In modern systems of geometric
modeling [1-3] and scientific visualization [4-8], the definition of surfaces
with complex shapes plays an important role. Traditionally, both domestic and
foreign geometric modeling systems distinguish 4 main form-building operations:
extrusion, rotation, kinematic operation and sectional operation (lofting). The
first 3 operations are usually classified as kinematic with a different
trajectory of the guide line, which can be rectilinear, circular or arbitrary.
A distinctive feature of the section operation is that the guide lines -
sections that determine the trajectory of the generative line - can be
different both in shape and position in space. The operation of shaping by
sections is widely used in modeling structures in aircraft and mechanical
engineering, and the lofting method often appears in domestic and foreign
scientific works [9,10]. However, all of the above operations are kinematic and
can be generalized by the moving simplex method [11, 12], the essence of which
is the use of local simplexes to define geometric objects. Local simplices,
moving in three-dimensional space, fill it with affine-equal lines, forming a
continuous surface. The method has a generalization to multidimensional space
and was effectively used for geometric modeling of multifactor processes and
phenomena in the form of hypersurfaces of a multidimensional affine space [13].
At the same time, special interpolation curves were developed based on
Bernstein polynomials, providing the shaping of surfaces and hypersurfaces of
multidimensional space using the moving simplex method in point calculus. As a
result of further research, these curves were improved [14] for more effective
adaptation to the original data and the elimination of unplanned oscillations. In
this case, the values of the current parameter and the distances between the
interpolation nodes are actually matched, which provides adaptation of the
interpolation curve to the original data on an irregular network of points. This
approach in some cases can be an effective alternative to piecewise spline
interpolation, which has found wide application in computer-aided design [15, 16],
computer graphics [17] and visualization systems [18, 19].
The definition of interpolation curves in point
calculus was initially implemented using the example of Bézier curves of
the
-order,
which are analytically
determined by Bernstein polynomials. The idea was to replace the control points
of Bezier curves with interpolation nodal points by composing and solving a
system of linear algebraic equations using the Cramer method. In this case, a
uniform distribution of the parameter was used, providing the necessary
replacement of points. In the case of a uniform distribution of interpolation
nodes on a regular network of points, this approach gave a significant increase
in productivity, since it made it possible to reduce most of the resulting
parametric equations to linear ones. In addition, the linear relationship
between coordinates (process factors) and parameters made it possible to easily
implement the replacement of variables.
At
the
same time,
the
starting
points
(interpolation
nodes)
are
not
always
located
equally.
This
leads
to
the
need
to
use
an
uneven
distribution
of the
parameter
to
determine
adaptive
interpolation
curves
[14].
Let's consider a method for determining interpolation
curves based on Bezier curves. Based on this, the point equation of the Bezier curve
arc of the
-order
will have
the following form:
|
|
(1)
|
where
– the current point of the Bezier curve, which with its movement
fills the space, forming a Bezier curve line;
– Bezier
curve control points;
– current
parameter, which varies from 0 to 1;
.
To replace control points with interpolation nodes, it
is necessary to determine the value of the
-parameter corresponding to each of the interpolation nodes. Let us
accept these values based on the uniform distribution of the current parameter
values
.
As a result, we
get:
In
accordance
with
the
methodology
described
in
[10],
points
are
replaced
by
points
in
the
point
equation
(1)
by
composing
and
solving
a
system
of
linear
algebraic
equations.
In
this case, the condition is accepted that the curve passes through a point
at a certain value of the
parameter
.
Thus, the
number of parameter
values
corresponds to the number of points
.
In [10], the distribution of the parameter was assumed to be
uniform from 0 to 1, and in [11] it was assumed to be uneven and dependent on
the coordinates of the initial points.
The use of Bezier curves is conditioned by the
peculiarities of determining the coefficients on the basis of Newton's binomial
for the current parameter
and its
addition to 1, based on which the condition is true
.
The
fulfillment of this condition ensures that the Bezier curve belongs to
-dimensional
space, which
allows us to generalize the interpolation curves to multidimensional space due
to the invariant properties of the point equations with respect to parallel
projection.
And
due
to the
invariant
properties
of
point
equations
with
respect
to
parallel
projection,
it
allowed
us to
determine
interpolation
curves
in
a
space
of
any
dimension.
At
the same time, further research has shown that in this way it is possible to
model interpolation curves, using as a prototype any continuous curves
parameterized in point calculus.
As a result, a special program was developed in the
Maple [20] computer algebra system to determine point equations of
interpolation curves. The program is implemented in the Maple internal
programming language. The
following
listing
of the program is shown using the example of determining the point equations of
an interpolation curve based on a Bezier curve of the
-order
in accordance with
the mathematical apparatus given above.
restart;
n:=5;
e:= 0;
for
i
from
0
to
n
do
od:
S:={}:
SA:={}:
SM:={}:
for
i
from
0
to
n
do
od:
R:= solve
(S, SA);
assign
(R);
collect
(e, SM);
In this example, the input data is exclusively the
curve order
,
which defines
the initial Bezier curve equation. In the first cycle, all necessary equations
of the system
eq
[i]
are formed on the basis of the initial Bezier
curve equation. The second cycle performs substitution of parameter values on
the interval from 0 to 1, as a result of which the system of linear algebraic
equations is compiled and solved using the
solve
operator. The
collect
operator for convenience of further use performs sorting of the obtained
equation by the interpolation curve represented in point form, where points are
coordinate vectors.
By Similarly, by specifying the equations of other
initial continuous curves
eq
[i] parameterized in pointwise
calculus, one can obtain the pointwise equations of interpolation curves based
on them.
As one example of the use of interpolation curves with
predetermined geometric properties, consider the process of modeling the
surface of the onion dome. According to the geometrical conditions of the
problem (Fig. 1), it is necessary that the forming line of the surface passes
through the points
,
,
and has a tangent
at the point
.
The point
is fixed and the points
and
are current, which form
closed lines. Thus, the image is defined by two flat closed sections and the
point at which the curve image touches the axis of the rotation surface. To
prevent self-intersections of the dome surface, it is necessary to fulfill the
condition of the tangency of the curve.
Proceeding from the
fact that it is necessary to provide 4 geometric conditions (incidence of 3
points and 1 tangency), as an initial one we use the arc of the Bezier curve of
the 3rd order, the flexibility of which is enough to provide the inflection of
the forming line of the modeled surface [21]:
|
|
(2)
|
Fig. 1.
Geometric scheme for modeling the surface of an onion dome
For
each
of
the
three
points
of the
forming
line
,
and
we
take the following values of the current parameter
equal
to
,
and
.
In
this
case,
it is
assumed
that
the
section
containing
the
line
is
in the
middle.
If
it
is
located
higher
or
lower,
it is
necessary
to
enter
the
corresponding
value
instead
of the
parameter
value.
After
substituting
the
parameter
values
into
equation
(2) and solutions to SLAEs,
we
obtain:
|
|
(3)
|
The
points
in
equation
(3)
are
understood
to be
coordinate
vectors
that
provide
hidden
parallelism
as
a
result
of the
construction
and
visualization
of the
surface.
When
using
parameterization,
the
corresponding
coordinate
vectors
can
be
changed
with
additional
parameters,
taking
into
account
geometric
conditions.
In
the
coordinate
form,
equation
(3)
will
take
the
following
form:
To
determine
the
lines
and
,
we
will
use
the
point
equation
of an
ellipse
with
two
conjugate
axes
[20]
This is
one of
the
simplest
parameterizations
of a
circle
in
the
case
when
the
conjugate
axes
and
are
equal.
where
–
the
current
angular
parameter
(Fig.
3).
In result is
we
obtain
a
computational
algorithm
for
modeling
the
surface
of the
bulbous
dome.
It
provides
for the
definition
equations of
moving points
and
,
followed
by the
construction
of the
forming
surface
of the
bulbous
dome.
This
surface
is
described
by a
continuous
curve
of order 3, incident to 3 points
and
having
a
tangent
at
the
beginning
of the
arc.
To
visualize
the
resulting
bulbous
dome
model,
we will
use
the
Maple
computer
algebra
system
(Fig.
2a).
Also,
as
an
example,
we
will
model
a
bulbous
dome,
in
which
the
line
will
not
be a
circle,
but
a
sine
wave
with
an
axis
in
the
form
of a
circle
(Fig.
2b).
Its
algorithm
for
constructing
which
and
the
geometric
scheme
are
described
in
[11].
At
the same time, the
equation
of the
generator
remains
unchanged.
|
|
|
|
a)
|
b)
|
Fig.
2.
Visualization
of the
bulbous
dome
surface:
a)
with
a
circle
in
the
plane
;
b)
with
a
sine
wave
in
the
plane
The
proposed
method
can
also
be used
to
simulate
the
composite
surfaces
of
the
dome
(Fig.
4).
In
this
case,
each
of
the
lobes
of the
surface
is
determined
by the
kinematic
operation
of the
movement
of the
forming
curve
of the
3rd
order,
which
passes
through
3
points
and
has
a
vertical
tangent
along
two
circles
that
are
located
in
planes
and
.
The
desired
number
of
circles
must
be
set
in
advance,
and
their
radius
is
calculated
depending
on
the
lengths
of the
arcs
of the
guiding
circles
and
in
such
a
way
that
the
surface
of the
bulbous
dome
is
closed
An
example
of
an
element
of
such
a
composite
surface
is
shown
in
Fig.
3.
The
algorithm
for
constructing
this
element
does
not
fundamentally
differ
from
the
geometric
scheme
shown
in
Fig.
1
and
its
analytical
description
given
above.
Only
the
coordinates
of the
points
differ
,
and
.
They
are
no
longer
incident
to a
single
vertical
line.
Fig.
3.
The
petal
element
of the
composite
surface
of the
bulbous
dome
Next,
to
obtain
a
composite
surface,
it is
necessary
to
fill
the
circular
array
with the
resulting
petal
elements (Fig. 3).
To
do
this,
an
external
cycle
is
formed.
The
results
of
this
operation
are
shown
in
Figures
4a
and
4b.
It
should
be
noted
that
in
all
cases,
a
continuous
interpolation
curve
of the
3rd
order
was
used
as a generative one.
It
has
a
tangent,
As the minimum possible algebraic curve to provide the necessary shape of the
surface with a minimum of geometric conditions and the simplest mathematical
apparatus, which is based on the use of coordinate vectors.
|
|
|
|
a)
|
b)
|
Fig. 4.
Visualization of modifications to the shape of the bulbous dome surface: a)
with a vertical petal; b) with a twisted petal.
As
another
example
of
modeling
a
surface
using
interpolation
curves,
consider
a
bowl
surface
model.
To
compare
the
results,
we
will build a
model
of
the form of a
bowl
surface
in
2
ways:
using
continuous
interpolation
curves
and
using
splines.
In
our
example,
all
horizontal
sections
of the
surface
consist
of
circles,
except
for the
plane
,
which
is
defined
by a
sinusoid
with
a
circular
axis
(Fig.5).
Fig. 5. Geometric
scheme of the bowl-shaped surface
Equations
of
points
differ
only
in the
indices
and have the following form:
|
|
(4)
|
The
equation
of the
point
–
the
sinusoid
with
a
circular
axis
described
in
the
work
[11]:
|
|
(5)
|
where
-
the
number
of
"waves"
of the
sine
wave,
and
angle
in
this
example
is
equal
to
.
After
determining
the
equations
of the
guide
lines,
in order
to
build
a
model
on
top
of the
news,
it is
necessary
to
substitute
equations (4)
and
(5)
into
the
equation
of the
generating
interpolation
curve, passing through 5 interpolation nodes
:
where
–
a
parameter
that
varies
from
0
to
1.
It should be noted that this
equation of the continuous interpolation curve of the 4th order passing through
5 interpolation nodes is obtained using a special program implemented in the
Maple computer algebra system, the listing of which is given above.
The
result
of
modeling
the
desired
surface
is
shown
in
Fig.
6a.
And
Fig.
6b
shows
the
result
of
modeling
the
surface
of the
bowl
based
on
the
same
geometric
scheme
(Fig.
5),
in
which
a
spline
of the
2nd
order
of
smoothness
was
used
as
a
generatrix
instead
of an
interpolation
curve,
which
forms
a
composite
spline
surface.
As
can be
seen
from
Fig.
6,
the
model based on interpolation curves inherits
traces
of the
"waves"
of the
sine
wave
to
the
base
of the
surface.
And
the
spline
surface
in
all
4
sections
has
retained
the
shape
of a
circle,
and
traces
of
"waves"
are
preserved
only
in
the
area
between
the
planes
and
.
|
|
|
|
a)
|
b)
|
Fig.
6.
Visualization
of
modifications
to the
shape
of the
vase
surface
:
a)
with
a
continuous
interpolation
curve
forming;
b)
with
a
spline
forming
Thus,
the
results
of
modeling
the
two
variants
of the
vase
surface
are
quite
different
from
each
other.
The
choice
between
using
interpolation
curves
and
splines
can
be
made
based
on
the
geometric
properties
that
the
simulated
surface
should
have.
But
it is
also
possible
to
combine
it
with
spline
surfaces,
given
the
possibility
of
controlling
the
geometric
properties
of
continuous
interpolation
surfaces.
In
addition,
continuous
interpolation
arcs
of the
outline
can
be
used
to
reduce
the
"piecemealness"
of
composite
curved
lines.
As
a
result
of the
conducted
research,
the
following
conclusions
can
be
drawn:
1.
The
process
of
surface
modeling
presented
in
the
paper
by the
method
of the
movable
simplex
of
point
calculus
,
taking
into
account
the
possibility
of
generalization
to
a
multidimensional
space,
is
an
expanded
analogue
of the cross-
section
operation
(lofting
operation),
which
is
widely
used
in
automated
design
and
visualization
systems.
2.
Both
continuous
and
composite
interpolation
curves
can
be
used
as
forming
lines
of
such
surfaces,
as
shown
in
the
examples.
The
choice
of the
interpolation
method
depends
on
the
conditions
of a
specific
practical
task.
At
the
same time,
in
our
opinion,
continuous
interpolation
curves,
as
well
as
piecewise
ones,
have
the
versatility
necessary
for
computer-aided
design
systems
and
could
significantly
complement
their
tools
along
with
splines.
3.
The
program
for
determining
the
point
equations
of
continuous
interpolation
curves
using
coordinate
vectors
is
shown
in
the
example
of
using
Bezier
curves
as
a
prototype.
But
after
minor
modifications,
as
shown
in
the
examples,
these
curves
can
be
adapted
both
to
the
initial
data
and
to
the
necessary
geometric
conditions.
This
significantly
expands
the
possibilities
of
their
use
in
CAD.
4.
The
proposed
approach
to
modeling
surfaces
of
complex
shape
provides
for
generalization
towards
parameterization
of
geometric
bodies
as
three-dimensional
objects
belonging
to three-
dimensional
space
[22].
5.
The
implementation
of
any
complex
shaping
tools
in
CAD
requires
a
fairly
large
amount
of
computing
resources.
The
features
and
benefits
of
point
calculus,
taking
into
account
hidden
parallelism,
can
provide
significant
performance
gains
for
projects
containing
large
volumes
of
geometric
elements.
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