Picture 1.
Computer Graphics & Geometry
Y.I. Bityukov
Moscow aviation institute (technical university).
Applied mechanics faculty
. Let’s built a twice-continuous differential curve, coming through these points.
Let 
. The straight line coming through the points 
,
is prescribed with a equation:
,
or in the coordinate form:
Let 
Choose on the straight line 
points
, 
and draw a straight line 
. Its equation in the coordinate form looks like:
Analogically choose on the straight-line points
, 
,
,
and draw a straight line 
.
Continuing this process will get straight lines 
and points on this straight line (picture 1):
Picture 1.
, 
, 
,
, where
Let’s consider 
, and as result will get polygonal line
, which approximated by the curve built by Â-splines:
where 
, 
are Â-splines built on the divisible units:
,
- is a different on the segment 
.
The obtained curve comes through points 
, i=0,1,…,n so that points 
i=2,…,n-1 lay on the same straight line, point 
coincides with 
and point 
coincides with point 
.
Now let’s consider building of closed curve coming through points 
, where 
. Let’s use the building consided building above to the point set 
and get the polygonal line 
. On the straight line 
let’s choose points:
,
,
,
, where
On the straight line 
let’s choose two points 
,
. Let’s consider
, as a result will get closed polygonal line
, which is aproximated by twice continuous deferential curve
Where 
à 
- Â – splines built on the uniform net.
,
is a parameter mobile on the segment
. The built curve is obviously to come through the set points.
2. Modeling of the surface, which comes through the specified points.
Let’s consider now the building of the surface with the given discrete frame. Let’s fix in the space Deckard’s frame of reference Oxyz. Discrete frame will be a point set of fragments of this surface, perpendicular to axis Oz:
(1)
where k+1 – is a number of segments, 
number of points in the segment-j, 
. With the help of consided above method modeling of closed curve let’s build section of the surface: 
:
,
where 
-is a polygonal line built from many points
à - Â –splines built on the uniform net 
.
Let’s build a linea surface
.
Let’s choose 
so that it will be appropriate for the òàê, ÷òîáû îíî óäîâëåòâîðÿëî inequality 
and built curves 
, 
,
. Analogically will build a linea surface
,
will get curves 
, 
,
. Continuing this process will get linea surfaces
j=2,…,k-1 and curves 
, 
, 
. Let’ s consider 
. With every fixed 
will get a polygonal line 
, 
, which is approximated by the curve coming through the ends of the vectors 
:
(2)
where 
 – splines built on the uniform net with divisible nets:
.
So the equation (2) determines the surface, coming through the fixed points (1).
Given above methods of building curves do not demand knowledge of derivatives in the units of spline, consequently you don’t have to spend time to solve systems of equations determing these systems of derivatives. It is easy to operate with the form of a curve on account of choise of 
, the quantity of which could be taken various for different straight lines
. Besides that, on the very straight line the points could be chosen standing not equally on the 
. All this is true for the surface. Besides that the equation of the surface has a simpler form in comparison with, for example, the equation of the surface, which is get with the method of bivariate interpolation. On the picture 2 is given an example of the surface built with the help of worked out method. Programming of the worked out algorithms was done on the language Delphi.
Picture 2
, 
,
, 
,
where 
, à 
- rectangles (picture 3)
,
,
at which connection 
.
Picture 3.
Let’s build a surface 
determined on the rectangle
where
,
so that 
. The given problem will solve in two stages. At the beginning will build a continuation of the surface 
of the surface 
. Let’s determing in the rectangle 
in the following way: if
, than let’s take =, and if 
, than
where the constants
are determined from the system of linea equations:
(3)
The determiner of the system (3) is different from zero (because this is Wandermond’s determiner), that’s why the system has a solution. Et’s prove that 
.
.
This means that
. Let 
any integer vector and 
, than for 
we have
.
Consequently, for 
,for the system (3),will get
.
This means that 
.
Analogically we built a continuation 
of the surface. The surface
will determine by a equality:
,
where the functions 
satisfy correlations:
.
The obtained surface satisfies the given demands 
.
The received computing models were realized. The examples of using of the given program are shown below.
Computer Graphics & Geometry