Abstract

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.

© 2013 Optical Society of America

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References

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  1. P. H. Pathak, W. D. Burnside, and R. J. Marhefkam, “A uniform GTD analysis of the diffraction of electromagnetic waves by a smooth convex surface,” IEEE Trans. Antennas Propag. 28, 631–642 (1980).
    [CrossRef]
  2. G. Farin, “Creeping waves on a perfectly conducting cone,” IEEE Trans. Antennas Propag. 25, 661–670 (1977).
    [CrossRef]
  3. S. W. Lee, E. K. Yung, and R. Mitta, “GTD solution of slot admittance on a cone or cylinder,” Proc. IEEE 126, 487–492 (1979).
    [CrossRef]
  4. E. O. Hosam, “Effect of H-wave polarization on laser radar detection of partially convex targets in random media,” J. Opt. Soc. Am. A 27, 1716–1722 (2010).
    [CrossRef]
  5. J. Pérez and M. F. Cátedra, “Application of physical optics to the RCS computation of bodies modeled with NURBS surfaces,” IEEE Trans. Antennas Propag. 42, 1404–1411 (1994).
    [CrossRef]
  6. M. Mohamed and O. Runborg, “A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems,” Commun. Math. Sci. 5, 617–648 (2007).
  7. S. Sefi, “Ray tracing tools for high frequency electromagnetics simulations,” Licentiate thesis (Royal Institute of Technology, Stockholm, 2003), pp. 46–47.
  8. N. Wang, Y. Zhang, and C. H. Liang, “Creeping ray-tracing algorithm of UTD method based on NURBS models with the source on surface,” J. Electromagn. Waves Appl. 20, 1981–1990(2006).
    [CrossRef]
  9. J. H. Mathews and K. D. Fink, Numerical Methods Using Matlab (Pearson, 2004).
  10. W. Bochm, “Generating the Bezier points of b-spline curves and surfaces,” Comput. Aided Des. 13, 365–366 (1981).
    [CrossRef]
  11. C. William, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice-Hall, 1971).
  12. J. M. Beck, R. T. Farouki, and J. K. Hinds, “Surface analysis methods,” IEEE Comput. Graph. Appl. 6, 18–36 (1986).
    [CrossRef]
  13. H. B. Keller, Numerical Methods for Two-Point Boundary-Value Problems (Dover, 1993).
  14. L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces (Dover Phoenix, 2004).

2010 (1)

2007 (1)

M. Mohamed and O. Runborg, “A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems,” Commun. Math. Sci. 5, 617–648 (2007).

2006 (1)

N. Wang, Y. Zhang, and C. H. Liang, “Creeping ray-tracing algorithm of UTD method based on NURBS models with the source on surface,” J. Electromagn. Waves Appl. 20, 1981–1990(2006).
[CrossRef]

1994 (1)

J. Pérez and M. F. Cátedra, “Application of physical optics to the RCS computation of bodies modeled with NURBS surfaces,” IEEE Trans. Antennas Propag. 42, 1404–1411 (1994).
[CrossRef]

1986 (1)

J. M. Beck, R. T. Farouki, and J. K. Hinds, “Surface analysis methods,” IEEE Comput. Graph. Appl. 6, 18–36 (1986).
[CrossRef]

1981 (1)

W. Bochm, “Generating the Bezier points of b-spline curves and surfaces,” Comput. Aided Des. 13, 365–366 (1981).
[CrossRef]

1980 (1)

P. H. Pathak, W. D. Burnside, and R. J. Marhefkam, “A uniform GTD analysis of the diffraction of electromagnetic waves by a smooth convex surface,” IEEE Trans. Antennas Propag. 28, 631–642 (1980).
[CrossRef]

1979 (1)

S. W. Lee, E. K. Yung, and R. Mitta, “GTD solution of slot admittance on a cone or cylinder,” Proc. IEEE 126, 487–492 (1979).
[CrossRef]

1977 (1)

G. Farin, “Creeping waves on a perfectly conducting cone,” IEEE Trans. Antennas Propag. 25, 661–670 (1977).
[CrossRef]

Beck, J. M.

J. M. Beck, R. T. Farouki, and J. K. Hinds, “Surface analysis methods,” IEEE Comput. Graph. Appl. 6, 18–36 (1986).
[CrossRef]

Bochm, W.

W. Bochm, “Generating the Bezier points of b-spline curves and surfaces,” Comput. Aided Des. 13, 365–366 (1981).
[CrossRef]

Burnside, W. D.

P. H. Pathak, W. D. Burnside, and R. J. Marhefkam, “A uniform GTD analysis of the diffraction of electromagnetic waves by a smooth convex surface,” IEEE Trans. Antennas Propag. 28, 631–642 (1980).
[CrossRef]

Cátedra, M. F.

J. Pérez and M. F. Cátedra, “Application of physical optics to the RCS computation of bodies modeled with NURBS surfaces,” IEEE Trans. Antennas Propag. 42, 1404–1411 (1994).
[CrossRef]

Eisenhart, L. P.

L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces (Dover Phoenix, 2004).

Farin, G.

G. Farin, “Creeping waves on a perfectly conducting cone,” IEEE Trans. Antennas Propag. 25, 661–670 (1977).
[CrossRef]

Farouki, R. T.

J. M. Beck, R. T. Farouki, and J. K. Hinds, “Surface analysis methods,” IEEE Comput. Graph. Appl. 6, 18–36 (1986).
[CrossRef]

Fink, K. D.

J. H. Mathews and K. D. Fink, Numerical Methods Using Matlab (Pearson, 2004).

Hinds, J. K.

J. M. Beck, R. T. Farouki, and J. K. Hinds, “Surface analysis methods,” IEEE Comput. Graph. Appl. 6, 18–36 (1986).
[CrossRef]

Hosam, E. O.

Keller, H. B.

H. B. Keller, Numerical Methods for Two-Point Boundary-Value Problems (Dover, 1993).

Lee, S. W.

S. W. Lee, E. K. Yung, and R. Mitta, “GTD solution of slot admittance on a cone or cylinder,” Proc. IEEE 126, 487–492 (1979).
[CrossRef]

Liang, C. H.

N. Wang, Y. Zhang, and C. H. Liang, “Creeping ray-tracing algorithm of UTD method based on NURBS models with the source on surface,” J. Electromagn. Waves Appl. 20, 1981–1990(2006).
[CrossRef]

Marhefkam, R. J.

P. H. Pathak, W. D. Burnside, and R. J. Marhefkam, “A uniform GTD analysis of the diffraction of electromagnetic waves by a smooth convex surface,” IEEE Trans. Antennas Propag. 28, 631–642 (1980).
[CrossRef]

Mathews, J. H.

J. H. Mathews and K. D. Fink, Numerical Methods Using Matlab (Pearson, 2004).

Mitta, R.

S. W. Lee, E. K. Yung, and R. Mitta, “GTD solution of slot admittance on a cone or cylinder,” Proc. IEEE 126, 487–492 (1979).
[CrossRef]

Mohamed, M.

M. Mohamed and O. Runborg, “A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems,” Commun. Math. Sci. 5, 617–648 (2007).

Pathak, P. H.

P. H. Pathak, W. D. Burnside, and R. J. Marhefkam, “A uniform GTD analysis of the diffraction of electromagnetic waves by a smooth convex surface,” IEEE Trans. Antennas Propag. 28, 631–642 (1980).
[CrossRef]

Pérez, J.

J. Pérez and M. F. Cátedra, “Application of physical optics to the RCS computation of bodies modeled with NURBS surfaces,” IEEE Trans. Antennas Propag. 42, 1404–1411 (1994).
[CrossRef]

Runborg, O.

M. Mohamed and O. Runborg, “A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems,” Commun. Math. Sci. 5, 617–648 (2007).

Sefi, S.

S. Sefi, “Ray tracing tools for high frequency electromagnetics simulations,” Licentiate thesis (Royal Institute of Technology, Stockholm, 2003), pp. 46–47.

Wang, N.

N. Wang, Y. Zhang, and C. H. Liang, “Creeping ray-tracing algorithm of UTD method based on NURBS models with the source on surface,” J. Electromagn. Waves Appl. 20, 1981–1990(2006).
[CrossRef]

William, C.

C. William, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice-Hall, 1971).

Yung, E. K.

S. W. Lee, E. K. Yung, and R. Mitta, “GTD solution of slot admittance on a cone or cylinder,” Proc. IEEE 126, 487–492 (1979).
[CrossRef]

Zhang, Y.

N. Wang, Y. Zhang, and C. H. Liang, “Creeping ray-tracing algorithm of UTD method based on NURBS models with the source on surface,” J. Electromagn. Waves Appl. 20, 1981–1990(2006).
[CrossRef]

Commun. Math. Sci. (1)

M. Mohamed and O. Runborg, “A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems,” Commun. Math. Sci. 5, 617–648 (2007).

Comput. Aided Des. (1)

W. Bochm, “Generating the Bezier points of b-spline curves and surfaces,” Comput. Aided Des. 13, 365–366 (1981).
[CrossRef]

IEEE Comput. Graph. Appl. (1)

J. M. Beck, R. T. Farouki, and J. K. Hinds, “Surface analysis methods,” IEEE Comput. Graph. Appl. 6, 18–36 (1986).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

P. H. Pathak, W. D. Burnside, and R. J. Marhefkam, “A uniform GTD analysis of the diffraction of electromagnetic waves by a smooth convex surface,” IEEE Trans. Antennas Propag. 28, 631–642 (1980).
[CrossRef]

G. Farin, “Creeping waves on a perfectly conducting cone,” IEEE Trans. Antennas Propag. 25, 661–670 (1977).
[CrossRef]

J. Pérez and M. F. Cátedra, “Application of physical optics to the RCS computation of bodies modeled with NURBS surfaces,” IEEE Trans. Antennas Propag. 42, 1404–1411 (1994).
[CrossRef]

J. Electromagn. Waves Appl. (1)

N. Wang, Y. Zhang, and C. H. Liang, “Creeping ray-tracing algorithm of UTD method based on NURBS models with the source on surface,” J. Electromagn. Waves Appl. 20, 1981–1990(2006).
[CrossRef]

J. Opt. Soc. Am. A (1)

Proc. IEEE (1)

S. W. Lee, E. K. Yung, and R. Mitta, “GTD solution of slot admittance on a cone or cylinder,” Proc. IEEE 126, 487–492 (1979).
[CrossRef]

Other (5)

J. H. Mathews and K. D. Fink, Numerical Methods Using Matlab (Pearson, 2004).

S. Sefi, “Ray tracing tools for high frequency electromagnetics simulations,” Licentiate thesis (Royal Institute of Technology, Stockholm, 2003), pp. 46–47.

H. B. Keller, Numerical Methods for Two-Point Boundary-Value Problems (Dover, 1993).

L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces (Dover Phoenix, 2004).

C. William, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice-Hall, 1971).

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Figures (12)

Fig. 1.
Fig. 1.

Point on the parametric surface patch.

Fig. 2.
Fig. 2.

Creeping ray on the union of several surface patches.

Fig. 3.
Fig. 3.

Tangent plane at the attachment point.

Fig. 4.
Fig. 4.

Transition between surface patches.

Fig. 5.
Fig. 5.

Creeping ray tracing on the cylinder.

Fig. 6.
Fig. 6.

Creeping ray tracing on the cone.

Fig. 7.
Fig. 7.

Creeping ray tracing on the sphere.

Fig. 8.
Fig. 8.

Creeping ray-tracing result on a satellite component.

Fig. 9.
Fig. 9.

Geodesic curvatures at the discrete points.

Fig. 10.
Fig. 10.

Creeping ray-tracing result on an asymmetric object.

Fig. 11.
Fig. 11.

Geodesic curvatures at the discrete points.

Fig. 12.
Fig. 12.

Comparison of analytical solution and UTD numerical solution.

Tables (3)

Tables Icon

Table 1. Detachment Point and Length of the Creeping Ray on the Cylinder

Tables Icon

Table 2. Detachment Point and Length of the Creeping Ray on the Cone

Tables Icon

Table 3. Detachment Point and Length of the Creeping Ray on the Sphere

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

r⃗(u,v)=i=0mj=0nωi,jp⃗i,jBmi(u)Bnj(v)i=0mj=0nωi,jBmi(u)Bnj(v),
Bsk(t)=s!k!(sk)!tk(1t)sk,t[0,1].
(r⃗(u,v)p⃗0)|r⃗(u,v)p⃗0|·n^(u,v)=0,
(r⃗(u,v)p⃗s)|r⃗(u,v)p⃗s|·n^(u,v)=0.
(d2uds2)+Γ111(duds)2+2Γ121(dudsdvds)+Γ221(dvds)2=0,
(d2vds2)+Γ112(duds)2+2Γ122(dudsdvds)+Γ222(dvds)2=0,
duds=δ,
dvds=ξ,
dδds=Γ111δ2Γ121δξΓ221ξ2,
dξds=Γ112δ2Γ122δξΓ222ξ2,
f(u,v,δ,ξ)=Γ111δ2Γ121δξΓ221ξ2,
g(u,v,δ,ξ)=Γ112δ2Γ122δξΓ222ξ2.
si+1=si+Δs,
{ui+1=ui+Δui=ui+Δs·δi+Δs6(k1+k2+k3)vi+1=vi+Δvi=vi+Δs·ξi+Δs6(l1+l2+l3),
{δi+1=δi+Δδi=δi+Δs6(k1+2k2+2k3+k4)ξi+1=ξi+Δξi=ξi+Δs6(l1+2l2+2l3+l4),
{k1=f(ui,vi,δi,ξi)l1=g(ui,vi,δi,ξi),
{k2=f(ui+Δs2δi,vi+Δs2ξi,δi+Δs2k1,ξi+Δs2l1)l2=g(ui+Δs2δi,vi+Δs2ξi,δi+Δs2k1,ξi+Δs2l1),
{k3=f(ui+Δs2δi+Δs24k1,vi+Δs2ξi+Δs24l1,δi+Δs2k2,ξi+Δs2l2)l3=g(ui+Δs2δi+Δs24k1,vi+Δs2ξi+Δs24l1,δi+Δs2k2,ξi+Δs2l2),
{k4=f(ui+Δs·δi+Δs22k2,vi+Δs·ξi+Δs22l2,δi+Δs·k3,ξi+Δs·l3)l4=g(ui+Δs·δi+Δs22k2,vi+Δs·ξi+Δs22l2,δi+Δs·k3,ξi+Δs·l3),
{r⃗u(0)=r⃗(u0,v0)ur⃗v(0)=r⃗(u0,v0)v,
|r⃗u(0)|·Δu0sinβ=|r⃗v(0)|·Δv0sinα=Δssin(παβ).
{δ0=Δu0Δs=|τ^0×r^v(0)||r⃗u(0)|·|r^u(0)×r^v(0)|ξ0=Δv0Δs=|r^u(0)×τ^0||r⃗v(0)|·|r^u(0)×r^v(0)|.
r⃗1(u,1)=r⃗2(u,0).
u=av+b.
{a=unun1vnvn1b=ununun1vnvn1vn.
uint=avint+b=a+b.
{δn+1=Δun+1Δs=|τ^n+1×r^v(n+1)||r⃗u(n+1)|·|r^u(n+1)×r^v(n+1)|ξn+1=Δvn+1Δs=|r^u(n+1)×τ^n+1||r⃗v(n+1)|·|r^u(n+1)×r^v(n+1)|,
{r⃗u(n+1)=r⃗2(un+1,vn+1)ur⃗v(n+1)=r⃗2(un+1,vn+1)v.

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