-
2008 International Conference on Emerging TechnologiesIEEE-ICET
2008Rawalpindi, Pakistan, 18-19 October, 2008
Interpolation with rational cubic spirals
Agha Ali Razaa, Zulfiqar Habibb and Manabu Sakaic
a,b Department ofComputer Science, National University
ofComputer & Emerging Sciences (FAST-NU), Lahore, Pakistan
C Department ofMathematics and Computer Science,University
ofKagoshima, Kagoshima, Japan 890-0065
aali. raza@nu. edu.pk, bzulfiqar. habib@nu. edu.pk,
[email protected]
2. Description of method
Let Pb i = 0, 1, 2, 3 be four given control points. Thecubic
Bezier curve defined by them has nine degrees offreedom and is
represented in form as
( )- (1-t)3 PO +3t(1-t)2p1+3t2(1-t)P2+ t3 P3 0 < t < 1
z t - (1-t)3+3w(1-t)2t+3W(1-t)t 2+t3 ' - -(2.1)
Note that the above rational cubic curve reduces to ausual cubic
one for a choice of w =1. Its signed curvature1(t) is given by
where "x", "." and II • II mean the cross, inner productof two
vectors and the Euclidean norm, respectively.Without loss of
generality, we define spirals to be planararcs with non-negative
curvature and continuous non-zeroderivative of the curvature. This
paper assumes that(i) Po = (-1, 0) and P3 = (1, 0), (ii) the
tangent anglebetween the tangent vector and the vector (1, 0) at t
= 0 is
(2.2)z' (t)xz" (t)
K(t) = IIz'(t)311
would become smaller or even empty as the tangentangles become
relatively equal. .
This paper considers more flexible rational CUbICspirals to
overcome the above mentioned problem. Sincethere is no closed form
for the roots of the derivative ofthe curvature (of degree 12 for
this rational cubic case),we first transform the unit interval [0,
1] to [0, (0) andthen derive a sufficient spiral region with
respect to theparameters introduced under the fixed tangent angles
~tthe endpoints [1]. Section 3 represents the parameters Interms of
the endpoint curvatures and next derives thespiral regions with
respect to the curvatures under thefixed tangent directions.
Section 4 gives the numericallydetermined spiral regions which help
us adjust theendpoint curvatures and illustrative numerical
examplesto demonstrate the flexibility of our rational cubic
splinemethod, especially for the case when the tangent anglesare
relatively equal.
Rational cubic curve, Curvature, Spiral, Path planning.
In visually pleasing curve and surface design, it isoften
desirable to have a spiral transition curve withoutextraneous
curvature extrema. The purpose may bepractical [Gibreel et
al.(1999)], e.g., in highway designs,railway routes and aesthetic
applications [Burchard et al.(1993)] [1]. Since it is not easy to
locate the zeros of thederivative of the curvature of a high degree
curve,therefore simplified cases have been developed to join
thefollowing pairs of objects: (i) straight line to circle,
(ii)circle to circle with a broken back C transition, (iii)
circleto circle with an S transition, (iv) straight line to
straightline and (v) circle to circle with one circle inside the
other([3] - [6]). For other works on spirals that are rational
andpolynomial functions and match end conditions please see[1].
Dietz and Piper have numerically computed viableregions for the
cubic spirals to aid in adjusting theselection of control points
without zero curvaturerestriction [1]. These viable regions allow
the constructionof spirals joining one circle to another circle
where onelies inside the other by Kneser's theorem (case (v»
[2].Their results have enabled us to use a single cubic curverather
than two segments for the benefit that designersand implementers
have fewer entities to be concernedwith. However, their numerically
obtained spiral regions
Abstract
Keywords
1. Introduction
Spiral curves of one-sided, monotone increasing ordecreasing
curvature have the advantage that theminimum and maximum curvature
is at their endpointsand they contain no inflection points or local
curvatureextrema. This paper derives a spiral condition for
arational cubic spline on the endpoint curvatures under thefixed
positional and tangential conditions and providesmoreflexible
spiral regions.
978-1-4244-2211-1/08/$25.00 ©2008 IEEE
2008 International Conference on Emerging TechnologiesIEEE-ICET
2008Rawalpindi, Pakistan, 18-19 October, 2008
Interpolation with rational cubic spirals
Agha Ali Razaa, Zulfiqar Habibb and Manabu Saka{a,b Department
ojComputer Science, National University. oj
Computer & Emerging Sciences (FAST-NU), Lahor~, PakistanC
Department ojMathematics and Computer SCience,
University ojKagoshima, Kagoshima, Japan 890-0065
[email protected], [email protected],
[email protected]
2. Description of method
Let Ph i = 0, I, 2, 3 be four given control points. Thecubic
Bhier curve defined by them has nine degrees offreedom and is
represented in form as
( )- (1-t)3PO +3t(1-t)2p1 +3t2(l-t)P2+t 3P3 0 < t < 1
z t - (1-t)3+3W(1-t)2t+3w(1-t)t2+t3 ' _ _(2.1)
Note that the above rational cubic curve reduces to ausual cubic
one for a choice of W = I. Its signed curvatureK(t) is given by
where "x", "." and II • II mean the cross, inner productof two
vectors and the Euclidean norm, respectively.Without loss of
generality, we define spirals to be planararcs with non-negative
curvature and continuous non-zeroderivative of the curvature. This
paper assumes that(i) Po = (-I, 0) and P3 = (l, 0), (ii) the
tangent anglebetween the tangent vector and the vector (I, 0) at t
= 0 is
(2.2)z' (t)xz" (t)
K(t) = IIz'(t)311
would become smaller or even empty as the tangentangles become
relatively equal. .
This paper considers more flexible rational cubiCspirals to
overcome the above mentioned problem. Sincethere is no closed form
for the roots of the derivative ofthe curvature (of degree 12 for
this rational cubic case),we first transform the unit interval [0,
I] to [0, (0) andthen derive a sufficient spiral region with
respect to theparameters introduced under the fixed tangent angles
atthe endpoints [I]. Section 3 represents the parameters interms of
the endpoint curvatures and next derives thespiral regions with
respect to the curvatures under thefixed tangent directions.
Section 4 gives the numericallydetermined spiral regions which help
us adjust theendpoint curvatures and illustrative numerical
examplesto demonstrate the flexibility of our rational cubic
splinemethod, especially for the case when the tangent anglesare
relatively equal.
Rational cubic curve, Curvature, Spiral, Path planning.
Abstract
1. Introduction
Keywords
In visually pleasing curve and surface design, it isoften
desirable to have a spiral transition curve withoutextraneous
curvature extrema. The purpose may bepractical [Gibreel et
al.(1999)], e.g., in highway designs,railway routes and aesthetic
applications [Burchard et al.(1993)] [I], Since it is not easy to
locate the zeros of thederivative of the curvature of a high degree
curve,therefore simplified cases have been developed to join
thefollowing pairs of objects: (i) straight line to circle,
(ii)circle to circle with a broken back C transition, (iii)
circleto circle with an S transition, (iv) straight line to
straightline and (v) circle to circle with one circle inside the
other([3] - [6]). For other works on spirals that are rational
andpolynomial functions and match end conditions please see[I].
Dietz and Piper have numerically computed viableregions for the
cubic spirals to aid in adjusting theselection of control points
without zero curvaturerestriction [I]. These viable regions allow
the constructionof spirals joining one circle to another circle
where onelies inside the other by Kneser's theorem (case (v))
[2].Their results have enabled us to use a single cubic curverather
than two segments for the benefit that designersand implementers
have fewer entities to be concernedwith. However, their numerically
obtained spiral regions
Spiral curves of one-sided, monotone increasing ordecreasing
curvature have the advantage that theminimum and maximum curvature
is at their endpointsand they contain no inflection points or local
curvatureextrema. This paper derives a spiral condition for
arational cubic spline on the endpoint curvatures under thefixed
positional and tangential conditions and providesmoreflexible
spiral regions.
978-1-4244-2211-1/08/$25.00 ©2008 IEEE
-
denoted by ¢O, while the tangent angle at t = 1 is denotedby ¢JI
as shown in Figure 1 where the values for ¢o and tPIare constrained
so that 0 < ¢o < ¢JI < 1[/2 [2]. Theseconstraints restrict
our curves to spiral arcs of increasingcurvature. Then, for PI
==(uI, VI) and pz ==(uz, vz), notethat
VI = -(UI + w) tan ¢O, Vz = (uz - w) tan ¢JI (2.3)
As in [3], the tangent lines for the parametric cubic att = 0
and t = 1 and the horizontal axis form a trianglewhere the lengths
of the lower two sides are
do =2sin¢JI/sin(¢o+¢JI) (for the side touching Po),andd l
==2sintA/sin(¢JI+¢JI) (for the side touchingpI).
Note that
Ilz'(O)11 = 3(UI + w)/cos¢o,
and
Ilz'(I)11 == 3(w - uz)/cos¢o.
This paper extensively uses the ratio parameters ifo, fi):
(2.4)
Therefore, we have
(w + Uv w - uz) = m (~,JL),tan."o tantP1
where
m == 2sin¢osin t/JI/{3sin(¢o+t/JI)} ,
, , ,,,,,\\\\\III
n/1'p3
P2PI
Figure 1 Spiral transition between two circles with given
endpointPo and P3, tangents, and angles.
In order to convert the parameterization from the
strictcondition (0 :::; t :::; 1) to a relatively relaxed
condition(s ~ 0) the parameter t is substituted as t = 1/(1 + s) to
get,
94(1+s)4sintPosintPl ~~z .H. . lZ-i{1-(1-3w)s+s2}8 s in3 ( tPo+
tPl) ~l=Om l l (fo, tv tPo' tP1)s
(2.6)Where Hi ifo,fi; ¢O, ¢JI), 0 :::; i :::; 12 are functions
of/o, fi and¢o,t/JI.
The problem of designing cubic spirals could beformulated
analytically as finding conditions on thecoefficients of the above
12th degree polynomial to ensurethe non-positive zeros (for w = 1,
the polynomial reducesto the quintic one). As the necessary and
sufficientsolution to this problem is difficult to be found, we
give asufficient one from (2.6):
The rational transition cubic curve of the form (2.1) isa spiral
if
By Descartes Rule ofSigns, the rational cubic curve ofthe form
(2.1) is a spiral if2.7 holds.
3. Spiral conditions on the endpointcurvatures
The algebraically simple parameters ifo, fi) do not givedirect
information on how to match or approximate givencurvature values at
the endpoints [1]. This sectionconsiders the problem of finding
useful approximation to
the control points Pb i == 1, 2 are given by
PI (= (uI, VI)) = -w(l, 0) + mfo(cot ¢O, -1),pz(= (Uz, vz)) =
w(l, 0) - mfi(cot t/JI, 1). (2.5)
The following figure (Figure-I) shows the transitioncurve, two
tangents and two osculating circles atPi, i = 0, 3 where the points
Pb 0 ~ i ~ 3 are denotedcounterclockwise by four small discs for
(¢o, t/JI) = (0.7, 1)with (w,fO,fi) = (0.8, 1.6, 1.2).
Hi ifo, fi; ¢o, ¢JI) ~ 0,H 12-i (Ii, /0, t/JI; ¢o) :::; 0,for, 0
:::; i:::; 6. (2.7)
denoted by ¢O, while the tangent angle at t = 1 is denotedby ¢JI
as shown in Figure 1 where the values for ¢o and tPIare constrained
so that 0 < ¢o < ¢JI < 1[/2 [2]. Theseconstraints restrict
our curves to spiral arcs of increasingcurvature. Then, for PI
==(uI, VI) and pz ==(uz, vz), notethat
VI = -(UI + w) tan ¢O, Vz = (uz - w) tan ¢JI (2.3)
As in [3], the tangent lines for the parametric cubic att = 0
and t = 1 and the horizontal axis form a trianglewhere the lengths
of the lower two sides are
do =2sin¢JI/sin(¢o+¢JI) (for the side touching Po),andd l
==2sintA/sin(¢JI+¢JI) (for the side touchingpI).
Note that
Ilz'(O)11 = 3(UI + w)/cos¢o,
and
Ilz'(I)11 == 3(w - uz)/cos¢o.
This paper extensively uses the ratio parameters ifo, fi):
(2.4)
Therefore, we have
(w + Uv w - uz) = m (~,JL),tan."o tantP1
where
m == 2sin¢osin t/JI/{3sin(¢o+t/JI)} ,
, , ,,,,,\\\\\III
n/1'p3
P2PI
Figure 1 Spiral transition between two circles with given
endpointPo and P3, tangents, and angles.
In order to convert the parameterization from the
strictcondition (0 :::; t :::; 1) to a relatively relaxed
condition(s ~ 0) the parameter t is substituted as t = 1/(1 + s) to
get,
94(1+s)4sintPosintPl ~~z .H. . lZ-i{1-(1-3w)s+s2}8 s in3 ( tPo+
tPl) ~l=Om l l (fo, tv tPo' tP1)s
(2.6)Where Hi ifo,fi; ¢O, ¢JI), 0 :::; i :::; 12 are functions
of/o, fi and¢o,t/JI.
The problem of designing cubic spirals could beformulated
analytically as finding conditions on thecoefficients of the above
12th degree polynomial to ensurethe non-positive zeros (for w = 1,
the polynomial reducesto the quintic one). As the necessary and
sufficientsolution to this problem is difficult to be found, we
give asufficient one from (2.6):
The rational transition cubic curve of the form (2.1) isa spiral
if
By Descartes Rule ofSigns, the rational cubic curve ofthe form
(2.1) is a spiral if2.7 holds.
3. Spiral conditions on the endpointcurvatures
The algebraically simple parameters ifo, fi) do not givedirect
information on how to match or approximate givencurvature values at
the endpoints [1]. This sectionconsiders the problem of finding
useful approximation to
the control points Pb i == 1, 2 are given by
PI (= (uI, VI)) = -w(l, 0) + mfo(cot ¢O, -1),pz(= (Uz, vz)) =
w(l, 0) - mfi(cot t/JI, 1). (2.5)
The following figure (Figure-I) shows the transitioncurve, two
tangents and two osculating circles atPi, i = 0, 3 where the points
Pb 0 ~ i ~ 3 are denotedcounterclockwise by four small discs for
(¢o, t/JI) = (0.7, 1)with (w,fO,fi) = (0.8, 1.6, 1.2).
Hi ifo, fi; ¢o, ¢JI) ~ 0,H 12-i (Ii, /0, t/JI; ¢o) :::; 0,for, 0
:::; i:::; 6. (2.7)
-
Thus, q and r being both positive from (3.5), the two
positive roots are fromlo2 - JPIo + r = 0, i.e.
the admissible set of endpoint curvatures(ICo, ICI) (= (IC(O),
IC(I))) of a cubic rational spiral under thefixed positional and
tangential end conditions. For thisend, we analytically present the
parameters (fo, fi) in termsof (ICo, ICI) with (¢o, t/JI) fixed.
The endpoint curvatures aregiven as
p = ;(cz + ...rx.cos~) (~ cz) > 0 (3.8)
(3.1)
(3.9)
Forlo = (JP -.Jp - 4r)/2 andfi from (3.1), the spiralregion on
the endpoint curvatures (ICo, ICI) is given by
From 3.1, we have a quartic equation oflo:
(3.2)
where, with the conditions D ~ 0 and ICI > dl3.
We need to factorize (3.2) into the quadratics:
where q+r ==P-C2J r-q ==cI/JP and qr ==Co. Hence wehave
4. Numerically determined spiral regions
The HRegionPlot" command of Mathematica® drawsthe spiral regions
of (ICo, ICI) denoted by the insides of theclosed curves in Figs
2a, 3a and 4a where using a wellknown result that the osculating
circle at the endpoint Pois completely included in the osculating
one at the otherendpoint P3, we can restrict the possible region
asICO :::; sin¢o and ICI ~ (ICosint/JI- sin2
{(¢o+t/JI)/2})/(ICo-sin¢o).The spiral regions with respect to (ICo,
ICI) move fromdarker to lighter regions as the parameter w
increases. Ourrational cubic method gives a spiral even for(¢O,
t/JI) = (0.99, 1) where "NVR" (no viable region) isgiven in Table 1
[1]. The figures 2b, 3b and 4b show theplots of the spiral regions
for some values chosen fromwithin the spiral regions depicted in
the part-a of therespective figures. Finally the figures 2c, 3c and
4c are theplots of the curvature IC, showing its
monotonicallyincreasing trend with parameter t.
(3.3)
The system (3.1) has a solution (fo, fi) ofpositive pair ifICI
> d/3, andD(=256ICo2ICI(3ICI-d)+32dIdICoICI(3d - 8ICI) - d2cf) ~
o.
I' - KO ( 2_1'2) - J3d1dJl - -- U JO'u - W
d 1 dw KO
(i)
(ii)
(iii)
1 ( C1 )q = - p - C2--2 -JP '1 ( Cl)r = 2: p - C2 - -JP '
pep - C2)2 - 4coP - c;. (3.5)
The three numerical cases are presented below:
Case 1: (¢o, t/JI) = (0.6, 1) (Figures 2a, 2b, 2c)
Case 2: (¢o, t/JI) = (0.99, 1) (Figures 3a, 3b, 3c)
The cubic equation 3.8 (iii) has at least one positiveroot and
its positive one is given by
p=i(ZCz+ zft+ m q--1),q=;Cp+.yfP-4iP)(3.6)
Finally we give one more spiral region of (ICo, ICI)
todemonstrate usefulness of our rational spline method for(¢o,
t/JI) = (1, 1.01) while in [1] "NVR" is denoted even for(¢o, t/JI)
= (1, 1.2).
with (A, J..l) = (12co + c~, 27c; + 72coC2 - 2c~).Case 3: (¢o,
t/JI) = (1, 1.01) (Figures 4a, 4b, 4c)
Since KgKt(Lf - 4A3) = -272d6d8W12D (~ 0) (3.7)let q =re i8 (r =
A~, 0 :::; (J :::; 1C) to obtain
Thus, q and r being both positive from (3.5), the two
positive roots are fromfo2 - JPfo + r = 0, i.e.
the admissible set of endpoint curvatures(ICo, ICI) (= (IC(O),
IC(I))) of a cubic rational spiral under thefixed positional and
tangential end conditions. For thisend, we analytically present the
parameters (fo, fi) in termsof (ICo, ICI) with (¢o, t/JI) fixed.
The endpoint curvatures aregiven as
p = ;(cz + ...rx.cos~) (~ cz) > 0 (3.8)
(3.1)
(3.9)
Forfo = (JP -.Jp - 4r)/2 andfi from (3.1), the spiralregion on
the endpoint curvatures (ICo, ICI) is given by
From 3.1, we have a quartic equation offo:
(3.2)
where, with the conditions D ~ 0 and ICI > dl3.
We need to factorize (3.2) into the quadratics:
where q+r ==P-C2J r-q ==cI/JP and qr ==Co. Hence wehave
4. Numerically determined spiral regions
The HRegionPlot" command of Mathematica® drawsthe spiral regions
of (ICo, ICI) denoted by the insides of theclosed curves in Figs
2a, 3a and 4a where using a wellknown result that the osculating
circle at the endpoint Pois completely included in the osculating
one at the otherendpoint P3, we can restrict the possible region
asleo :::; sin¢o and ICI ~ (ICosint/JI- sin2
{(¢o+t/JI)/2})/(ICo-sin¢o).The spiral regions with respect to (ICo,
ICI) move fromdarker to lighter regions as the parameter w
increases. Ourrational cubic method gives a spiral even for(¢O,
t/JI) = (0.99, 1) where "NVR" (no viable region) isgiven in Table 1
[1]. The figures 2b, 3b and 4b show theplots of the spiral regions
for some values chosen fromwithin the spiral regions depicted in
the part-a of therespective figures. Finally the figures 2c, 3c and
4c are theplots of the curvature IC, showing its
monotonicallyincreasing trend with parameter t.
(3.3)
The system (3.1) has a solution (fo, fi) ofpositive pair ifICI
> d/3, andD(=256ICo2ICI(3ICI-d)+32dIdICoICI(3d - 8ICI) - d2cf) ~
o.
I' - KO ( 2_1'2) - J3d1dJl - -- U JO'u - W
d 1 dw KO
(i)
(ii)
(iii)
1 ( C1 )q = - p - C2--2 -JP '1 ( Cl)r = 2: p - C2 - -JP '
pep - C2)2 - 4coP - cf· (3.5)
The three numerical cases are presented below:
Case 1: (¢O, t/JI) = (0.6, 1) (Figures 2a, 2b, 2c)
Case 2: (¢O, t/JI) = (0.99, 1) (Figures 3a, 3b, 3c)
The cubic equation 3.8 (iii) has at least one positiveroot and
its positive one is given by
p=i(ZCz+ zft+ m q--1),q=;Cp+.yfP-4iP)(3.6)
Finally we give one more spiral region of (ICo, ICI)
todemonstrate usefulness of our rational spline method for(¢o,
t/JI) = (1, 1.01) while in [1] "NVR" is denoted even for(¢o, t/JI)
= (1, 1.2).
with (A, J..l) = (12co + c~, 27cf + 72coC2 - 2c~).Case 3: (¢O,
t/JI) = (1, 1.01) (Figures 4a, 4b, 4c)
Since KgKt(Lf - 4A3) = -272d6d8W12D (~ 0) (3.7)let q =re i8 (r =
A~, 0 :::; (J :::; 1C) to obtain
-
2.5
~2.0
1.5
0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55
10Figure-2a Spiral regions from darker to lighter as
w=0.6(0.1)1.0
0.860
0.855
0.850
0.845
0.810 0.815 0.820 0.825 0.830 0.835
10Figure-3a Spiral regions from darker to lighter as
w=0.64(0.02)0.72
0.840
Figure-3b Plot of the Spiral with(Ko, Kb w) = (0.825, 0.85,
0.72)
Figure-2b Plot of the Spiral with(Ko, Kb w) = (0.42, 1.6,
0.6)
-0.5-0.1
-0.2
-0.3
-0.4
-0.5
0.5 .0
1.4
1.2
1.0
0.8
0.6
0.0 ~2 0.4 O~ O~
Figure-2c Curvature plot with w=0.61.0
K(t)
0.850
0.845
0.840
0.835
0.830
0.2 0.4 0.6 0.8Figure-3c Curvature plot with w=0.72
1.0
0.8603.0
2.5
~2.0
1.5
0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55
0.855
0.850
0.845
0.810 0.815 0.820 0.825 0.830 0.835 0.840
toFigure-2a Spiral regions from darker to lighter as
w=O.6(0.1 )1.0
toFigure-3a Spiral regions from darker to lighter as
w=0.64(0.02)0.72
Figure-3b Plot of the Spiral with(Ko. KI, w) = (0.825, 0.85,
0.72)
.0-0.1
-0.2
-0.3
-0.4
-0.5
-0.05-0.10-0.15-0.20-0.25-0.30
Figure-2b Plot of the Spiral with(KO' Kj, w) = (0.42, 1.6,
0.6)
1.00.2 0.4 0.6 0.8Figure-3c Curvature plot with w=O.72
0.830
0.835
0.840
0.845
K(t)
0.850
1.00.2 0.4 0.6 0.8
Figure-2c Curvature plot with w=0.60.0
1.2
1.0
K(t)
1.6
1.4
0.6
0.8
-
0.860
0.855
0.850
0.820 0.825 0.830 0.835 0.840
10Figure-4a Spiral regions from darker to lighter as
w=0.65(0.025)0.7
5. Conclusion
The work presented in this paper is an attempt toimprove the
flexibility currently available in numericalspiral design
techniques as spirals are a useful part ofdesign and path planning
applications. The flexibilityprovided by the rational cubic spirals
is far greater thanthe normal cubic spiral which is used in [1]. As
a result,this method finds satisfactory solutions in many
caseswhere the method of [1] reports no viable region
(NVR),especially as ¢o--+¢JI. Case 3 presented in section 4 is
suchan example.
However, there is still a lot of room for future work inthis
area. For example, the behavior of spiral design when¢o = ¢JI
should be investigated. The possibility of circulararcs using this
approach should be scrutinized as rationalcurves allow circular
arcs. Besides, the question of singlespiral segment joining
concentric circles still stands as amajor challenge (previous works
allow this using multiplespiral segments, or non-spiral curves
([9], [10])). Andfinally, the formulation of necessary and
sufficientconditions for spiral arcs conforming to given spatial
andtangential end conditions remain as a significantmilestone.
Figure-4b Plot of the Spiral with(KO' Kb w) = (0.83, 0.855,
0.7)
-0.5-0.1
-0.2
-0.3
-0.4
-0.5
0.5 .06. References
[1] D. Dietz & B. Pier (2004), Interpolation with cubic
spirals,Computer Aided Geometric Design 21, 165-180.
[2] H. W. Guggenheimer (1963), Differential geometry, NewYork:
MacGraw-Hill.
[3] D. Walton & D. Meek (1996), A planar cubic
Bezierspiral,Journal of Computational and Applied Mathematics,
72,85-100.
K(t)
0.855
0.850
0.845
0.840
0.835
0.2 0.4 0.6 0.8Figure-4c Curvature plot with w=0.7
1.0
[4] D. Meek & D. Walton (1998), Planar spirals that match
G2hermite data, Computer Aided Geometric Deometric Design
15,103-126.
[5] D. Walton & D. Meek (1999), Planar G2 transition
betweentwo circles with a fair cubic Bezier curve, Computer
AidedDesign, 31, 857-866.
[6] D. Walton & D. Meek (2002), Planar G2 transition with
afair Pythagorean hodograph quintic curve, Journal ofComputational
and Applied Mathematics, 138, 109-126.
[7] M. D. Walton & D. Meek (2007), G2 curve design with
apair of Pythagorean Hodograph quintic spiral segments,Computer
Aided Geometric Design, 24, 267- 285.
[8] T.N.T. Goodman & D. Meek (2007), Planar
interpolationwith a pair of rational spirals, Journal of
Computational andApplied Mathematics, 201, 112-127.
0.820 0.825 0.830 0.835
toFigure-4a Spiral regions from darker to lighter as
w=0.65(0.025)0.7
-0.1
-0.2
-0.3
-0.4
-0.5
Figure-4b Plot ofthe Spiral with(KO, Kb w) = (0.83, 0.855,
0.7)
K(t)
0.855
0.850
0.845
0.840
0.835
02 0.4 0.6 0.8Figure-4c Curvature plot with w=0.7
0.840
.0
1.0
5. Conclusion
The work presented in this paper is an attempt toimprove the
flexibility currently available in numericalspiral design
techniques as spirals are a useful part ofdesign and path planning
applications. The flexibilityprovided by the rational cubic spirals
is far greater thanthe normal cubic spiral which is used in [I]. As
a result,this method finds satisfactory solutions in many
caseswhere the method of[l] reports no viable region
(NVR),especially as tAJ-tPJ. Case 3 presented in section 4 is
suchan example.
However, there is still a lot of room for future work inthis
area. For example, the behavior of spiral design whentAJ = tPJ
should be investigated. The possibility of circulararcs using this
approach should be scrutinized as rationalcurves allow circular
arcs. Besides, the question of singlespiral segment joining
concentric circles still stands as amajor challenge (previous works
allow this using multiplespiral segments, or non-spiral curves
([9], [10])). Andfinally, the formulation of necessary and
sufficientconditions for spiral arcs conforming to given spatial
andtangential end conditions remain as a significantmilestone.
6. References
[I] D. Dietz & B. Pier (2004), Interpolation with cubic
spirals,Computer Aided Geometric Design 21, 165-180.
[2] H. W. Guggenheimer (1963), Differential geometry, NewYork:
MacGraw-Hill.
[3] D. Walton & D. Meek (1996), A planar cubic
Bezierspiral,Joumal of Computational and Applied Mathematics,
72,85-100.
[4] D. Meek & D. Walton (1998), Planar spirals that match
G2hermite data, Computer Aided Geometric Deometric Design
15,103-126.
[5] D. Walton & D. Meek (1999), Planar G2 transition
betweentwo circles with a fair cubic Bezier curve, Computer
AidedDesign, 31, 857-866.
[6] D. Walton & D. Meek (2002), Planar G2 transition with
afair Pythagorean hodograph quintic curve, Journal ofComputational
and Applied Mathematics, 138, 109-126.
[7] M. D. Walton & D. Meek (2007), G2 curve design with
apair of Pythagorean Hodograph quintic spiral segments,Computer
Aided Geometric Design, 24, 267- 285.
[8] T.N.T. Goodman & D. Meek (2007), Planar
interpolationwith a pair of rational spirals, Journal of
Computational andApplied Mathematics, 201,112-127.
-
[9] Habib, Z., Sakai, M., 2005. Spiral transition curves and
theirapplications. Scientiae Mathematicae Japonicae 61 (2),
195-206. e2004, 251-262.http://www.jams.or.jp/semj0112004.html.
[10] Habib, Z., Sakai, M., 2007. On PH quintic spirals
joiningtwo circles with one circle inside the other.
Computer-AidedDesign 39 (2), 125-132.http://dx.doi.org/10.1016/j
.ead.2006.1 0.006.
[11] Habib, Z., Sakai, M., 2007. G2 Pythagorean hodographquintic
transition between circles with shape control. ComputerAided
Geometric Design 24,
252-266.http://dx.doi.org/lO.1016/j.eagd.2007.03.004.
[12] Habib, Z., Sakai, M., Sarfraz, M., 2004. Interactive
shapecontrol with rational cubic splines. Computer-Aided Design
&Applications 1 (1-4),709-718.
[13] Habib, Z., Sarfraz, M., Sakai, M., 2005. Rational
cubicspline interpolation with shape control. Computers &
Graphics29 (4),
594-605.http://dx.doi.org/lO.1016/j.eag.2005.05.010.
[14] Sakai, M., 1999. Inflection points and singularities
onplanar rational cubic curve segments. Computer AidedGeometric
Design 16, 149-156.
[15] Sarfraz, M., 2005. A rational cubic spline for
thevisualization of monotone data: An alternate approach.Computer
& Graphics 27, 107-121.
[9] Habib, Z., Sakai, M., 2005. Spiral transition curves and
theirapplications. Scientiae Mathematicae Japonicae 61 (2),
195-206.e2004,251-262.
http://www.jams.or.jp/scmjoI/2004.html.
[10] Habib, Z., Sakai, M., 2007. On PH quintic spirals
joiningtwo circles with one circle inside the other.
Computer-AidedDesign 39
(2),125-132.http://dx.doi.org/10.1016/j.cad.2006.10.006.
[11] Habib, Z., Sakai, M., 2007. G2 Pythagorean hodographquintic
transition between circles with shape control. ComputerAided
Geometric Design 24,
252-266.http://dx.doi.org/10.1016/j.cagd.2007.03.004.
[12] Habib, Z., Sakai, M., Sarfraz, M., 2004. Interactive
shapecontrol with rational cubic splines. Computer-Aided Design
&Applications 1 (1--4),709-718.
[13] Habib, Z., Sarfraz, M., Sakai, M., 2005. Rational
cubicspline interpolation with shape control. Computers &
Graphics29 (4),
594-605.http://dx.doi.org/lO.1016/j.cag.2005.05.010.
[14] Sakai, M., 1999. Inflection points and singularities
onplanar rational cubic curve segments. Computer AidedGeometric
Design 16, 149-156.
[15] Sarfraz, M., 2005. A rational cubic spline for
thevisualization of monotone data: An alternate approach.Computer
& Graphics 27,107-121.