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Creeping Ray Tracing Algorithm for Arbitrary NURBS Surfaces Based on Adaptive Variable Step Euler Method

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Although the uniform theory of diffraction (UTD) could be theoretically applied to arbitrarilyshaped convex objects modeled by nonuniform rational B-splines (NURBS), one of the great challenges in calculation of the UTD surface diffracted fields is the difficulty in determining the geodesic paths along which the creeping waves propagate on arbitrarilyshaped NURBS surfaces. In differential geometry, geodesic paths satisfy geodesic differential equation (GDE). Hence, in this paper, a general and efficient adaptive variable step Euler method is introduced for solving the GDE on arbitrarilyshaped NURBS surfaces. In contrast with conventional Euler method, the proposed method employs a shape factor (SF) ξ to efficiently enhance the accuracy of tracing and extends the application of UTD for practical engineering. The validity and usefulness of the algorithm can be verified by the numerical results.
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Research Article
Creeping Ray Tracing Algorithm for Arbitrary NURBS
Surfaces Based on Adaptive Variable Step Euler Method
Song Fu,1Yun-Hua Zhang,1Si-Yuan He,1Xi Chen,2and Guo-Qiang Zhu1
1School of Electronic Information, Wuhan University, Wuhan 430072, China
2No. 29 Research Institute, China Electronics Technology Group Corporation, Chengdu 610000, China
Correspondence should be addressed to Yun-Hua Zhang; zhangyunhua@whu.edu.cn
Received  April ; Revised  July ; Accepted  July 
Academic Editor: Ana Alejos
Copyright ©  Song Fu et al. is is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Although the uniform theory of diraction (UTD) could be theoretically applied to arbitrarilyshaped convex objects modeled
by nonuniform rational B-splines (NURBS), one of the great challenges in calculation of the UTD surface diracted elds is the
diculty in determining the geodesic paths along which the creeping waves propagate on arbitrarilyshaped NURBS surfaces. In
dierential geometry, geodesic paths satisfy geodesic dierential equation (GDE). Hence, in this paper, a general and ecient
adaptive variable step Euler method is introduced for solving the GDE on arbitrarilyshaped NURBS surfaces. In contrast with
conventional Euler method, the proposed method employs a shape factor (SF) to eciently enhance the accuracy of tracing
and extends the application of UTD for practical engineering. e validity and usefulness of the algorithm can be veried by the
numerical results.
1. Introduction
In the high-frequency electromagnetic problems, the UTD
analysis of the diraction of electromagnetic (EM) waves
[]bysmoothconvexsurfacesisofgreatimportance,such
as the EM compatibility, the analysis of coupling between
two antennas [], and the estimation of scattering properties.
And, as known to all, ray tracing of creeping waves plays
an important role in the process of obtaining the solution
of UTD surface diracted ied. erefore it is essential to
investigate the creeping ray tracing (geodesic paths) on sur-
faces rstly. In fact, however, except some typically geometric
objects which have analytical solution of the geodesic path, it
is a great challenge in determining the geodesic paths along
which the creeping waves propagate on arbitrarily shaped
smooth convex surfaces.
Jha et al. []putforwardanoveltechniqueknown
as the geodesic constant method, an analytical method,
and it is applicable to the creeping ray tracing on general
paraboloids of revolution and hybrid surfaces of revolution.
Usually, for some slightly more complex models such as
simple aircra, satellite, they are usually approximated by
some typically geometric objects, such as boards, cylinders,
cones, and spheres, which already have analytical solution
in tracing the creeping rays, for engineering application.
But it is dicult to use such typically geometric objects
to approximate arbitrarily shaped objects, which apparently
limits the application of UTD method.
Hence it is very essential to introduce the numerical
ray tracing methods of creeping waves to handle arbitrarily
shaped convex objects. In [], the models are characterized
by many discrete triangular meshes. And creeping ray tracing
numerically on discrete triangular meshes is presented by
Surazhsky et al. However, such methods cannot be used
directly for the UTD solution. Except modeling arbitrarily
shaped convex object by discrete triangular meshes, it can
also be described in terms of free-form parametric surfaces
such as NURBS surfaces []. Moreover, due to the advantages
ofhighprecisioninmodelingwithlownumberofpatchesand
great exibility in simulation algorithms, NURBS has been
introduced into the area of high-frequency electromagnetic
analysis [,]. When object is described by NURBS surfaces,
geodesic path can be obtained by solving the geodesic
dierential equation []. Hence, some numerical schemes
wereemployedtohandletheGDE.In[], to obtain the
tracks of creeping rays on NURBS surfaces, the geodesic
Hindawi Publishing Corporation
International Journal of Antennas and Propagation
Volume 2015, Article ID 604861, 12 pages
http://dx.doi.org/10.1155/2015/604861
International Journal of Antennas and Propagation
u
v
x
y
z
Mapping
oo
S
(a) uv-parameter domain (b) x
y
z coordinates
0
u0
P󳰀:(u0,
0)
=
0u=u
0
P: (u0,
0
)
r
󳰀
u
r
󳰀
r
F : Parametric surface.
dierential equations are solved by Euler method which is
the rst-order scheme for dierential equations. And in [],
the Runge-Kutta method of order  was applied to solving
the GDE for determining the tracing on NURBS surfaces.
Euler method is highly ecient but with low precision. On
the contrary, the Runge-Kutta method is of high precision but
time-consuming in the process of creeping ray tracing.
In this paper, aiming at improving the accuracy and
eciency of creeping ray tracing on arbitrarily shaped
NURBS surface, we present a creeping ray tracing algorithm
basedonanoveladaptivevariablestepEulermethod.Since
theadaptivevariablestepEulermethodisbasedonthe
conventional Euler method, the eciency is ensured. And in
the process of numerically iteratively solving the GDE, due
to the introduction of shape factor , the discrete step sizes
are adaptively corrected timely. Hence,compared with the
conventional Euler method, the proposed method can easily
ensure the accuracy of creeping ray tracing on arbitrarily
shaped NURBS surface. Namely, it is more suitable for
engineering applications. e eciency and accuracy of the
proposed method will be analyzed and veried by comparing
the results of dierent methods.
2. NURBS Modeling for Arbitrarily
Shaped Convex Objects
NURBS are generations of nonrational B-splines and rational
and nonrational Bezier curves and surfaces []. And NURBS
surface is the rational generalization of the nonrational B-
spline surface. It is dened as follows:
(,V)[0,1]2→
(,V)
=𝑚
𝑖=0𝑛
𝑗=0𝑖𝑗P𝑖𝑗𝑝
𝑖()𝑞
𝑗(V)
𝑚
𝑖=0𝑛
𝑗=0𝑖𝑗𝑝
𝑖()𝑞
𝑗(V),()
where 𝑖𝑗 are the weights and P𝑖𝑗 are the control points;
𝑝
𝑖() are the normalized B-spline basis functions of
degree dened recursively as
0
𝑖()=
1,
𝑖≤≤𝑖+1,
0,others,
𝑝
𝑖()=−𝑖
𝑖+𝑝 −𝑖𝑝−1
𝑖()+𝑖+𝑝+1−
𝑖+𝑝+1−𝑖+1𝑝−1
𝑖+1(),
()
where 𝑖are the so called knots forming a knot vector =
{0,1,...,𝑚}. e degree, number of knots, and number of
control points are related by the formula =++1.
In many applications of computational geometry, NURBS
surfaces are oen expanded in terms of rational Bezier
patches by applying the Cox-De Boor transform algorithm
[] since Bezier patches are numerically more stable than
NURBS surfaces, while NURBS are more ecient for the
storage and representation of a model. e surface points of
a rational Bezier surface are given by
(,V)[0,1]2→
(,V)
=𝑚
𝑖=0𝑛
𝑗=0𝑖𝑗P𝑖𝑗𝑚
𝑖()𝑛
𝑗(V)
𝑚
𝑖=0𝑛
𝑗=0𝑖𝑗𝑚
𝑖()𝑛
𝑗(V),()
where P𝑖𝑗 are the control points and 𝑖𝑗 are the weights. 𝑚
𝑖()
and 𝑛
𝑗(V)are Bernstein polynomials of degrees and ,
respectively, which are dened for all integers (= 0),
(= 0)by
𝑚
𝑖()=!
!(−)!𝑖(1−)𝑚−𝑖 ,
𝑛
𝑗(V)=!
!!V𝑗(1V)𝑛−𝑗 .()
A NURBS or Bezier surface can be dened as a vector-
valued mapping from two-dimensional Vparameter domain
to a set of three-dimensional coordinates. And they can
be generally described as
:
(,V)=(,V),(,V),(,V)𝑇.()
e mapping of V-parameter domain to three-space can be
seen in Figure .
International Journal of Antennas and Propagation
(a) Cylinder (b) Cone (c) Sphere
F : NURBS models.
In Figure (b), when we x V=V0and let vary, then
(,V0)depends on one parameter and is, therefore, a curve
called a -parameter curve. Similarly, if we x =
0,then
the curve
(0,V)is a V-parameter curve. Note that both
curves pass through :
(0,V0)in R3.Tangentvectors
for the -parameter and V-parameter curves are given by
dierentiating the components of
(,V)with respect to and
V,respectively.Wewrite
󸀠
𝑢(,V)=
(,V)
 =󸀠
𝑢(,V),󸀠
𝑢(,V),󸀠
𝑢(,V)𝑇,
󸀠
V(,V)=
(,V)
V=󸀠
V(,V),󸀠
V(,V),󸀠
V(,V)𝑇.()
In this paper, if |
󸀠
𝑢(,V)|and |
󸀠
V(,V)|are both constant
on parametric surfaces, we call the surface uniform grid sur-
face, otherwise nonuniform grid surface. Here some typical
NURBS models are given in Figure . e surfaces of cylinder
are uniform grid surfaces, and others are nonuniform grid
surfaces.
e surfaces of models above are digitized by networks of
V-isoparametric curves. And the cylinder, cone, and sphere
are, respectively, constructed by  Bezier patches highlighted
in dierent colors. And in the following, for convenience,
dierent Bezier patches will be shown in the same color.
3. Algorithm of Creeping Ray Tracing on
Arbitrary NURBS Surfaces
According to the positions of the source and the observation
points, the diraction problems of convex surfaces can be
classied as three types. First, the source and the observation
points are both out of the surfaces and the observation points
are far away from the surfaces. is case belongs to the
scattering problem of the smooth convex surfaces. Second,
the source is directly on the surfaces, and the observation
points are far away from the surfaces. is can be called
theradiationproblemofantennas.ird,thesourceandthe
observationpointsarebothonthesurfaces.isisthemutual
coupling problem among antennas on the surface. Hence,
the ray tracing should also be divided into three cases, as
in Figures . In all of these cases, the ray trajectories on
the surfaces are called creeping rays, which are restrained
to propagate along geodesic paths. And this paper mainly
focuses on the tracing of creeping rays since it is the most
dicult part in the process of tracing.
3.1. e Start Points and Exit Points of the Creeping Rays.
As shown in Figure ,atthestartingpoint
0, the grazing
incident condition satises
0,V0−
0
0,V0−
0
0,V0=0,()
where
(0,V0)is the unknown
0on the surface
(,V)and
(0,V0)is the corresponding unit normal vector of at
0.
At the exiting point
𝑠:
(𝑖,V𝑖),

𝑖,V𝑖−
𝑠

𝑖,V𝑖−
𝑠
𝑖,V𝑖=0.()
According to () and (),wecanobtainanumberof
the eective starting and exiting points of creeping rays on
NURBS surfaces.
3.2. Creeping Ray Tracing by Solving GDE with Adaptive Vari-
able Step Euler Method. As we know, the paths of creeping
rays on arbitrarily shaped surfaces satisfy GDE. en the
problem of obtaining creeping rays is changed into solving
GDE. Generally, GDE are solved by Euler method which is
a simple and fast method. However, according to study, in
most of the cases conventional Euler method does not work
because of its low accuracy and stability.
For the nonlinear problems, the step controlled correc-
tion procedure is essentially needed. Hence, in this paper, in
order to improve the accuracy of ray tracing and make sure
of the eciency, an adaptive variable step Euler method is
presented to solve the GDE.
International Journal of Antennas and Propagation
Incident rays
Creeping rays
Emergent rays
Plane waves
(a) Side view
Incident rays
Creeping rays
Emergent rays
Pn
P0
P1
P0,...,P
n: observation points
(b) Top view
F : Ray tracing of the plane wave incidence.
Q: source point
Q
Creeping rays
Emergent rays
P0,...,P
n: observation points
Pn
P0
(a) Side view
Creeping rays
Emergent rays
Q
Q: source point
P0,...,P
n: observation points
Pn
P0
(b) Top view
F : Ray tracing of source on surface.
Firstly, GDE are given as follows:
2
2+Γ1
11 
2+2Γ1
12 
 V
+Γ1
22 V
2
=0, ()
2V
2+Γ2
11 
2+2Γ2
12 
 V
+Γ2
22 V
2
=0, ()
where Γ𝑘
𝑚𝑛 are the Christofel symbols of the second kind, is
arc length along the geodesic path, and (,V)are parametric
coordinates of creeping rays.
en, the rst and second derivatives of in () can be
approximately expressed by
󸀠()=(+)−()
,()
󸀠󸀠 ()=󸀠(+)−󸀠()

=(++)−(+)−(+)+()
2,()
where is the step size between two adjacent discrete points,
the determination of which is of great importance. is shape
factor (SF), which is introduced to adaptively control every
discrete step size. And the value of is subject to the shape
International Journal of Antennas and Propagation
Creeping rays
Q: source point
P: observation point
P
Q
(a) Side view
Creeping rays
Q: source point
P: observation point
Q
P
(b) Top view
F : Ray tracing of source and observation point on surface.
Incident
Creeping ray
Exiting point
Starting point Q0
Qs
direction ̂
t
Source p0
Observation point ps
Surface S:r(u,  )
F : Creeping ray on surface .
of the object; more details about will be presented in
Section ..
In numerical calculation, a number of discrete points
(𝑖,V𝑖)can be computed to express the creeping ray, where
=0,1,2,....So, according to discretization, () and ()
canberewrittenas
󸀠
𝑖=𝑖+1−𝑖
𝑖,()
󸀠󸀠
𝑖=𝑖+2
𝑖+1−𝑖+1+𝑖
𝑖+1,𝑖𝑖2,()
where
𝑖+1 =(+𝑖+1,𝑖𝑖),whichisatestpoint.Since
󸀠
𝑖=𝑖+1−𝑖
𝑖
𝑖+1−𝑖
𝑖+1,𝑖𝑖,()
then
𝑖+1=𝑖+𝑖+1,𝑖 ⋅𝑖+1−𝑖. ()
Next, substituting () into () we obtain
󸀠󸀠
𝑖=𝑖+2−𝑖+1−𝑖+1,𝑖 ⋅𝑖+1−𝑖
𝑖+1,𝑖𝑖2.()
Similarly, the rst and second derivatives of Vcan be
V󸀠
𝑖=V𝑖+1V𝑖
𝑖,
V󸀠󸀠
𝑖=V𝑖+2V𝑖+1−𝑖+1,𝑖 ⋅V𝑖+1V𝑖
𝑖+1,𝑖𝑖2.()
Finally, substituting (),(),and() into the GDE ()
and (),wehave
𝑖+2=𝑖+1+𝑖+1,𝑖 ⋅𝑢,
V𝑖+2=V𝑖+1+𝑖+1,𝑖 ⋅V,()
where
𝑢=𝑖+1−𝑖−Γ1
11 𝑖+1−𝑖2
+2Γ1
12 𝑖+1−𝑖V𝑖+1V𝑖+Γ1
22 V𝑖+1V𝑖2,
V=V𝑖+1V𝑖−Γ2
11 𝑖+1−𝑖2
+2Γ2
12 𝑖+1−𝑖V𝑖+1V𝑖+Γ2
22 V𝑖+1V𝑖2.
()
According to (),tocalculate(𝑖+2,V𝑖+2)on the creeping
path, the previous two points (𝑖+1,V𝑖+1),(𝑖,V𝑖)and 𝑖+1,𝑖 are
needed. erefore the rst two points (0,V0),(1,V1)and 1,0
should be calculated at the beginning of creeping ray tracing.
Here, the starting point 0(0,V0)canbeobtainedby().
And according to the dierential geometry [], the second
International Journal of Antennas and Propagation
v-parameter curve
S
u-parameter curve Q0
0
u0
r(u0,
0)·(
1−
0)̂
t·h
0
ru(u0,
0)·(u
1−u
0)
Incident direction ̂
t
F : e determination of the second tracing point.
point is approximated by the sum of the two tangent vectors
on tangent plane at the starting point 0shown in Figure .
Hence, the second point can be expressed by
1=0+0⋅
𝑢0,V0⋅

󸀠
𝑢0,V0,
V1=V0+0⋅
V0,V0⋅

󸀠
V0,V0,()
where 0is initial step size and
𝑢(0,V0)and
V(0,V0)
are the unit tangent vectors for the -parameter and V-
parameter curves at 0(0,V0). And tangent vectors
󸀠
𝑢(0,V0)
and
󸀠
V(0,V0)for the -parameter and V-parameter curves
can be obtained by ().
is the unit vector along the incident
direction.
e general expression of shape factor is derived in
Section .. According to the expression, 1,0can be obtained.
Aer the calculation of (0,V0),(1,V1)and 1,0,() can
iterate step by step with the increase of .
3.3. e Derivation of Shape Factor .As we know, the smaller
the step size is, the more accurate that derivative approx-
imate expression is. However, in the process of iteratively
solving the discrete points on creeping ray path, the eciency
of algorithm will decrease with the increase of the number
of discrete points. More importantly, the more the discrete
points, the larger the accumulative error which may lead to
wrong results.
Clearly, for the approximation of 󸀠
𝑖or V󸀠
𝑖,wecould
achieve greater eciency if we could provide more discrete
points where the variables are changing rapidly to achieve
accuracy, while less discrete points should be provided in
regions where variables are changing slowly. erefore it is
necessary to allocate discrete points during the process of
Euler approximation.
However the fact is that the parametric expressions
((),V())ofthecreepingraypathareunknown.Wecannot
directly obtain the variable change speed degree which the
policy of allocating discrete points is based on. So, the shape
factor is introduced to determine the discrete step size.
e discrete value of shape factor 𝑖+1,𝑖 is dened to reect
therelativechangerateofparametersattwoadjacentpoints
on the creeping path. Since the creeping rays include two
parametric expressions () and V(), we should consider
C
P
S
v-parameter curve
u-parameter curve
0
u0
r
̂
t
ru
F : Curve on the parametric surface.
the change of ,Vsimultaneously for well determining 𝑖+1,𝑖.
Hence,
𝑖+1,𝑖 =
󸀠
𝑖
󸀠
𝑖+1
,󸀠
𝑖
󸀠
𝑖+1
V󸀠
𝑖
V󸀠
𝑖+1
,
V󸀠
𝑖
V󸀠
𝑖+1
,else,()
where 󸀠
𝑖+1,󸀠
𝑖,V󸀠
𝑖+1,andV󸀠
𝑖are unknown, which need to be
solved out. And the relationship between the SF and step
size is
𝑖+1=𝑖+1,𝑖 ⋅𝑖.()
Let be an arc length parametric curve (creeping ray) on
surface :
(,V)which pass through point as shown in
Figure  and denote that
()=
((),V()).()
Let
be a unit tangent vector of at ,andletthenumberof
the discrete points be .
Tangent vector for the creeping ray at the point can
be obtained by dierentiating equation () with respect to
the arc length
𝑖=
((),V())
 =
𝑢,𝑖 𝑖
 +
V,𝑖 V𝑖
,()
where
𝑢,𝑖 and
V,𝑖 canbecalculatedby().Since|
𝑖|=1,
𝑢,𝑖 𝑖
 +
V,𝑖 V𝑖
=1.()
However, 𝑖/and V𝑖/cannot be obtained from ().
Now, we approximate 𝑖/ and V𝑖/ at point on
by 𝑖/at point on -parameter curve (V=V0)and
V𝑖/at point on V-parameter curve, respectively. For
on -parameter curve,
V,𝑖 =0,() can be
󸀠
𝑢,𝑖 𝑖
 =1. ()
For on V-parameter curve,
𝑢,𝑖 =0,() can be
󸀠
V,𝑖 V𝑖
=1.()
International Journal of Antennas and Propagation
T : Comparison of creeping rays on cylinder obtained from three dierent methods.
Source point (, , )
Creeping ray number
Last point
Analytical method (., ., .) (., ., .) (., ., .)
Euler method (., ., .) (., ., .) (., ., .)
Proposed method (., ., .) (., ., .) (., ., .)
Length (m)
Analytical method . . .
Euler method . . .
Proposed method . . .
So, according to () and (),
𝑖
 =1
󸀠
𝑢,𝑖,
V𝑖
=1
󸀠
V,𝑖.()
Similarly, at the +1point on ,
𝑖+1
 =1
󸀠
𝑢,𝑖+1,
V𝑖+1
 =1
󸀠
V,𝑖+1.()
Substituting () and () into (), nally the shape factor
expression is
𝑖+1,𝑖 =
󸀠
𝑢,𝑖+1
󸀠
𝑢,𝑖
,
󸀠
𝑢,𝑖+1
󸀠
𝑢,𝑖
󸀠
V,𝑖+1
󸀠
V,𝑖
,
󸀠
V,𝑖+1
󸀠
V,𝑖
,else.
()
So, based on the 𝑖+1,𝑖, we can obtain discrete variable step
adaptively.
4. Results and Discussions
4.1. Creeping Ray Tracing Algorithm Validation. e creeping
rays can be calculated theoretically on some typical objects,
such as cylinder, cone, and sphere. erefore, the proposed
method can be veried from the analytical results of these
canonical objects.
(1) Creeping Ray Tracing on Cylinder (Uniform Grid Surfaces).
e cylinder which has a radius of  m and height of  m
is considered for creeping ray tracing. We consider three
dierent incident wave directions, all of which incident start
fromthesamepoint(1,0,0). In order to validate the
accuracy of the proposed creeping ray tracing method, it is
compared against the analytical method and the conventional
Euler method. Figure  shows the creeping ray tracing results
obtained from the above mentioned methods, for three
dierent incident directions, respectively. It is shown that,
0
0
0.5
1
1.5
2
2.5
3
Analytical method
Euler method
Proposed method
1
3
2
−1
1
1
0
−1
Q
F : Creeping ray tracing on cylinder.
for each incident direction, the creeping ray obtained from
theproposedmethodoverlayswiththosefromtheother
two methods. To further validate the proposed method, the
creeping ray ending point and the length of the creeping rays
are compared, as shown in Tab l e  .
From Table ,thenumericalresultsbytheproposed
method are in good agreement with the analytical method,
which can conrm its rationality. And we can nd that the
numerical results by the proposed method and Euler method,
respectively, are identical, and this is because cylinder is
so special that the () and V() of creeping ray on its
surfaces are rst-order linear. Namely, the discrete step sizes
canbechosenequally.Andinthisspecialcase(uniform
grid surfaces), 𝑖+1,𝑖 =1,theproposedmethodisreduced
to conventional Euler method. (Comparing to the Euler
method, the advantages of the proposed method can be
demonstrated in the following examples.)
(2) Creeping Ray Tracing on Cone (Nonuniform Grid Surfaces).
e cone which has a radius of  m and height of  m is consid-
ered for creeping ray tracing. We consider three dierent inci-
dent wave directions, all of which incident start from the same
International Journal of Antennas and Propagation
0
0.5
1
1
0
0
0.5
1
1.5
2
Analytical method
Euler method
Proposed method
−1
0.5 −1
Q
(a) Side view
0
0.2
0.4
0.6
0.8
1
Analytical method
Euler method
Proposed method
Q
1
2
3
−0.2
−0.4
−0.6
−0.8
−1
−1 −0.5 0 0.5 1
(b) Top view
F : Creeping ray tracing on cone.
T : Comparison of creeping rays on cone obtained from three dierent methods.
Source point (, , )
Creeping ray number
Last point
Analytical method (., ., .) (., ., .) (., ., .)
Euler method (., ., .) (., ., .) (., ., .)
Proposed method (., ., .) (., ., .) (., ., .)
Length (m)
Analytical method . . .
Euler method . . .
Proposed method . . .
Time (s) Euler method . . .
Proposed method . . .
point (1,0,0).InFigure , the creeping ray tracing results
on a cone are shown. e numbers , , and  in Figure (b)
are three creeping rays of dierent incident directions.
Firstly, from Tab l e  ,wecanndthenumericaltracing
results by proposed method are in good agreement with the
analytical method. As a comparison, the results obtained by
theEulermethodareinloweraccuracy.Moreover,thecreep-
ing ray results obtained by the Euler method deviate from
that of the analytical method when the creeping ray trajectory
is closer to cone-tip. is is clearly shown in Figure  and
Table , the Euler method results deviate from the analytical
resultsfortheraynumber,anditevenfallsintofalseresults
when the ray runs too close to the cone-tip as for the ray
number . is indicates that the conventional Euler method
cannot be applied to nonuniform grids surfaces while the
proposed method is capable of handling the nonuniform
grids surfaces. Secondly, the proposed method is as ecient
as conventional Euler method.
(3) Creeping Ray Tracing on Sphere (Nonuniform Grid Sur-
faces). espherewhichhasaradiusofmisconsidered
for creeping ray tracing. According to dierential geometry,
the geodesic paths on sphere are orthodromes. Hence, the
numerical results of sphere can be validated by the theoretical
results.
From Figure (a), the result by proposed method is
in good agreement with the theoretical results. And from
Table  and Figures (b) and (c), although, with the
decrease of discrete step size, the error by Euler method is
reduced, time consumption is increased a lot. More impor-
tantly, if the discrete step size continues to decrease, from
Figure (d), the result is totally wrong, which means Euler
method is unstable.
4.2. Electromagnetic Calculation with UTD on the Base of
Creeping Ray Tracing Algorithm. To assess the accuracy of
UTD diraction based on the proposed creeping ray tracing
International Journal of Antennas and Propagation
0
0.5
0
0.5
0
0.5
1
Q
0.5
0.5
0.5
−1
(a) Proposed method
0
0.5
00.5
0
0.5
1
Q
0.5
−1
0.5
0.5
(b) Euler method
0
0.5
0
0.5
0
0.5
1
Q
0.5
0.5
0.5
−1
(c) Euler method
0
0.5
0
0.5
0
0.5
1
Q
0.5
0.5
0.5
−1
(d) Euler method
F : Creeping ray tracing on sphere.
T : Results of three creeping rays tracing on sphere.
Source point (, , )
Methods (a) Proposed method (b) Euler method (c) Euler method (d) Euler method
Initial step size (m) . . . .
Last point (., ., .) (., ., .) (., ., .) (., ., .)
Time (s) . . . .
algorithm, the analytical results of canonical targets are given.
Figures  and  show the bistatic scattering electric elds of
PEC sphere and cylinder in shadow region, respectively, and
the UTD solutions are compared with the analytical results.
Moreover, in reality, lots of targets are low detectable.
When analyzing the scattering properties of those targets
especially for some constructed by smooth convex curved
surfaces, the contributions of creeping waves cannot be
neglected. Here, taking an antiradar stealth screen compo-
nent of a radar stealth satellite as example, since the antiradar
stealth screen component is constructed by some smooth
convex curved surfaces (see Figure ), the contributions of
creeping waves will play important roles in this case. In
the following example, the monostatic RCS of the antiradar
stealth screen are studied by comparing the PO + UTD
solution with the numerical solution to demonstrate the
contribution of creeping waves.
() e bistatic diracted electric elds of a PEC cylinder
inshadowregionarecalculatedbyUTD.eradiusofthe
cylinder is  m, and the distance between the observation
 International Journal of Antennas and Propagation
x
y
R
Diracted ray
𝜌
30
Ps
−30
Incident rays
Creeping rays
Emergent rays
Field points
F : Diracted contribution in shadow region.
0 1010 2020 3
0
30
0
0.2
0.2
0.4
0.
6
Bistatic angle (deg.)
Magnitu
d
e o
f
e
l
ectric
e
ld
E (V/m)
Analytical result
UTD
F : Scattered electric eld of PEC cylinder.
point and the origin is  m. e frequency of incident plane
waveisGHz.esketchmapofthediractedcontribution
is given in Figure .
From Figure , the UTD results of PEC cylinder in the
shadow region based on the proposed creeping ray tracing
algorithm are in good agreement with the analytical results,
and the good agreement conrms the validity of the creeping
ray tracing algorithm.
() e bistatic diracted electric elds of a PEC sphere
inshadowregionarecalculatedbyUTD.eradiusofthe
sphere is  m, and the distance between the observation
point and the origin is  m. e frequency of incident plane
waveisGHz.esketchmapofthediractedcontribution
is given in Figure .
As shown in Figure , the UTD results of PEC sphere
in the shadow region based on the proposed creeping ray
tracing algorithm are in good agreement with the analytical
results, which denotes the validity of the creeping ray tracing
algorithm.
() Monostatic RCS of the antiradar stealth screen com-
ponent of a radar stealth satellite are calculated by the PO
method combined with the UTD method based on the
proposed creeping ray tracing algorithm, and these results
are then compared with that of the numerical method to
validatetheaccuracyoftheproposedmethod.esideand
D view of the antiradar stealth screen component are shown
in Figure ,where=1.12m, =0.3m, and =3.0m.
Figure  shows the trajectories of creeping rays on the
antiradar stealth screen component. In space, the conical tip
of the antiradar stealth screen is directed to the earth. Hence
we focus on the analysis of scattering property about the
antiradar stealth screen nearby region of the conical tip. Here,
the RCS will be calculated in the range of =90180.e
frequency of incident plane wave is  GHz.
Next, we compare the monostatic RCS of the target in the
range of =90180by three dierent methods, namely,
the numerical method Multilevel Fast Multipole Algorithm
(MLFMA), the PO method, and the PO + UTD method. In
International Journal of Antennas and Propagation 
x
y
Diracted ray
Ps
R
𝜌
150
210
Incident rays
Creeping rays
Emergent rays
Field points
1
0.5
0
−0.5
−1
2
1
0
−1
−2 1
0.5
0
−0.5
−1
F : Diracted contribution in shadow region.
150 160 170 180 190 200 21
0
0
0.2
0.4
0.6
0.8
1
Bistatic angle (deg.)
Magnitu
d
e o
f
e
l
ectric
e
ld
E (V/m)
Analytical result
U
TD
F : Scattered electric eld of PEC sphere.
Incident rays
Creeping rays
r
r
r
r
h
R
x
z
113
23
F : Creeping ray tracing on an antiradar stealth screen.
 International Journal of Antennas and Propagation
90 100 110 120 130 140 150 160 170 18
0
0
10
20
30
RCS (
d
Bsm)
MLFMA
PO
PO + UTD
Monostatic angle 𝜃(deg.)
−10
−20
−30
−40
−50
0
90
180
270
F : Monostatic RCS of the antiradar stealth screen.
thePO+UTDscheme,theUTDsolutionbasedonthepro-
posed algorithm is used to obtain the diracted contribution
of creeping rays. In the following, the monostatic RCS of the
antiradar stealth screen are given in Figure .
From Figure , it is observed that the contribution of
creeping waves is very obvious when the incident direction
=145175.Inthisrange,thankstothecontributionsof
the creeping waves, the PO + UTD results are about  dBsm
higher than the PO results. So, the contributions of creeping
waves cannot be neglected. And compared with PO, the PO
+ UTD solution has a better agreement with the MLFMA
solution.
Hence, the rationality and validity of the creeping ray
tracing algorithm presented in this paper are proved.
5. Conclusion
An accurate and ecient creeping ray tracing algorithm
basedonanadaptivevariablestepEulermethodispre-
sented in this paper. e proposed algorithm is applicable to
arbitrary NURBS surface. is adaptive variable step Euler
method, based on the conventional Euler method, employs
ashapefactor(SF)to solve the GDE. e SF can
timely reect the relative change rate of the creeping ray
parameters at two discretely adjacent points on surfaces.
And according to , every adaptive step size, which is very
important for the accuracy of creeping ray tracing, can be
obtained in turn. Finally based on the fast and accurate
creeping ray tracing, the contribution of UTD diraction can
be calculated. Bearing in mind the accuracy and eciency,
the algorithm of NURBS-UTD will be very useful in practical
engineering.
Conflict of Interests
e authors declare that there is no conict of interests
regarding the publication of this paper.
Acknowledgments
is work was supported by the National Natural Sci-
ence Foundation of China (Grant no. , Grant no.
, and Grant no. ), the China Postdoctoral
Science Foundation, and the Fundamental Research Funds
for the Central Universities.
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Article
An efficient approach to analyze the electromagnetic (EM) scattering from coated objects above a three-dimensional (3-D) dielectric random rough surface is proposed in this paper. The electric and magnetic current densities on the sea surface are analytically realized using the semi-deterministic facet scattering model (SDFSM). Considering Fresnel reflection coefficient, a scheme combining the shooting and bouncing rays (SBR) and the physical theory of diffraction (PTD) is developed to calculate the scattering of the coated objects. Simulation results obtained with the combined method are verified by comparison with exact simulations using the multilevel fast multiple method (MLFMM). Scattering of large coated objects over sea surface composite model are too large and complex to be calculated using exact methods. With the asymptotic approach in this paper, the scattering performance can be analyzed with rather moderate computational efforts. Numerical results of composite scattering from an electric large three-dimensional model, e.g. an airplane target over a randomly rough surface with Pierson Moskowitz (PM) spectrum are presented and discussed.
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Abstract Keywords: CEM, GTD, Ray Tracing, NURBS, Software Design, Shadowing. Over the past 20 years, the development in Computational Electromagnetics has produced a vast choice of methods based on the large number of existing mathematical formulations of the Maxwell equations. None of them dominate over the others, instead they complement each other and the choice of method depends on the frequency range of the electromagnetic waves. This work is focused on the most popular method in the high frequency scenario, namely the Geometrical Theory of Diraction,( GTD). The main advantage of GTD is the ability to predict the electromagnetic field asymptotically in the limit of vanishing wavelength, when other methods, such as the Method of Moments, become computationally too expensive. The low cost of GTD is due to both the fact that there is no runtime penalty in increasing the frequency and that the ray tracing, which GTD is based on, is a geometrical technique. The complexity is then no longer dependent on electrical size of the problem but instead on geometrical sub problems which are manageable. For industrial applications the geometrical structures, with which the rays interact, are modelled by trimmed Non-Uniform Rational B-Spline (NURBS) surfaces, the most recent standard used to represent complex free-form geometries.
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An accurate creeping ray-tracing algorithm is presented in this paper to determine the tracks of creeping waves (or creeping rays) on arbitrarily shaped free-form parametric surfaces [nonuniform rational B-splines (NURBS) surfaces]. The main challenge in calculating the surface diffracted fields on NURBS surfaces is due to the difficulty in determining the geodesic paths along which the creeping rays propagate. On one single parametric surface patch, the geodesic paths need to be computed by solving the geodesic equations numerically. Furthermore, realistic objects are generally modeled as the union of several connected NURBS patches. Due to the discontinuity of the parameter between the patches, it is more complicated to compute geodesic paths on several connected patches than on one single patch. Thus, a creeping ray-tracing algorithm is presented in this paper to compute the geodesic paths of creeping rays on the complex objects that are modeled as the combination of several NURBS surface patches. In the algorithm, the creeping ray tracing on each surface patch is performed by solving the geodesic equations with a Runge-Kutta method. When the creeping ray propagates from one patch to another, a transition method is developed to handle the transition of the creeping ray tracing across the border between the patches. This creeping ray-tracing algorithm can meet practical requirements because it can be applied to the objects with complex shapes. The algorithm can also extend the applicability of NURBS for electromagnetic and optical applications. The validity and usefulness of the algorithm can be verified from the numerical results.
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The computation of geodesic paths and distances on triangle meshes is a common operation in many computer graphics applications. We present several practical algorithms for computing such geodesics from a source point to one or all other points efficiently. First, we describe an implementation of the exact "single source, all destination" algorithm presented by Mitchell, Mount, and Papadimitriou (MMP). We show that the algorithm runs much faster in practice than suggested by worst case analysis. Next, we extend the algorithm with a merging operation to obtain computationally efficient and accurate approximations with bounded error. Finally, to compute the shortest path between two given points, we use a lower-bound property of our approximate geodesic algorithm to efficiently prune the frontier of the MMP algorithm. thereby obtaining an exact solution even more quickly.
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The well-known algorithm by de Boor for calculating a point of a B-spline curve can also be used to produce the Bézier points of a B-spline curve or surface.
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System requirements: IBM and compatibles with DOS 2.0 or higher or UNIX. This book offers an introduction to the field that emphasizes Bernstein-Bezier methods and presents subjects in an informal, readable style, making this an ideal text for an introductory course at the advanced undergraduate or graduate level. This 3rd edition includes several new section and numerical examples, a treatment of the new blossoming principle, and new C programs. All C programs are available on a disk included with the book. The Problems Sections at the end of each chapter have also been extended.
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A technique is presented for the RCS computation of electrically large conducting bodies, modelled by NURBS patches, using physical optics (PO) and asymptotic expansion of integral methods. The NURBS surfaces are transformed in Bezier patches to apply PO. Excellent accuracy is obtained.
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A generalized approach to analytical surface-ray tracing in three dimensions, and a review of its application to convex conducting bodies, is presented, using the Eisenhart coordinate system. The ray-parameters so obtained, for quadric cylinders (QUACYLs) and surfaces of revolution (QUASORs), are in a one-parameter form for UTD mutual-coupling applications. The ray analysis is also extended to the hybrid QUACYLs (e.g. aircraft wings) and hybrid QUASORs (e.g., satellite-launch vehicles), by introducing Hertz's principle of particle dynamics to EM theory. This mathematical formulation is applicable even to other important non-Eisenhart surfaces, such as the ogive. A summary of the mathematical formulations is included
Article
An approximate asymptotic solution is presented for the electromagnetic fields which are induced on an electrically large perfectly conducting smooth convex surface by an infinitesimal magnetic or electric current moment on the same surface. This solution can be employed to calculate the mutual coupling between antennas on a convex surface in an efficient and accurate manner. In this solution, the surface fields propagate along Keller's surface ray paths, and their description remains uniformly valid within the shadow boundary transition region including the immediate vicinity of the source. Furthermore, the effect of surface ray torsion on the surface fields is indicated in the present solution, through a factor T/k , where T denotes the surface ray torsion and k is the surface curvature in the ray direction. This solution is deduced from the asymptotic solutions to simpler canonical problems. Numerical results for the mutual coupling between slots in cylinders and cones are presented, and are shown to compare very well with experiments.