- 1. Introduction. BSD in engineering applications. Tensorial
Representation. Comparison BSD and T-BSD. Conclusions. A Tensor
Calculus Approach for Bzier Shape e Deformation L. Hilario, N.
Monts, M.C.Mora, A. Falceo June 18-22, 2012 Valencia SIAM
Conference on Applied Linear AlgebraL. Hilario, N. Monts, M.C.Mora,
A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo
e1/40
2. Introduction. BSD in engineering applications. Tensorial
Representation. Comparison BSD and T-BSD. Conclusions. 1
Introduction. 2 Bzier Shape Deformation (BSD) in engineering
applications.e 3 Tensorial Representation of the BSD Algorithm
(T-BSD). 4 Comparison BSD and T-BSD. 5 Conclusions.L. Hilario, N.
Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape
Deformation eo e2/40 3. Introduction. BSD in engineering
applications. Tensorial Representation. Comparison BSD and T-BSD.
Conclusions. Tensorial Structure of the BSD The objective of this
work: The objective of this work is the reformulation of an
algorithm using Tensorial Notation T-BSD This technique is called
Tensor-Bzier Shape Deformation (T-BSD) .e Computational Cost One of
the most important facts in engineering applications is the cost in
computational time because some algorithms are applied in
real-time.L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus
Approach for Bzier Shape Deformation eo e3/40 4. Introduction. BSD
in engineering applications. Tensorial Representation. Comparison
BSD and T-BSD. Conclusions. Tensorial Structure of the BSD
Computational cost There are an increased in numerical methods that
make use of tensors. It is useful to reduce the numerical cost.
(See Falc A. 2010, see o Hackbusch W., see Kolda T.G. et al
2009...) BSD The BSD computes the deformation of a Bzier curve
through a eld ofe vectors. Applications There are two applications
of the BSD: Mobile Robots and Liquid Composite Moulding.L. Hilario,
N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier
Shape Deformation eo e4/40 5. Introduction. BSD in engineering
applications. Tensorial Representation. Comparison BSD and T-BSD.
Conclusions. Parametric curves The parametric curves (Bzier,
B-Splines, NURBS, RBC) are the e most widely used in computer
graphics and geometric modelling since points on the curve are
easily computed. The representation of this kind of parametric
curves is a SMOOTH CURVE. Our algorithm BSD is developed with Bzier
curves. They are a polynomial curves and e they possess a number of
mathematical properties which facilitate their manipulation and
analysis.L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus
Approach for Bzier Shape Deformation eo e5/40 6. Introduction. BSD
in engineering applications. Tensorial Representation. Comparison
BSD and T-BSD. Conclusions. Problems in Mobile Robots Trajectory
generation problem This problem consists in computing a feasible
trajectory between a start and a goal state-time, for a given
robotic system. The trajectories should be a continuous and a
smooth curve. It is necessary to avoid slipping of the wheels.
Collision Avoidance problem The smooth and continuous trajectory
should be free of collisions. CPU time The algorithms are applied
in real-time, for that reason the cost in computational time of the
algorithms must be the lowest and the best. Realistic cluttered
scenarios A realistic scenario is considered to be unknown, dynamic
and sometimes cluttered with mobile obstacles. For that reason, the
reduction of the execution time is necessary in limit situations.L.
Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for
Bzier Shape Deformation eo e6/40 7. Introduction. BSD in
engineering applications. Tensorial Representation. Comparison BSD
and T-BSD. Conclusions. Trajectory Generation Problem The
smoothness of the parametric curve is a useful property for the
Trajectory Generation Problem in Mobile Robots. The parametric
curves represent in an appropiate manner the Trajectory of the
Robot. A lot of researchers consider parametric curves in the
construction of trajectories for wheeled robots, (see for example,
Choi et. al, 2008-2009, Skrjanc and Klancar, 2007), etc.L. Hilario,
N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier
Shape Deformation eo e7/40 8. Introduction. BSD in engineering
applications. Tensorial Representation. Comparison BSD and T-BSD.
Conclusions. Collision Avoidance Problem Collision avoidance is a
fundamental problem in many areas such as robotics. An extreme
situation of collision avoidance.......L. Hilario, N. Monts,
M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape
Deformation eo e8/40 9. Introduction. BSD in engineering
applications. Tensorial Representation. Comparison BSD and T-BSD.
Conclusions. Collision Avoidance Problem The generation of the path
can be properly done using reactive path planning methods adapting
to environmental changes. One of the most popular reactive methods
is Articial Potential Fields(APF) (see Khatib, 1986), that is the
basis of the Potential Field Projection method (PFP) (see Mora and
Tornero, 2007) used in this work.APF consists in lling the robots
workspacewith an articial potential eld in which therobot is
attracted by the goal and repelledby the obstacles.APF produces a
eld of vectors that guidesthe robot to non-collision positions.L.
Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for
Bzier Shape Deformation eo e9/40 10. Introduction. BSD in
engineering applications. Tensorial Representation. Comparison BSD
and T-BSD. Conclusions. Trajectory G.+ Collision A. Design and
Modify a Parametric Curve is an important research issue, (Wu et
al.2005, Xu et al. 2002) One of these techniques (Wu et al., 2005)
has been adapted for its use in path planning for Holonomic Robots.
BSD modies the parametric curve through a eld of vectors.The shape
of the Bzier curve is modied.eThe changes of the shape are
minimized from the original one. These vectors are computed with
PFP. The Repulsive Forces will modify the Original Trajectory to
avoid every obstacle. The First Technique joining: Trajectory
Generation using Parametric Curves Avoiding the Obstacles using
Potential Field methodsL. Hilario, N. Monts, M.C.Mora, A. Falc A
Tensor Calculus Approach for Bzier Shape Deformation eo e10/40 11.
Introduction. BSD in engineering applications. Tensorial
Representation. Comparison BSD and T-BSD. Conclusions. Liquid
Composite Moulding (LCM) The geometric line between the dry and the
wet area of the preform is dened as FLOW FRONT. The ow front
advance computation is used in Liquid Composite Moulding (LCM)
simulation because is a common tool to compute the control actions
in advanced composite manufacturing during lling to take decision
on-line.L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus
Approach for Bzier Shape Deformation eo e11/40 12. Introduction.
BSD in engineering applications. Tensorial Representation.
Comparison BSD and T-BSD. Conclusions. Liquid Composite Moulding
Finite Element Method (FEM) techniques are used to compute the ow
fronts representation. The result is a discrete set of points
(nodes). However, the resins ow front is a continuous smooth curve.
A continuous ow front is proposed using parametric curves, in this
case Bzier curve. eL. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor
Calculus Approach for Bzier Shape Deformation eo e12/40 13.
Introduction. BSD in engineering applications. Tensorial
Representation. Comparison BSD and T-BSD. Conclusions. Liquid
Composite Moulding The ow front is updated with the BSD algorithm
because the ow front is modied through a eld of vectors, in this
application, velocity vectors. These velocity vectors are obtained
throughout the Darcys Law applying Finite Element Methods
Simulation. Darcys Lawk v=P(1)L. Hilario, N. Monts, M.C.Mora, A.
Falc A Tensor Calculus Approach for Bzier Shape Deformation eo
e13/40 14. Introduction. BSD in engineering applications. Tensorial
Representation. Comparison BSD and T-BSD. Conclusions. Denitions
Denition A Bzier Curve is dened as,en(t) =Pi Bi,n (t)(2)i=0 n is
the Order of the Bzier curve.e n Bi,n (t) =i t i (1 t)ni Bernstein
Basis t [0, 1] is the Intrinsic Parameter. (n + 1) Control Points,
Pi such that i = 0, 1, , n.L. Hilario, N. Monts, M.C.Mora, A. Falc
A Tensor Calculus Approach for Bzier Shape Deformation eo e14/40
15. Introduction. BSD in engineering applications. Tensorial
Representation. Comparison BSD and T-BSD. Conclusions. Denitions
Denition A Modied Bzier curve is dened as,en S ((t)) := (Pi + i )
Bi,n (t); t [0, 1](3)i=0 To deform a given Bzier curve describing a
Trajectory or a Flowe Front, the control points must be changed and
the perturbation, i , of every control point must be computed.L.
Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for
Bzier Shape Deformation eo e15/40 16. Introduction. BSD in
engineering applications. Tensorial Representation. Comparison BSD
and T-BSD. Conclusions. Constrained optimization problem. This
problem is solved dening a constrained optimization problem. It is
solved with the Lagrange Multipliers Theorem. The optimization
function minimizes the distance between the orginal Bzier curve,
(t), and the modied Bzier curve, S ((t)).ee Thus, this function
minimizes the changes of the shape.(Wu et al.2005) Denition The
optimization function is dened as, 1 2(t) S ((t)) 2 dt (4) 0L.
Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for
Bzier Shape Deformation eo e16/40 17. Introduction. BSD in
engineering applications. Tensorial Representation. Comparison BSD
and T-BSD. Conclusions. Optimization Function Disadvantage A Bzier
curve is numerically unstable if the Bzier curve has a largee e
number of control points. It is necessary to concatenate some Bzier
curves to obtain the e complete trajectory or the complete ow
front. So the optimization function is redened. Denition The
optimization function using k-Bzier curves is dened as,e k1 2 g :=
l (t) S (l (t)) 2dt (5) l=10L. Hilario, N. Monts, M.C.Mora, A. Falc
A Tensor Calculus Approach for Bzier Shape Deformation eo e17/40
18. Introduction. BSD in engineering applications. Tensorial
Representation. Comparison BSD and T-BSD. Conclusions. The set of
Constraints First Constraint The Modied Bzier, S (i (t)), passes
through the Target Point, Ti .e Mathematical Formulation krl(l)
(l)r1 = , Tj S (l (tj ))(6) l=1 j=1L. Hilario, N. Monts, M.C.Mora,
A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo
e18/40 19. Introduction. BSD in engineering applications. Tensorial
Representation. Comparison BSD and T-BSD. Conclusions. The set of
Constraints First Constraint Mobile Robots In Mobile Robots, this
constraint means that the robot is guided to non-collision
positions. The vectors joining the Start Point and the Target Point
are the eld of forces computed through the PFP. LCM In LCM, this
constraint means that the ow front is modied during lling the
mould.The ow front evolution is updated by the BSD. In this case,
the led of vectors are the velocity vectors obtained with the
Darcys Law.L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor
Calculus Approach for Bzier Shape Deformation eo e19/40 20.
Introduction. BSD in engineering applications. Tensorial
Representation. Comparison BSD and T-BSD. Conclusions. The set of
Constraints Second Constraint Continuity and derivability is
necessary to impose on the joined points of the concatenated
curves. Mathematical Formulationk1(l) (l+1) r2 =, S (l (tf )) S
(l+1 (t0)) (7)l=1k1(l) (l+1) r3 =, S (l (tf )) S (l+1 (t0))
(8)l=1L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus
Approach for Bzier Shape Deformation eo e20/40 21. Introduction.
BSD in engineering applications. Tensorial Representation.
Comparison BSD and T-BSD. Conclusions. The set of Constraints
Second Constraint Mobile Robots In Mobile Robots, the Trajectory of
the robot must be a smooth Trajectory, for that reason it is
imposed this constraint. LCM In LCM, the actual resins ow front is
a continuous smooth curve, so it is necessary this restriction.L.
Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for
Bzier Shape Deformation eo e21/40 22. Introduction. BSD in
engineering applications. Tensorial Representation. Comparison BSD
and T-BSD. Conclusions. The set of Constraints Third Constraint
Therefore, derivative constraints on the start and end points of
the resulting concatenated curves are imposed. Mathematical
Formulation(1)(1)(k)(k)r4 = , 1 (t0 ) S (1 (t0 )) + , k (tf ) S (k
(tf )) (9)L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus
Approach for Bzier Shape Deformation eo e22/40 23. Introduction.
BSD in engineering applications. Tensorial Representation.
Comparison BSD and T-BSD. Conclusions. The set of Constraints Third
Constraint Mobile Robots In Mobile Robots, this constraint is
necessary because the continuity between the Present position and
the predicted Future position is ensured. LCM In LCM, this
constraint is useful to maintain the derivative property of the
curve.L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus
Approach for Bzier Shape Deformation eo e23/40 24. Introduction.
BSD in engineering applications. Tensorial Representation.
Comparison BSD and T-BSD. Conclusions. The Lagrange Multipliers
Lagrange Multipliers The Lagrange Multipliers Theorem has been
applied to solve the constrained optimization problem. The idea is
to minimize the function dened in 5 including the set of
constraints dened below. Lagrange FunctionL((1) , , (k) , ) = g +
r1 + r2 + r3 + r4 (10) The solution of the problem In order to
obtain the Minimum of this convex function, we only to compute the
stationary point of the Lagrangian derivative. L= 0; (l) = 1, ,
k(11)(l) L=0 (12) L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor
Calculus Approach for Bzier Shape Deformation eo e 24/40 25.
Introduction. BSD in engineering applications. Tensorial
Representation. Comparison BSD and T-BSD. Conclusions. The solution
A square linear system of equations is obtained: A X = b. It is
solvable and the solution X = (, ) computes the perturbation of
every control point. ExampleExample Mobile RobotsLCM !L. Hilario,
N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier
Shape Deformation eo e25/40 26. Introduction. BSD in engineering
applications. Tensorial Representation. Comparison BSD and T-BSD.
Conclusions. Tensorial Notation The BSD model has been reformulated
using Tensorial Structure (T-BSD) reducing the critical point in
engineering applications: The cost computational time to get a
suitable Real-Time control. We introduce some of the notation used
in this presentation. Denition The Kronecker Product of A Rn1 n1
and B Rn2 n2 , written A B, is the tensor algebraic operation dened
as a11 B a12 B a1n1 B a21 B a22 B a2n B Rn1 n2 n1 n2 .1AB = ..
..... . . .. . an1 1 B an1 2 B an1 n1 BL. Hilario, N. Monts,
M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape
Deformation eo e26/40 27. Introduction. BSD in engineering
applications. Tensorial Representation. Comparison BSD and T-BSD.
Conclusions. Tensorial Notation Let A = [A1 An ] be an m n matrix
where Aj is its j-th column vector. Then vec A is the mn 1 vector
A1 vec A = . . . .An Thus the vec operator transform a matrix into
a vector by stacking the columns of the matrix one underneath the
other.L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus
Approach for Bzier Shape Deformation eo e27/40 28. Introduction.
BSD in engineering applications. Tensorial Representation.
Comparison BSD and T-BSD. Conclusions. Mathematical Formulation
Denition The denition of the Bzier curve, (1), it is written in
equivalent matrix e form,n (u) = Pn (t) Bn (u); u [0, 1] t (13)
where Pn (t) = P0 (t) nPn (t) n R2(n+1)(14) T Bn (u) =B0,n (u) Bn,n
(u) R(n+1)1 .(15) Denition Its standard euclidean norm is dened asn
(u) t 2 2 = (Pn (t) Bn (u))T Pn (t) Bn (u)(16)L. Hilario, N. Monts,
M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape
Deformation eo e 28/40 29. Introduction. BSD in engineering
applications. Tensorial Representation. Comparison BSD and T-BSD.
Conclusions. Mathematical Formulation Denition For each xed t, the
energy of the u-parametrized curve n int L2 ([0, 1], R2 ) is given
by, 11/2 11/2 nt 2 = n (u) 2 dut 2 = (Pn (t)Bn (u))T Pn (t)Bn (u)du
. 00 (17) We consider a nite set of Target Points T0 , . . . , Tr
D, a rr connected and compact set in R2 . We move from an initial
Bzier curve, denoted by n andet characterized by the set of its
control points Pn (t), to a curve, denoted by n t+t by means a set,
of perturbations for each control point, namelyXn =X0 n Xnn R2(n+1)
. (18)L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus
Approach for Bzier Shape Deformation eo e29/40 30. Introduction.
BSD in engineering applications. Tensorial Representation.
Comparison BSD and T-BSD. Conclusions. Mathematical Formulation
Denition The resultant Bzier curve net+t is given by,n t+t (u) =
(Pn (t) + Xn ) Bn (u);u [0, 1].(19) To compute Xn :We minimize the
energy used by the curve to move from n totn .t+tMoreover, this
transformed curve passes through the targetpoints for a given set 0
= u1 < u2 < < urr 1 < urr = 1, ofrrparameter values.
Optimization Problem solved with Lagrange Multipliers Theorem min n
nt+t t22 (20) s. t. nrjt+t (uj ) = Tr for 1 j r and r n 1.L.
Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for
Bzier Shape Deformation eo e 30/40 31. Introduction. BSD in
engineering applications. Tensorial Representation. Comparison BSD
and T-BSD. Conclusions. Mathematical Formulation We write (20) in
equivalent matrix form as follows. Let Tr =T1r Tr r R2r . (21) andr
Bn = r Bn (u1 ) rBn (ur ) R(n+1)r(22) Finally, we consider the
matrix function 1n (Xn ) = Bn (u)T Xn Xn Bn (u) du,T (23) 0 then
(20) can be written in matrix form as: Matrix Form of the
Optimization Problem minXn R2(n+1) n (Xn ) (24)r s. t. (Pn (t) + Xn
)Bn = TrL. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus
Approach for Bzier Shape Deformation eo e 31/40 32. Introduction.
BSD in engineering applications. Tensorial Representation.
Comparison BSD and T-BSD. Conclusions. Mathematical Formulation If
we take the vec operator and we reformulate the constraint with the
Kronecker Product in (24) we obtain the equivalent minimization
program, Matrix Form of the Optimization Problem with the vec
operatormin(vec Xn )R2(n+1)1 n (vec Xn ) (25) rs. t. ((Bn )T I2 )
vec Xn = vec Tr vec (Pn (t)Bn ) r We note that the set of
constrains of the problem (25) is linear, in consequence the map n
is dened over a convex set. Thus, by proving the convexity of n ,
each critical point of (25) will give us an absolute minimum.L.
Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for
Bzier Shape Deformation eo e 32/40 33. Introduction. BSD in
engineering applications. Tensorial Representation. Comparison BSD
and T-BSD. Conclusions. A Concatenate T-BSD Now, we consider that
the curve t in now described by a nite set of concatenate Bzier
curves n1 , . . . , nk of degrees n1 , . . . , nk , ett
respectively. Thus, it is necessary to include the constraints
explained before, in the slide 18. Optimization Problem
Concatenating k Bzier Curves e We would to compute Xni R2(ni +1)
for 1 i k satisfyingk min(Xn1 ,...,Xnk ) (Xn1 , . . . , Xnk ) = i=1
ni (Xni ) rs. t.(Pni (t) + Xni )Bnii = Tri 1 i k ni X ni = X0i+1 ,
1 i k 1nni (Xnii Xnii 1 )nn= ni+1 (X1i+1 X0i+1 ), 1 i k 1n nn1 (X11
X01 ) n n=0 nk (Xnk Xnk 1 ) nk n k=0 (26)L. Hilario, N. Monts,
M.C.Mora, A. Falc A Tensor Calculus Approach for Bzier Shape
Deformation eo e 33/40 34. Introduction. BSD in engineering
applications. Tensorial Representation. Comparison BSD and T-BSD.
Conclusions. Now, we would to write (26) with a more compact
notation. To this end we use the following four block matrices. For
1 i k we deneRni = 0 0 I2 R22(ni +1) , (27) Rni =0 0 I2I2 R22(ni
+1) ,(28)Lni =I2 0 0 R22(ni +1)(29) andLi = n I2I2 0 0 R22(ni +1)
.(30)L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus
Approach for Bzier Shape Deformation eo e 34/40 35. Introduction.
BSD in engineering applications. Tensorial Representation.
Comparison BSD and T-BSD. Conclusions. Lagrangian Function The
Lagrangian function associated to (26) can be written as follows,k
L= i=1 ni (vec Xni )k ri T i=1 (i )((Bnii )T I2 )vec Xni vec Tri +
vec (Pni (t)Bnii )r rk1 i=1T [Rni vec Xni Lni+1 vec Xni+1 ]i T n1
L1 vec Xn1 k n T nk Rnk vec Xnk k+1k1 i=1Ti+1+k ni Rni vecXni ni+1
Lni+1 vec Xni+1 (31)L. Hilario, N. Monts, M.C.Mora, A. Falc A
Tensor Calculus Approach for Bzier Shape Deformation eo e 35/40 36.
Introduction. BSD in engineering applications. Tensorial
Representation. Comparison BSD and T-BSD. Conclusions. L Making
zero the partials, (vec Xn )T = 0; 1 i k, of thei Lagrangian
Functions is obtained a linear system equation dened as follow,
Linear System using Tensorial Structure The linear system, Az =
f(32) The A matrix is dened as follows, A Rpp(33)k k k p=2(ni + 1)
+ 2 ri + 2(k 1) + 4 + 2(k 1) = 2(ni + ri ) + 6k.i=1 i=1i=1 (34)L.
Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus Approach for
Bzier Shape Deformation eo e 36/40 37. Introduction. BSD in
engineering applications. Tensorial Representation. Comparison BSD
and T-BSD. Conclusions. Solution of the T-BSD The solution of this
system is the follow vector, vec Xn1 .. . vec Xnk r 1 1 . z=. Rp1
.(35) . rk k 1 .. . 2k The solution computes the perturbation,
vecXni , of every control point.L. Hilario, N. Monts, M.C.Mora, A.
Falc A Tensor Calculus Approach for Bzier Shape Deformation eo e
37/40 38. Introduction. BSD in engineering applications. Tensorial
Representation. Comparison BSD and T-BSD. Conclusions. Comparison A
picture is worth a thousand words If the number of the Bzier curves
is increased, BSD grows exponentially, e whereas T-BSD grows
linearly.L. Hilario, N. Monts, M.C.Mora, A. Falc A Tensor Calculus
Approach for Bzier Shape Deformation eo e38/40 39. Introduction.
BSD in engineering applications. Tensorial Representation.
Comparison BSD and T-BSD. Conclusions. Conclusions With T-BSD the
reduction of the computational cost of the BSD algorithm is
achieved.Tensorial Algebra reduces drastically the cost
computational time to apply the BSD model in Real-Time. (see Ammar
et al., 2009, Falc A., 2010, Kolda et al., 2009) o The BSD
algorithm has been devoloped to compute the deformation of a
parametric curve through a eld of vectors. This algorithm needs a
set of vectors.In Mobile Robots, the eld of forces necessary to
modify the Bzier ecurve are obtained by PFP. It is the FIRST
technique joining PFPwith the Parametric Curves.In LCM, the eld of
velocity vectors are obtained by Darcys Law. Itis the FIRST time
that the ow front is represented with acontinuous curve and it is
updated with the velocity vectors.L. Hilario, N. Monts, M.C.Mora,
A. Falc A Tensor Calculus Approach for Bzier Shape Deformation eo
e39/40 40. Introduction. BSD in engineering applications. Tensorial
Representation. Comparison BSD and T-BSD. Conclusions.Thank you for
your attention! Questions?L. Hilario, N. Monts, M.C.Mora, A. Falc A
Tensor Calculus Approach for Bzier Shape Deformation eo e40/40