One-Parametric Deformations
of Solids

**1. Introduction**

Solid cell *T* is a non-empty subset of the projectively extended Euclidean space over the set of real
numbers _{¥}E^{3}
that is the image of a simply connected region W = [0, 1]^{ 3 }Ì R^{3} under a local
homeomorphism, mapping analytically represented by the point function

**s**(*u ^{i}*) = (

In the solid cell regular point *P *= **s**(*a, b*, *c*), (*a*,
*b*, *c*) Î W, the tangent
trihedron is uniquely determined and describing solid intrinsic geometric
properties, see in [1],
[2]. Vertex of the
tangent trihedron is in the regular point*
P*, the common point of the tangent trihedron edges in semi-lines determined
by the regular point *P* and three direction vectors

**s*** _{i}*(

which are linearly independent tangent vectors to
the solid cell iso-parametric curve segments located
in the point *P*. Faces of the tangent
trihedron are parts of three tangent planes to the solid iso-parametric
surface patches sharing the common regular point *P*.^{ }Twist vectors

**s*** _{ij}*(

to these solid cell iso-parametric surface patches are located in the tangent
trihedron.

Interior distribution of the solid cell is indicated by the solid cell
density vector

**s**_{123}(*u*^{1}, *u*^{2},* u*^{3}) _{}.

The first differential form of a
solid cell in the regular point P=**s**(*a,b*,*c*), (*a*,*b*,*c*)ÎW, can be determined (see in [3])

(*d***s**)^{2}^{ }= (**s**_{i}*du ^{i}*)

and the discriminant of the solid cell
first differential form can be represented as

^{ }

*D *= *EGG** + 2*FF***F*** - *GF**^{2 }- *F*^{2}*G** - *EF***^{2 }= *A
*+ 2*C *- *B*,

*A *= *EGG**, *C = FF***F***, *B* = *GF**^{2} + *F*^{2}*G** + *EF***^{2}.

The mixed triple scalar product

*s** *=* *[**s**_{1}** s**_{ 2}** s**_{ 3}] =* *(**s**_{1}´**s**_{2})**.s**_{3 }=* *[**s**_{2
}**s**_{3 }**s**_{1}] = (**s**_{2}´**s**_{3})**.s**_{1}=_{ }[**s**_{3 }**s**_{1 }**s**_{2}] = (**s**_{3}´**s**_{1})**.s**_{2}

of the tangent vectors **s**_{1, }**s**_{2, }**s**_{3}_{ }to the solid cell iso-parametric curve segments satisfies the equation (as
proved in [4])

*s*^{2}^{ }= *D.*

Approximation Bèzier tri-cubic solid cell is determined by the
ordered basic grid of 64 real points in the space. Map of the basic figure is
the matrix M_{4x4x4 }with elements in the homogeneous co-ordinates of
points in the basic grid

M_{4x4x4}
= (* ^{ijk}P*),

Tri-cubic
approximation is determined by Bernstein polynomials for a = 1, 2, 3

*Be*_{0}(*u*_{a})=(1 -* u*_{a})^{3 }* Be*_{1}(*u*_{a})*=*3*u*_{a}(1 -* u*_{a})^{2 }* Be*_{2}(*u*_{a})*=*3*u*_{a}^{2}(1 *- u*_{a})* Be*_{3}(*u*_{a})*=u*_{a}^{3}

and interpolation matrices are in the
form

^{a}I(*u*_{a}) = (*Be*_{0}(*u*_{a}), *Be*_{1}(*u*_{a}), *Be*_{2}(*u*_{a}), *Be*_{3}(*u*_{a})), a=1, 2, 3.

Modelled Bèzier cell has vertices in the vertices of the
basic grid, while no other point of the basic figure is located in the cell.
Point function of the cell is in the form

**s**(*u*^{1}, *u*^{2},
*u*^{3}) =^{ 3}I(*u*^{3})** .** (^{1}I(*u*^{1})** .** M_{4x4x4} **.** ^{2}I(*u*^{2})^{T})=^{ ijk}PBe_{i}(u^{1}*)Be _{j}(u*

Density vector **s**_{123}(*a*, *b*,
*c*) in the cell arbitrary point *P*(*a*,
*b*, *c*), for *a*, *b*, *c*
Î [0, 1] is determined by points of
the basic grid and can be calculated from the density function

**s**_{123}(*u*^{1},
*u*^{2}, *u*^{3}) =^{ 3}I**´**(*u*^{3})**.**(^{1}I**´**(*u*^{1})**.**M_{4x4x4}**.**^{2}I**´**(*u*^{2})^{
T}) =^{ ijk}PBe_{i}**´***(u*^{1}*)Be _{j}*

^{a}I´(*u*_{a})=(*Be*_{0}**´**(*u*_{a}), *Be*_{1}**´**(*u*_{a}), *Be*_{2}**´**(*u*_{a}), *Be*_{3}**´**(*u*_{a})), a=1, 2, 3

*Be*_{0}**´**(*u*_{a}) = -3(1 -* u*_{a})^{2}, *Be*_{1}**´**(*u*_{a}) = 3(1-*u*_{a })(1 -* *3*u*_{a}),^{ }*Be*_{2}**´**(*u*_{a}) *= *3*u*_{a }(2 -* *3*u*_{a}),^{ }*Be*_{3}**´**(*u*_{a}) = 6*u*_{a}^{2}

which is the polynomial function of
degree two in three variables, where ^{ijk}P* *are
points of the basic grid, elements of the map M_{4x4x4}.

Density vectors
in the vertices of the Bèzier cell (with
parametric coordinates equal to 0 or 1) do not change their position while
considering interior deformations, and therefore only 8 interior points (one
for each vertex) of the basic grid determine entire interior solid cell
geometry and density. These points form interior density core, determined as
interior density hypercube, which influences all interior density. Density core
can be deformed arbitrarily (as discussed in [5]) to create a desirable distribution
in the net of iso-parametric curve segments and
surface patches, and to determine the interior density of the modelled solid
cell and its in-homogeneity in this way. Minor deformations of the homogeneous
distribution of iso-parametric curve segments in the
cell can be reached only, while moving all vertices of the interior density
hypercube within the convex envelope of the Bèzier
solid cell basic grid of 64 real points in the matrix M_{4x4x4} .
Extreme changes can be adopted by moving these points outside the Bèzier solid cell basic grid envelope. A special
class of deformations is discussed, called one-parametric deformations, based
on the translation of control points in the direction of one from parameters *u ^{i}^{
}*Î [0, 1],
for

Let the solid cell
point function can be rewritten in the way

** s**(

The three
dimensional matrix __M___{4x4x4}^{ }contains control points
of the solid basic grid, including 8 interior points forming the interior density
hypercube, but these points are attached a weight, so cold density parameter
influencing the solid interior density. This means, points^{ ijkP},* i*,* j*,*
k *= 2, 3 are substituted by points* ^{
ijkP}*,
=

We
can write therefore

**s**(*u*^{1}, *u*^{2}, *u*^{3})
= _{} , (*u*^{1},
*u*^{2}, *u*^{3}) Î W,

where the function *h*(*u*^{1}, *u*^{2}, *u*^{3}) can be expressed as

_{}, (*u*^{1}, *u*^{2}, *u*^{3}) Î W.

Inserting
extremal values of the weights to the vertices of the solid interior density
hypercube, a special interior density can be reached.

In the figure 1 below, there are illustrations
of the inhomogeneous interior distribution of the iso-parametric
net of surfaces in the solid cell, which is an in-homogeneous cube, a rational^{
}Bèzier solid cell. There had been chosen
a special position of the vertices of the interior density hypercube. Its four
"bottom" vertices, 4 control points in the second level of the solid
basic grid, were located into one point, while in the same way, the remaining
four "top" vertices of the hypercube, it
means 4 control points in the third level of the solid basic grid were also
located in one point only. Also the corresponding weights were chosen equal,
only their different and changing signs had been tested with respect to the
possible combinations and their influence on the interior density distribution
of the iso-parametric net of surfaces.

If
* ^{ijk}h*.> 0 for

Fig. 1

In the case of
* ^{ijk}h*.>
0 for

More details on
the possible modelling of the solid interior density with respect to the given
form of the iso-parametric surfaces and their common
position will be obtained in the future further study of the relevant problems
with relations to the solid first differential form and its discriminant.