Free-Form Deformations of
Solid Cells

**1. Introduction**

Solid cell *T* is a non-empty subset of the projectively extended Euclidean space over the real numbers
_{¥}E^{3} that is the image of a
simply connected region W = <0, 1>* ^{ }*Ì R

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

Separate
components *x*(*u ^{i}*),

Point *P *= **s**(*a, b*,
*c*), for (*a*, *b*, *c*) Î W is the regular point of the solid cell, when
all three partial derivatives of the solid cell point function with respect to
the particular variables

_{}

determine linearly independent vectors **s*** _{i}*(

In the solid
cell regular point *P *= **s**(*a, b*, *c*),
(*a*, *b*, *c*) Î W, the tangent trihedron is uniquely
determined and it describes solid intrinsic geometric properties. Vertex of the
tangent trihedron is the regular point* P*.
Edges are semi-lines determined by the point *P* and three direction vectors **s*** _{i}*(

t^{12 }= *P***s**_{1}**s**_{2 }, t^{13 }= *P***s**_{1}**s**_{3 },. t^{23 }= *P***s**_{2}**s**_{3 }.

Second mixed
partial derivatives of the solid cell point function denoted

_{}

determine twist vectors
**s*** _{ij}*(

The third mixed triple partial derivative of the solid cell point
function denoted

_{}

determines the solid cell density vector **s**_{123}(*a*, *b*, *c*) in the solid regular point *P*.

Fig. 1

**2. Bèzier
solid cells**

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) are determined with respect to the form
of the Bernstein polynomials derivatives by 8 interior points of the cell basic
grid: vertex, its 3 neighbouring points in the directions of parameters *u*^{1}, *u*^{2}, *u*^{3},
common neighbouring points of these three points in the basic grid, and their
common neighbour in the grid. Configuration is illustrated for the vertex ^{000}*P* in Fig. 2.

Considering
interior deformations, in which boundary of the cell remains invariant (it
means shape of the curve-like edges and surface like facets is preserved),
position of points in the grid influencing form of the boundary must be kept
unchanged. 8 interior points (one for each vertex) of the basic grid determine
interior solid cell geometry and density and they form interior density core,
determined as interior density hypercube illustrated in Fig. 3.

Fig. 2
Fig. 3

Density core can
be deformed arbitrarily (in Fig. 4 vertices of the Bèzier
solid cell and vertices of the interior density hypercube are mapped) to create
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-homogenity in this way.

Density vectors
can be calculated in an arbitrary solid cell regular point for different
position of interior density hypercube points, and these can be compared to
estimate the results of the deformation. Several illustrations of the
in-homogeneous distribution of the degenerated solid cell interior are given in
the following, where one face collapses to one edge, while 4 quadruples of the
solid cell basic grid points coincide as illustrated in Fig. 5,

^{i}^{00}*P = ^{ }*

Fig.
4
Fig.5

In Fig. 6, all
vertices of the interior density hypercube are moved within the convex envelope
of the Bèzier solid cell basic grid of 64 real
points in the matrix M_{4x4x4} . Minor
deformations of the homogeneous distribution of iso-parametric
curve segments in the cell can be reached only, in this way. Illustration in
the upper left corner of the Fig. 6 shows the homogeneous position of the iso-parametric net of curve segments in the cell.
Deformation in the direction of the parameter *u*^{1} is performed by moving the vertices of the interior
density hypercube towards the vertices of the the Bèzier solid cell basic grid (model in upper right
corner of the Fig. 6). The first curvature of *u*^{3}-parametric curve segments determined by point
functions* ***s**(*a*, *b*,* u*^{3}), *u*^{3} Î <0, 1> can be influenced by the singular
position of the interior density hypercube vertices in the vertices of the Bèzier solid cell basic grid, as seen on models in
the bottom left and right corners of the Fig. 6.

In Fig. 7,
vertices of the interior density hypercube are moved outside the convex
envelope of the Bèzier solid cell basic grid
of 64 real points in the matrix M_{4x4x4}. Their position is extremely
distant from the vertices of the Bèzier solid
cell basic grid, but the situation in the upper left corner of the Fig. 7 seems
to be very homogeneous, either. This is the result of the symmetric position of
the transformed interior density hypercube vertices with respect to the
vertices of the deformed Bèzier solid cell.
Model in the bottom left corner of the Fig. 7 was created by moving the
interior density hypercube vertices extremely far in the direction of the
parameter *u*^{1}, while in the
model in the bottom right corner of the Fig 7 they are located in the similar
position but in the direction of the parameter *u*^{2}.

Extreme in-homogeneity of the
distribution of iso-parametric curve segments in the
cell can be modelled, when the interior density hypercube vertices of the same
face are moved in the opposite directions and to the different distances.
Interior holes, bubbles and new edges can be created by this deformation,
influencing meaningfully the first curvature of the solid cell iso-parametric curve segments with respect to the expecting
results of the performed deformation, as shown in Fig. 8. Model in the bottom
left corner looks homogeneously either, because of the symmetric position of
the interior density hypercube vertices moved outside the Bèzier
solid cell basic grid, thought to the extreme distance and in the opposite
directions of the separate parameters.

Fig. 6

Fig. 7

Fig. 8