Interpolation with rationalcubic spirals Interpolation...
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2008 International Conference on Emerging Technologies IEEE-ICET 2008 Rawalpindi, Pakistan, 18-19 October, 2008 Interpolation with rational cubic spirals Agha Ali Raza a , Zulfiqar Habib b and Manabu Sakai c a,b Department of Computer Science, National University of Computer & Emerging Sciences (FAST-NU), Lahore, Pakistan C Department of Mathematics and Computer Science, University ofKagoshima, Kagoshima, Japan 890-0065 aali. raza@nu. edu.pk, b[email protected], [email protected] 2. Description of method Let Pb i = 0, 1, 2, 3 be four given control points. The cubic Bezier curve defined by them has nine degrees of freedom and is represented in form as ( ) - (1-t)3 PO +3t(1-t)2 p1 +3t 2 (1-t)P2+ t3 P3 0 < t < 1 z t - (1-t)3+3w(1-t)2t+3W(1-t)t 2 +t 3 ' - - (2.1) Note that the above rational cubic curve reduces to a usual cubic one for a choice of w = 1. Its signed curvature 1(t) is given by where "x", "." and II • II mean the cross, inner product of two vectors and the Euclidean norm, respectively. Without loss of generality, we define spirals to be planar arcs with non-negative curvature and continuous non-zero derivative of the curvature. This paper assumes that (i) Po = (-1, 0) and P3 = (1, 0), (ii) the tangent angle between 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 tangent angles become relatively equal. . This paper considers more flexible rational CUbIC spirals to overcome the above mentioned problem. Since there is no closed form for the roots of the derivative of the curvature (of degree 12 for this rational cubic case), we first transform the unit interval [0, 1] to [0, (0) and then derive a sufficient spiral region with respect to the parameters introduced under the fixed tangent angles the endpoints [1]. Section 3 represents the parameters In terms of the endpoint curvatures and next derives the spiral regions with respect to the curvatures under the fixed tangent directions. Section 4 gives the numerically determined spiral regions which help us adjust the endpoint curvatures and illustrative numerical examples to demonstrate the flexibility of our rational cubic spline method, especially for the case when the tangent angles are relatively equal. Rational cubic curve, Curvature, Spiral, Path planning. In visually pleasing curve and surface design, it is often desirable to have a spiral transition curve without extraneous curvature extrema. The purpose may be practical [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 the derivative of the curvature of a high degree curve, therefore simplified cases have been developed to join the following pairs of objects: (i) straight line to circle, (ii) circle to circle with a broken back C transition, (iii) circle to circle with an S transition, (iv) straight line to straight line and (v) circle to circle with one circle inside the other ([3] - [6]). For other works on spirals that are rational and polynomial functions and match end conditions please see [1]. Dietz and Piper have numerically computed viable regions for the cubic spirals to aid in adjusting the selection of control points without zero curvature restriction [1]. These viable regions allow the construction of spirals joining one circle to another circle where one lies inside the other by Kneser's theorem (case (v» [2]. Their results have enabled us to use a single cubic curve rather than two segments for the benefit that designers and implementers have fewer entities to be concerned with. However, their numerically obtained spiral regions Abstract Keywords 1. Introduction Spiral curves of one-sided, monotone increasing or decreasing curvature have the advantage that the minimum and maximum curvature is at their endpoints and they contain no inflection points or local curvature extrema. This paper derives a spiral condition for a rational cubic spline on the endpoint curvatures under the fixed positional and tangential conditions and provides moreflexible spiral regions. 978-1-4244-2211-1/08/$25.00 ©2008 IEEE 2008 International Conference on Emerging Technologies IEEE-ICET 2008 Rawalpindi, Pakistan, 18-19 October, 2008 Interpolation with rational cubic spirals Agha Ali Raza a , Zulfiqar Habib b and Manabu Saka{ a,b Department ojComputer Science, National University. oj Computer & Emerging Sciences (FAST-NU), Pakistan C Department oj Mathematics 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. The cubic Bhier curve defined by them has nine degrees of freedom and is represented in form as ( ) - (1-t)3 PO +3t(1-t)2 p1 +3t 2 (l-t)P2+t 3 P3 0 < t < 1 z t - (1-t)3+3W(1-t)2t+3w(1-t)t 2+t 3 ' _ _ (2.1) Note that the above rational cubic curve reduces to a usual cubic one for a choice of W = I. Its signed curvature K(t) is given by where "x", "." and II • II mean the cross, inner product of two vectors and the Euclidean norm, respectively. Without loss of generality, we define spirals to be planar arcs with non-negative curvature and continuous non-zero derivative of the curvature. This paper assumes that (i) Po = (-I, 0) and P3 = (l, 0), (ii) the tangent angle between 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 tangent angles become relatively equal. . This paper considers more flexible rational cubiC spirals to overcome the above mentioned problem. Since there is no closed form for the roots of the derivative of the curvature (of degree 12 for this rational cubic case), we first transform the unit interval [0, I] to [0, (0) and then derive a sufficient spiral region with respect to the parameters introduced under the fixed tangent angles at the endpoints [I]. Section 3 represents the parameters in terms of the endpoint curvatures and next derives the spiral regions with respect to the curvatures under the fixed tangent directions. Section 4 gives the numerically determined spiral regions which help us adjust the endpoint curvatures and illustrative numerical examples to demonstrate the flexibility of our rational cubic spline method, especially for the case when the tangent angles are relatively equal. Rational cubic curve, Curvature, Spiral, Path planning. Abstract 1. Introduction Keywords In visually pleasing curve and surface design, it is often desirable to have a spiral transition curve without extraneous curvature extrema. The purpose may be practical [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 the derivative of the curvature of a high degree curve, therefore simplified cases have been developed to join the following pairs of objects: (i) straight line to circle, (ii) circle to circle with a broken back C transition, (iii) circle to circle with an S transition, (iv) straight line to straight line and (v) circle to circle with one circle inside the other ([3] - [6]). For other works on spirals that are rational and polynomial functions and match end conditions please see [I]. Dietz and Piper have numerically computed viable regions for the cubic spirals to aid in adjusting the selection of control points without zero curvature restriction [I]. These viable regions allow the construction of spirals joining one circle to another circle where one lies inside the other by Kneser's theorem (case (v)) [2]. Their results have enabled us to use a single cubic curve rather than two segments for the benefit that designers and implementers have fewer entities to be concerned with. However, their numerically obtained spiral regions Spiral curves of one-sided, monotone increasing or decreasing curvature have the advantage that the minimum and maximum curvature is at their endpoints and they contain no inflection points or local curvature extrema. This paper derives a spiral condition for a rational cubic spline on the endpoint curvatures under the fixed positional and tangential conditions and provides moreflexible spiral regions. 978-1-4244-2211-1/08/$25.00 ©2008 IEEE
Transcript of Interpolation with rationalcubic spirals Interpolation...
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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
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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)
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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.
