composite curve from two plane 6-th
order parametric Bezier curves 
and 
. The main design requirement is passing curve
through two given points and conditions of a smoothness in these points. We assume that
the reader is familiar with the concepts of Bezier polynomial approximation. Shortly we
shall remind it. Bezier curve of 
-th order in terms of Bernstein polynomials is
given by [2]:
, 
, (2)
where 
- Bernstein polynomials,
,
- control (or Bezier) points.
Equation (2) can be interpreted in matrix form for a further computing convenience:
, ,
where 
- control points matrix,
- parameters matrix,
, 
.
One of the main properties of Bezier curve is convex hull property. This follows, since
for 
, the Bernstein polynomials are
nonnegative. Their sum is one. It means that the curve 
completely lies inside convex hull of Bezier points (see a Fig. 2).
Fig. 2. Bezier curve of 
-th
order and its control polygon
The selection of 6-th order Bezier curves is explained as follows. For design of
curvature continuous composite Bezier curve from two segments of the orders 
and 
it is necessary also to fix five control points of these segments: their
joint and two points additionally from both sides [3]. The fixing of ten control points is
necessary for closed C2-continuous composite curve constructed from two
segments. It is known, that number of control points of -th order curve is equal to
, and
for closed composite curve is equal to 
.
Therefore for obtaining of even one free control point for local modification of a segment
of such composite curve it is necessary to take curves of the order 
.
Suppose that we have the pieces of composite curve 
and 
of order ,
respectively. Each of these pieces is given by an equation (2).
The control points of a curve can be free
selected, but we shall use conditions of a smoothness in joints for computing of the
control points of a curve . Let's enter some
denotations:
(3)
The continuity conditions of a closed composite curve are:
We shall compute a control point 
from a
tangent continuity condition in a point 
.
Because of (3), vector is given by
, (4)
where 
- arbitrary constant.
Let’s 
. Since 
, equation (4) will be written as
.
We shall compute control point 
from a
curvature continuity condition in a point .
Because of (3), vector is given by
. (5)
Since and again, equation (5) will be written as
.
The next control point 
of second segment
can be free chosen. The control points 
and 
are computed according to conditions (3), (4), (5):
,
.
The computed control points and closed curvature continuous composite curve are illustrated in a Fig. 3.
Fig. 3. Closed curvature continuous composite curve
Curvatures of the curves and are given by:
. (6)
From a curvature plot shown in a Fig. 4 one can see that curvature of the constructed closed composite curve is continuous in joints.
Fig. 4. Curvature plot of closed composite curve
It is evidently that now for local modification of the composite curve it is possible
to select freely only 
, 
, that is obviously insufficiently. In this
connection we shall use a method of compound functions discussed in section 2.
We define blending function 
as 
-th order polynomial:
, , (7)
where 
, 
and 
- arbitrary constants, 
, 
.
In that case, the equation (1) is as follows:
(8)
Note two special cases: 1) 
, then 
and resulting curve is 
; 2) 
, then 
and resulting curve is
.
Other values of parameters 
and 
also can be selected from interval .
The plots of blending function for some arbitrary values of parameters and
are illustrated in a Fig. 5.
In a Fig. 5a: 
, 
varies from 0 up to 1 with a step 0.25.
In a Fig. 5b: 
, varies from 0 up to 1 with a step 0.25.
à)
b)
Fig. 5. Plots of function 
for 
, à) , b) 
.
The influence of parameter 
on a view of
a blending function is illustrated in a Fig. 6.
Fig. 6. Plots of function 
,
, 
.
The results of local modification of the constructed composite curve with usage (8) are illustrated in a Fig. 7.
Fig. 7. Example of local modification of the composite curve
The curvature of modified composite curve also is continuous in joints (see a Fig. 8).
Fig. 8. Comparison of curvature plots of initial and modified composite curves
Now we shall obtain equation (8) in Bernstein-Bezier form, which is useful for further computing convenience and control of the shape of the curve (see a Fig. 9).
Firstly, we convert polynomial (8) to the standard form polynomial with monomial basis 
:
, 
.
Secondly, with knowing coefficients 
of
polynomial function we shall obtain the
control points 
by dot product of matrixes:
,
where 
- transformation matrix with
dimensions 
,
and 
- row and column numbers,
- coefficients matrix.
For instance, we list the 
case
explicitly:
.
The polynomial blending function (7) allows to modify composite curve with preserving of the given continuity conditions. The shape of composite curve can vary from some given curve up to a straight line. Some examples of dependence of the shape of modifiable composite curve from function (7) parameters are illustrated in a Fig. 10.
Fig. 9. Control polygons of initial and modifiable composite curves
Fig. 10. Dependence of the modifiable curve from parameters of function
4. Design of surface of dependent cross-sections
Definition. The surface formed by motion by a plane curve of the variable shape is called a surface of dependent cross-sections.
It is necessary to say that mentioned above definition is not generally accepted. It is entered by professor I.I. Kotov to point out their difference from the surfaces with a grid of congruent (similar - for cyclical surfaces) of cross-sections [4]. Some authors call such surfaces as continuously-topographic, others - as complex surfaces [5]. Unfortunately, all these titles do not uncover an essence of these surfaces by virtue of their large variety in the nature and engineering. The attempts of more detail classification of these surfaces did not give positive results. All this reveals impossibility of the unified form of their determinants, and therefore, methods of definition them on a drawing. It is obviously that determinant of the surfaces of dependent cross-section should contain three components:
Let's construct a surface of dependent cross-sections 
with generatrix curve 
.
For obtaining an equation of this generatrix we shall consider symmetrical closed
composite C2-continuous curve consisting of four segments 
, where 
. The control points of this curve are computed according to methods described
in previous section 3 (see Fig. 11).
For data reduction we shall consider introduced composite curve as the whole curve,
using unified parameterization 
. The
parametric equation of each curve segment 
is defined over the parameter interval 
and can be inserted in an equation (8).
Fig. 11. Symmetrical closed composite curve (generatrix of surface)
If the cross-sections of a surface have the constant shape, but the different sizes,
all of them will be proportional to initial generatrix. The most simple condition
specifying “proportionality” generatrix during its motion is the linear change of
parameter
. Let's consider some patch of a
surface with two boundary curves (cross-sections applicable to option values
and 
).
The is generatrix of the surface of
dependent cross-section. As a result of the motion the curve reaches a plane of the curve
. The equation of such surface is given by
. (9)
At linear change of parameter the
surface of dependent cross-sections represents a ruled surface, because the surface
contains a family of straight lines. The input curves of this ruled surface are 
and 
. Every isoparametric line 
is
a straight line segment. In other words, we interpolate to whole curves, not just points.
The parameter interval 
can be expanded to
some other interval 
. The shape of both
input curves can varied from curvilinear to straight line.
In that specific case both input curves 
and 
also have the same shape, therefore
all cross-sections of the constructed ruled surface will be identical. Such ruled surface
is called cylinder (see Fig.12). The cross-sections of cylindrical surface but not
generatrices are illustrated in this figure.
Fig. 12. Example of linear interpolation of two identical curves (cylindrical ruled surface)
If all control points of a curve 
coincide, Bezier curve is degenerated to a
point due to convex hull property. In this case the constructed ruled surface will be
cone.
It is necessary to note, that ruled surfaces are both simple and fundamental to geometric design. They are considerably important, in particular for design of aerodynamics surfaces in aircraft engineering.
4.3. Non-linear interpolation of cross-section
If to accept the non-linear change of parameter during motion generatrix as a condition of formation of a surface of
dependent cross-sections, the equation of a surface (9) will be written in a general form:
, (10)
where 
– arbitrary function.
Limitations by selection of the function
are the following: 
and 
.
The cross-section of a surface is a geometrical place of points dividing sections
between curves 
and in ratio 
for any
given parameter 
. It is possible to use
cubic parametric Bezier curves as input curves , as shown in [2]. Then for the description of a surface the designer needs to
set two cross-sections and control polygon of a profile curve.
For these purposes we offer to use functions (7) or to interpolate to cross-sections
with the functions 
and 
. Besides, it is possible to design the varied
shapes of surfaces of dependent cross-sections due to local modification of one of
cross-sections with usage of a method of compound functions (8). Some examples of such
surfaces are illustrated in Fig. 13a and Fig.13b.
à)
b)
Fig. 13. Examples of non-linear interpolation of two cross-sections
In conclusion it is necessary to note, that at designing of surfaces of dependent cross-sections with the described method there is a capability of their local modification in a wide range of shapes - from simple cylindrical and conical surfaces up to polyhedrons.
[1] Coons S.A. Modification of the Shape of Piecewise Curves, Computer-Aided Design 9,
3, pp. 178-180, 1977.
[2] Faux I.D., Pratt M.J. Computational Geometry for Design and Manufacture. Ellis
Horwood, Chichester, 1979.
[3] Forrest A.R. Interactive Interpolation and Approximation by Bezier Polynomials.
Computer J. 15, 1, pp. 71-79, 1972.
[4] Kotov I.I. Descriptive Geometry. Moscow, 1970 (in Russian).
[5] Ivanov G.S. Descriptive Geometry. Moscow, 1995 (in Russian).
Computer Graphics & Geometry