Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol102.Issue1.1999:
Journal of Computational and Applied Mathematics 102 (1999) 37–47
The generalised Cornu spiral and its application
to span generation
J.M. Ali a , R.M. Tookey b , J.V. Ball b , A.A. Ball b; ∗
b
a
School of Mathematical Sciences, The University of Science Malaysia, 11800 Minden, Penang, Malaysia
School of Manufacturing and Mechanical Engineering, The University of Birmingham, Edgbaston, West Midlands,
B152TT, United Kingdom
Received 12 January 1998; received in revised form 23 June 1998
Abstract
A Cornu spiral is a plane curve having a linear curvature pro le. This paper considers plane curves having rational linear
curvature pro les. These curves are de ned to be generalised Cornu spirals (GCS) and are quality curves in the sense
that they are continuous and smooth, can contain one in
ection at most, and have a bounded and monotonic curvature
pro le. In addition, the GCS has an extra degree of freedom over the Cornu spiral that is available for shape control.
Starting from the intrinsic equation of the GCS, the technique of curve synthesis is used to design a quality curve that
c 1999 Elsevier Science B.V. All rights reserved.
can be applied to a wide range of span generation problems.
Keywords: Generalised Cornu spiral; Curve synthesis; Span generation
1. Introduction
Over the last 25 years, there has been an increasing interest in the synthesis of curves and
surfaces for Computer-Aided Design. This has stemmed from engineers wanting to design surfaces
with guaranteed geometric properties, rather than the more customary, and time-consuming, practice
of constructing a surface and then interrogating and re ning it until it has the necessary characteristics
[9]. In other words, engineers want to ensure quality by design.
Much of the current research was initiated by Nutbourne et al. in 1972 [10]. They developed a
technique for constructing plane curves by integrating their curvature pro les. The work is based
upon the Fundamental Theorem for Space Curves which states that if two single-valued continuous
functions (s) and (s), s ¿ 0, are given, then there exists a unique space curve r(s), determined up
to a rigid-body motion, for which s is the arc-length, the curvature and the torsion [14]. Setting
∗
Corresponding author. Tel: 121 414 4150; fax: 121 414 3515.
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 2 0 7 - 6
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J.M. Ali et al. / Journal of Computational and Applied Mathematics 102 (1999) 37–47
(s) = 0, the theorem can be applied to plane curves. Various curvature pro les were investigated in
[10] and applied to several span generation problems.
There has been particular interest in linear curvature segments which correspond to arcs of Cornu
spirals. Pal and Nutbourne [12] proved that at least two linear curvature segments are required to
interpolate end points, tangents and curvatures, and discussed quality issues for solutions with two
and three segments. Schechter [13] presented an interactive approach requiring up to ve segments
for a satisfactory span synthesis. More recently Meek and Walton [7] have presented an alternative
solution with conditions for existence and uniqueness.
This paper presents a practical engineering solution to the span generation problem, using a single
arc from a generalised Cornu spiral (GCS) [2]. The GCS has a rational linear curvature pro le and,
by de nition, is more versatile than the Cornu spiral in shape description. It is a quality curve in the
sense that it has a monotonic curvature pro le [3, 6]. However it is not in general possible for a GCS
(nor any curve with a monotonic curvature pro le) to interpolate speci ed end points, tangents and
curvatures. For example if the two end curvatures are equal then the blend curve must be a circular
arc which is unlikely to be geometrically consistent with the endpoints and tangents. In engineering
applications it is often preferable to resolve such inconsistencies by introducing small discontinuities
in curvature [4] rather than comprising the monotonic variation of curvature. Consequently, the
GCS construction in this paper matches the end points and tangents but only approximates the end
curvatures.
2. Intrinsic equation of a curve
Without any loss of generality, this paper considers arc-length s parametrised curves lying in the
xy plane, i.e., r(s) = [x(s) y(s)], 0 6 s 6 S, where S corresponds to the total arc-length. For plane
curves, it is straightforward to de ne the unit tangent vector t(s), the unit normal vector n(s) and
the (signed) curvature (s) directly from r(s) using
dx(s) dy(s)
t(s) =
;
ds
ds
dy(s) dx(s)
n(s) = −
ds
ds
and
dx(s) d 2y(s) dy(s) d 2 x(s)
−
;
ds
ds2
ds ds2
where a positive (s) corresponds to a counter clockwise rotation of the tangent.
Conversely, for a given curvature pro le (s), 0 6 s 6 S, it is possible by successive integrations
to determine t(s) and r(s) up to a rigid-body motion [14]. This process is referred to as curve
synthesis.
Let (s), 0 6 s 6 S, be de ned as the (signed) angle (in radians) from the positive x axis to t(s)
such that
(s) =
t(s) = [cos (s) sin (s)];
where
(s) = (0) +
Z
0
s
(t) dt:
(1)
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J.M. Ali et al. / Journal of Computational and Applied Mathematics 102 (1999) 37–47
Integrating again gives
r(s) = x(0) +
Z
s
cos (0) +
0
Z
t
(u) du dt y(0) +
0
Z
0
s
sin (0) +
Z
0
t
(u) du
dt :
(2)
Eq. (2) thus provides the equation of the curve from its curvature pro le. The integrals contained in
the above equation, called Fresnel integrals, cannot be integrated in general and numerical techniques
are required to generate points on the curve (cf. [10, 1]).
3. Generalised Cornu spiral
A GCS is de ned to be a curve having a rational linear curvature pro le [2]. In its normalised
form, (s), 0 6 s 6 S, is given by
(s) =
p + qs
;
S + rs
(3)
where p, q, r (¿ − 1) and S (¿ 0) are the free parameters of the arc. The above form degenerates
to Logarithmic and Cornu spirals when q = 0 and r =0, respectively. Note that r ¿ − 1 ensures (s)
is well behaved and continuous over 0 6 s 6 S. The GCS can in
ect at most once, at s =−p=q.
Di erentiating Eq. (3) with respect to s gives
d(s)
Sq − pr
=
:
ds
(S + rs)2
Since d(s)=ds, 0 6 s 6 S, does not change sign then (s) is monotonic with respect to arc-length
and hence bounded by its end values p=S and (p + qS)=(1 + r)S, respectively. In addition, (s) is
di erentiable (smooth) since d(s)=ds is continuous.
As noted at the end of the previous section, there is no explicit curve equation for the GCS unless
the curvature pro le is linear [10], i.e., r = 0. Suppose r 6= 0, then Eq. (3) can be integrated to give
(pr − Sq)
rs
(s) = (0) +
ln 1 +
2
r
S
+
qs
;
r
(4)
and Eq. (2) has to be integrated numerically. The four available degrees of freedom, namely p, q,
r and S, together with the predictable nature of the in
ection, should provide sucient
exibility
for a wide range of shapes. In the next section, the span generation problem is framed with respect
to the GCS.
4. Span generation
The GCS has proven shape qualities and is therefore an ideal candidate as a blend or transition
curve [6]. This section formulates the span generation problem with respect to the GCS.
Consider two distinct points r0 and r1 with speci ed unit tangent vectors t0 and t1 and speci ed
curvatures 0 and 1 . Suppose 0 and 1 are associated with t0 and t1 respectively. The ideal solution
to the span generation problem is a single curve segment r(s), 0 6 s 6 S, that matches the data at its
40
J.M. Ali et al. / Journal of Computational and Applied Mathematics 102 (1999) 37–47
Fig. 1. Family of GCSs with various values of the shape factor r.
ends. Note that, as yet, the length of the span S has not been speci ed. The problem is approached
from an intrinsic perspective, hence the curvature constraints are rst considered, followed by the
tangent and then the position constraints.
For the GCS to match the end curvatures, (s) must satisfy (0) = 0 and (S) = 1 . Substituting
into Eq. (3) gives
(s) =
(S − s)0 + (1 + r)s1
S + rs
(5)
which eliminates p and q and leaves only r and S as free parameters. Since r controls the curvature
pro le and hence the shape of the GCS, it is referred to as the shape-factor.
Fig. 1 shows a family of GCSs with 0 = 0:010, 1 = 0:005, S = 100 and r =−0:9999; −0:9; −0:5;
0:0; 1:0; 2:0; 10:0 and 10 000. The most highly curved segment corresponds to r =−0:9999 and the
attest to r = 10 000. The gure demonstrates the extra versatility of shape control with a GCS
compared to the Cornu spiral (r = 0:0) whose ends are marked with small circles.
Integrating Eq. (5) (cf. Eq. (4)) gives
(s) = (0) + Sm(s)0 + (s − Sm(s))1 ;
(6)
where
S(1 + r)ln (1 + (rs=S)) − rs
(7)
Sr 2
is a continuous function of s provided r 6= 0. Expanding ln (1 + rs=S) as a Maclaurin series and then
substituting into the above expression gives
m(s) =
m(s) =
s
S
1−
s
2S
−
rs2
S2
1
s
−
2 3S
+
r 2 s3
S3
1
s
−
3 4S
+ o(r 2 )
for |rs=S| ¡ 1. It is clear that the limit of m(s) is well de ned as r approaches zero, hence m(s) is
a continuous function.
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J.M. Ali et al. / Journal of Computational and Applied Mathematics 102 (1999) 37–47
Provided 0 6= 1 , then Eqs. (1) and (6) can be equated to give an alternative expression for m(s),
i.e.,
m(s) =
Rs
0
(t) dt − s1
:
S(0 − 1 )
Now both (s) and m(s) can be considered to be continuous functions of r as well as s, i.e., (r; s)
and m(r; s). Partial di erentiation of Eq. (5) with respect to r gives
@(r; s)
s(S − s)(0 − 1 )
;
=−
@r
(S + rs)2
which is also a continuous function of r and s, hence Leibniz’s rule [15] can be applied to give
@m(r; s)
=−
@r
Z
s
0
t(S − t)
dt;
S(S + rt)2
i.e., @m(r; s)=@r ¡ 0, r ¿ − 1 and 06s6S. This implies m(r; s) is strictly single valued (monotonic)
with respect to r and since the limits of m(r; s) as r approaches −1 and ∞ are s=S and 0 respectively,
then 0 ¡ m(r; s) ¡ s=S.
For the GCS to match the end tangents, (s) must satisfy (0) = 0 and (S) = 1 . Substituting
into Eq. (6) gives
1 = 0 + Sm(S)0 + S(1 − m(S))1 :
(8)
This expression constrains the arc-length S. When 0 = 1 ,
S = (1 − 0 )=1 ;
which corresponds to a circular solution, i.e., a GCS with r = 0; when 0 6= 1 , S must satisfy
0 −
1 − 0
S
1 − 0
− 1
S
¿ 0;
(9)
since 0 ¡ m(S) ¡ 1, and the GCS matching the end tangents and curvatures has the unique shapefactor r satisfying
(1 + r) ln(1 + r) − r 1 − 0 − S1
=
:
r2
S(0 − 1 )
(10)
Integrating Eq. (6) (cf. Eq. (2)) gives
r(s) =
x(0) +
y(0) +
Rs
cos(0 + Sm(t)0 + (t − Sm(t))1 ) dt
Rs
sin(0 + Sm(t)0 + (t − Sm(t))1 ) dt
0
0
:
(11)
Finally, for the GCS to match the end points, r(s) must satisfy r(0) = [x0 y0 ] and r(S) = [x1 y1 ].
Substituting into Eq. (11) gives
x1 = x0 +
Z
0
S
cos(0 + Sm(t)0 + (t − Sm(t))1 ) dt
(12)
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J.M. Ali et al. / Journal of Computational and Applied Mathematics 102 (1999) 37–47
and
y1 = y0 +
S
Z
sin(0 + Sm(t)0 + (t − Sm(t))1 ) dt:
(13)
0
Eqs. (12) and (13) can both be rewritten as nonlinear equations in r, using Eqs. (7) and (10).
They are unlikely to have a common solution, since x1 and y1 can be varied independently. This
con rms that a single GCS cannot in general match the speci ed end points, tangents and curvatures.
However, a practical solution is presented in the next section in which the end curvatures are only
approximated. In curvature approximation it is more meaningful to consider percentage errors rather
than absolute errors. As a rule of thumb, a 5% discrepancy in curvature is just visible irrespective
of the curvature value. Consequently, the general aim of the GCS construction is to minimise the
percentage discrepancies in the end curvatures, and the expectation is that for well-conditioned data
the discrepancies will be small.
0 and 1 were introduced as the speci ed end curvatures but throughout this section they have
also denoted the end curvatures of the GCS. Now that GCSs are to be constructed which do not
necessarily match the speci ed end curvatures, there is a need to clarify the notation. For consistency
with all the numbered equations in this section, 0 and 1 are de ned as the end curvatures of the
GCS. Indeed, Eq. (5) can be taken as the de nition of a GCS; with 0 , 1 , r ( ¿ − 1) and S ( ¿ 0)
being the free parameters of the arc. Consequently, in the next section, the values of 0 and 1 are
initially assigned to match the speci ed end curvatures but are subsequently adjusted by a minimal
amount so that the GCS matches the end points and tangents.
5. A practical solution to span generation
Without any loss of generality, the problem is approached via the special case where r0 is the
origin and t0 is the unit vector in the direction of the positive x axis, i.e., x0 = y0 = 0 = 0. It is
noted that the span data can be translated and rotated so that these conditions are satis ed and, after
construction, the GCS can be transformed back to the original datum. For convenience, it is assumed
that the span data satis es (r1 − r0 ) · t0 ¿ 0 and t0 · t1 6= − 1 and the GCS does not turn through
more than radians, hence x1 ¿ 0 and − ¡ 1 ¡ .
Let , −=2 ¡ ¡ =2, be the (signed) angle (in radians) from the start tangent vector t0 to the
chord vector r(S) − r0 on the GCS. Then, using x0 = y0 = 0 = 0,
= tan
−1
y(S)
:
x(S)
(14)
Suppose
tan−1
y(S)
x(S)
= tan−1
y1
x1
(15)
and x(S) ¿ 0, then the arc-length S and the curvature pro le (s) can be scaled to ensure Eqs. (12)
and (13) are satis ed. Let
=
s
x(S)2 + y(S)2
;
x12 + y12
J.M. Ali et al. / Journal of Computational and Applied Mathematics 102 (1999) 37–47
43
then the scaled values are given by S= and (s) [10]. The shape-factor r is invariant under
scaling [2].
Consider the constant curvature pro le corresponding to 0 = 1 = . From Eqs. (11) and (14),
this curvature pro le corresponds to
= tan−1
1 − cos 1
sin 1
=
1
2
and is not dependent on . This suggests the following algorithm for span generation. If 0 = 1
and 1 =2 = tan−1 (y1 =x1 ), i.e., the end points, tangents and curvatures are consistent with a circular
solution, then the GCS can be constructed. Now suppose 0 = 1 but 1 =2 6= tan−1 (y1 =x1 ), then either
0 or 1 needs adjusting to ensure there is a solution since it can be assumed both the end points
and tangents are xed. When 0 6= 1 , ranges for S and r can be determined from Eqs. (9) and (10),
respectively. The m¿0 pairs of values of r and S satisfying Eq. (15) could possibly be determined
using a search routine. If m ¿ 0, construct the GCS with nearest to unity. This corresponds to the
one that minimises the percentage discrepancies in end curvatures. If m = 0, adjust the end curvatures
by the minimum amount until m ¿ 0.
Although the above algorithm is conceptually simple, there are computational aspects that make
it dicult to implement. These usually result from the span data being ill-conditioned. This has led
to the development of a practical two-part solution. It rst attempts to nd an exact solution up to
a constant scaling of S and (s) by xing 0 and 1 and calculating a value for S. If unsuccessful,
it then nds an approximate solution by assigning S and calculating new values for 0 and 1 .
The algorithm rst checks for a circular solution, that is, if 0 = 1 and 1 =2 = tan−1 (y1 =x1 ), then
the GCS is constructed. If there is no circular solution and 0 6= 1 , then a range for the arc-length
Sa 6S6Sb is calculated using the bounds in [8]. This range is based on the two circular arcs that
interpolate {r0 ; r1 ; t0 } and {r0 ; r1 ; t1 } respectively and gives reasonable bounds on S. For stability
reasons, the algorithm calculates 0 = 0:90 + 0:11 and 1 = 0:10 + 0:91 . Provided 0 and 1 span
1 =S, i.e., (0 −1 =S)(1 =S −1 )¿0, for all values of S in Sa 6S6Sb , then a range for the shape-factor
ra 6r6rb can be calculated using Eq. (10) and the values of corresponding to ra and rb , i.e., a
and b , can be calculated. If ( a −tan−1 (y1 =x1 ))(tan−1 (y1 =x1 )− b )¿0, a successive bisection routine
is used to nd the value of ra 6r6rb satisfying Eq. (15) and then a GCS can be constructed.
If an exact solution has not been found, then an iterative routine is used to adjust 0 and 1
until a solution is found. Appropriate values for S = (Sa + Sb )=2, 0 and 1 are rst assigned before
the iteration can begin. Since there is no circular solution, then either 0 or 1 needs adjusting if
(0 − 1 =S)(1 =S − 1 ) ¡ 0 or 0 = 1 = 1 =S. If 0 61 ¡ 1 =S or 1 =S ¡ 1 ¡ 0 , then 0 and 1 are
adjusted to 0′ and 1′ respectively using
0′ = (8:2S0 + 1:8S1 − 21 )=8S;
1′ = (181 − 1:8S0 − 8:2S1 )=8S:
These are the necessary values of 0′ and 1′ to give 0′ = 0 and 1′ = 21 =S − 1 , and hence (0′ −
1 =S)(1 =S −1′ ) ¿ 0. Similarly, if 1 ¡ 0 ¡ 1 =S or 1 =S ¡ 0 61 , then 0 and 1 are adjusted using
0′ = (181 − 8:2S0 − 1:8S1 )=8S;
1′ = (1:8S0 + 8:2S1 − 21 )=8S:
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J.M. Ali et al. / Journal of Computational and Applied Mathematics 102 (1999) 37–47
Finally, if 0 = 1 = 1 =S, then 0 and 1 are adjusted using
0′ = 1 =S − ;
1′ = 1 =S + ;
where ¿ 0 is some user-set curvature tolerance. Experiments have shown that = 0:000005
gives reasonable results. When 0 = 1 , the adjustments have arbitrarily ensured 0′ ¡ 1 =S ¡ 1′ . It
is easy to prove that these adjustments give (0′ − 1 =S)(1 =S − 1′ ) ¿ 0.
It is possible that 0 and 1 are still inappropriate for span generation. If ( − 1 =2)(1 =2−
tan−1 (y1 =x1 )) ¿ 0, then the curvature pro le needs re
ecting about 1 =S, i.e., 0 and 1 need adjusting
to 0′ and 1′ using
0 = 21 =S − 0 ;
1′ = 21 =S − 1 :
In essence, the condition checks whether the current chord angle and the desired chord angle
tan−1 (y1 =x1 ) span the circular chord angle 1 =2. If this is the case, then re
ection of the curvature
pro le ensures and tan−1 (y1 =x1 ) are to the same side of 1 =2. It is proved in the appendix that
this re
ection gives ( − 1 =2)(1 =2 − tan−1 (y1 =x1 )) ¡ 0.
The iteration loop is now described for nding the GCS. It aims to balance the changes to 0 and
1 so that they are both adjusted by proportionally the same amount. 0 is rst xed and a value 1′
found using the method of false position to construct a GCS whose chord angle lies nearer the
desired chord angle tan−1 (y1 =x1 ). The method returns a value for 1′ in the range (S1 + 0 )=2S and
(2S1 − 1 )=S which ensures 0 and 1′ span 1 =S. Similarly, 1 is xed and 0′ determined. Provided
either 0 6= 0′ or 1 6= 1′ , then ! is calculated using
v1 =(v0 + v1 );
0;
!=
1;
1=2;
0 6= 0; 1 6= 0;
0 = 0′ = 0; 1′ 6= 0 or 1 = 0′ = 1′ = 0;
1 = 1′ = 0; ′0 6= 0 or 0 = 0′ = 1′ = 0;
otherwise;
where v0 = |(0′ − 0 )=0 | and v1 = |(1′ − 1 )=1 |, and new values for 0 and 1 determined using
0′′ = 0 + !(0′ − 0 );
1′′ = 1 + (1 − !)(1′ − 1 )
and the iterative process repeated. The loop terminates if 0 = 0′ and 1 = 1′ : this suggests that
either the point or tangent data is inappropriate for the problem. Note that at any stage the loop can
be exited if Eq. (15) is satis ed and the GCS constructed.
6. Example
Table 1 gives seven examples of the GCS construction algorithm using r1 = [90:1879 38:0816],
t1 = [0:7542 0:6567]. The table gives the initial curvatures for each GCS, together with the nal
curvatures, the shape-factor and the overall arc-length. GCS 1 is an exact solution, while the other
six are approximate solutions. As a sequence, GCSs 2, 1 and 3 illustrate that a proportional change
to both initial curvatures results in an approximately proportional change to both nal curvatures.
This is to be expected since the algorithm attempts to balance the proportional changes to the
curvatures at each stage. As a sequence, GCSs 3, 4 and 5 have the same initial 0 , but decreasing
J.M. Ali et al. / Journal of Computational and Applied Mathematics 102 (1999) 37–47
45
Table 1
Examples of constructed GCSs
GCS
Initial curvatures (mm−1 )
0
1
Final curvatures (mm−1 )
0
1
Shape-factor
Arc-length
(mm)
1
2
3
4
5
6
7
0:01000
0:01050
0.00950
0.00950
0.00950
0.00500
−0.01000
0.01000
0.01046
0.00962
0.00945
0.00933
0.00932
0.00867
0:5000
1:2083
−0:0082
−0:1903
−0:3294
−0:3368
−0:8449
100.0000
99.9628
100.0410
100.0609
100.0798
100.0807
100.2146
0.00500
0.00525
0.00475
0.00450
0.00425
0.01000
0.00500
0.00500
0.00523
0.00469
0.00452
0.00432
0.00432
0.00179
values for 1 . Again, the algorithm behaves in a predictable manner with the nal curvatures varying
almost proportionally. It can be deduced therefore that the algorithm is stable with respect to small
changes to the initial curvatures provided the curvature estimates are reasonable. The last two GCSs
demonstrate that the algorithm is robust and can cope with ill-conditioned span data. Ideally, the
curvature pro le needs re
ecting for GCS 6 and the sign of 0 needs changing for GCS 7.
Finally it should be remarked that for all the well-conditioned sets of span data, the extra shape
control of the GCS enables the speci ed end curvatures to be matched within 2%, and the discrepancies are not detectable by eye. However the discrepancies using a Cornu spiral would be much
larger and visible in GCSs 4 and 5.
7. Conclusions
This paper has introduced the generalised Cornu spiral. In particular, the curvature pro le of a
Cornu spiral has been generalised from a linear to a rational linear function of arc-length. This
gives one additional degree of freedom that can be used to control the shape of the curve. It has
been demonstrated that the GCS provides a one-parameter family of solutions to the simpli ed span
generation problem when the curvature constraint is relaxed.
The algorithm described in this paper has been extended to three dimensions by considering two
planar views of the span data and constructing GCSs in each of these planes. When combined using
parallel projections, the resulting space curve is expected to be a quality curve since the two planar
GCSs are themselves quality curves. The algorithm works well in practice provided the span data is
nearly planar, that is, the torsion pro le is insigni cant. However, no consideration has been given to
the resulting curvature pro le on the three-dimensional GCS due to the construction based approach
rather than an analytic one.
The Fundamental Theorem for Space Curves states there is a unique space curve, determined up
to a rigid body motion, corresponding to given curvature and torsion pro les [14]. In essence, this
curve is the solution to the system of three simultaneous rst-order di erential equations known
as the Serret–Frenet equations. A more direct approach to the three-dimensional problem therefore
would be to solve the so-called Ricatti equation that is the general solution to the Serret–Frenet
equations. This was established over a hundred years ago by Darboux [5] and seems worthy of
further consideration.
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J.M. Ali et al. / Journal of Computational and Applied Mathematics 102 (1999) 37–47
Acknowledgements
The authors are pleased to acknowledge the support of the UK’s Engineering and Physical Sciences Research Council (GR/K 31268). The rst author would like to express his gratitude to The
University of Science Malaysia for a study grant under the Academic Sta Higher Education Scheme
that allowed him to complete his Ph.D. at The University of Birmingham.
Appendix. Re
ection of the curvature pro le
It is proved that if ( −1 =2)(1 =2−tan−1 (y1 =x1 )) ¿ 0, then re
ection of the curvature pro le about
1 =S, i.e., ′ (s) = 21 =S − (s), gives ( ′ − 1 =2)(1 =2 − tan−1 (y1 =x1 )) ¡ 0, where ′ corresponds to
the GCS with the re
ected curvature pro le. In essence, it is proved that ( − 1 =2)(1 =2 − ′ ) ¿ 0,
and hence the desired result follows immediately.
Consider the three curvature pro les a (s) = (s), b (s) = 1 =S and c (s) = 21 =S − (s) each
de ned over 06s6S, where S ¿ 0. a (s) is the original and c (s) the re
ected curvature pro le.
b (s) is a constant curvature pro le corresponding to the mean tangent rotational angle [11]. Recall
the de nition of , −=2 ¡ ¡ =2, from Eq. (14), and let a , b and c correspond to the curves
with a (s), b (s) and c (s) respectively. Since a = , b = 1 =2 and c = ′ , then ( −1 =2)(1 =2−
′ ) ¿ 0 if, and only if, ( a − Rb )( b − c ) ¿ 0.
Rt
Rt
t
For convenience, let fa (t) = 0 a () d, fb (t) = 0 b () d and fc (t) = 0 c () d, then there
are four cases to consider to complete the proof of ( a − b )( b − c ) ¿ 0, namely
(i) − =2 ¡ fa (t) ¡ fb (t) ¡ fc (t)60;
(ii) − =2 ¡ fa (t) ¡ fb (t)60 ¡ fc (t) ¡ =2;
(iii) − =2 ¡ fa (t) ¡ 0 ¡ fb (t) ¡ fc (t) ¡ =2;
(iv) 06fa (t) ¡ fb (t) ¡ fc (t) ¡ =2:
The rst two cases are proven here and then the other two follow mutatis mutandis.
(i) − =2 ¡ fa (t) ¡ fb (t) ¡ fc (t)60:
Let t ¿ 0, then 0 ¡ cos fa (t) ¡ cos fb (t) ¡ cos fc (t) and sin fa (t) ¡ sin fb (t) ¡ sin fc (t)60, since
the cosine function is positive, the sine function is negative and both functions are strictly increasing
in (−=2; 0]. Hence, 0 ¡ cos fa (t) ¡ cos fb (t) ¡ cos fc (t) and sin fa (t) ¡ sin fb (t) ¡ sin fc (t)60
for all t ¿ 0. This implies, since S ¿ 0, that
0¡
Z
S
cos fa (t) dt ¡
Z
S
cos fb (t) dt ¡
S
cos fc (t) dt
0
0
0
Z
and
Z
0
S
sin fa (t) dt ¡
Z
0
S
sin fb (t) dt ¡
Z
0
S
sin fc (t) dt60;
J.M. Ali et al. / Journal of Computational and Applied Mathematics 102 (1999) 37–47
47
and hence
RS
0
RS
0
sin fa (t) dt
cos fa (t) dt
¡
Rs
0
RS
0
sin fb (t) dt
cos fb (t) dt
RS
¡ R 0S
0
sin fc (t) dt
60;
cos fc (t) dt
i.e., tan a ¡ tan b ¡ tan c 60. Since tan is strictly increasing in (−=2; 0], then a ¡ b ¡ c
and ( a − b )( b − c ) ¿ 0.
(ii) − =2 ¡ fa (t) ¡ fb (t)60 ¡ fc (t) ¡ =2:
Let t ¿ 0, then 0 ¡
R S cos fc (t) and 0 ¡ sinRfSc (t). Hence, 0 ¡ cos fc (t) and
R S 0 ¡ sin fc (t)R Sfor all t ¿ 0.
This implies 0 ¡ 0 cos fc (t) dt and 0 ¡ 0 sin fc (t) dt, and hence 0 ¡ 0 sin fc (t) dt= 0 cos fc (t) dt,
i.e., 0 ¡ tan c . Now, tan a ¡ tan b 60 from case (i), and therefore it follows that tan a ¡ tan b
¡ tan c . Since tan is strictly increasing in (−=2; =2), then a ¡ b ¡ c and ( a − b )( b −
c ) ¿ 0.
References
[1] J.A. Adams, The intrinsic method for curve de nition, Comput.-Aided Des. 7 (1975) 243 – 249.
[2] J.M. Ali, Geometric control of planar curves, Ph.D. Thesis, The University of Birmingham, UK, 1994.
[3] A.A. Ball, CAD: Master or Servant of Engineering? The Mathematics of Surfaces VII, T. Goodman, R. Martin
(Eds.), Information Geometers, Winchester, UK, 1997, pp. 17 – 23.
[4] J.V. Ball, Generation of smooth curves and surfaces, Ph.D. Thesis, The University of Birmingham, UK, 1997.
[5] G. Darboux, Lecons sur la theorie generale des surfaces, vol. I, Gauthier-Villars, Paris, 1887.
[6] R.T. Farouki, Pythagorean-hodograph quintic transition curves of monotone curvature, Comput.-Aided Des. 29 (1997)
601– 606.
[7] D.S. Meek, D.J. Walton, Clothoid spline transition spirals, Math. Comp. 59 (1992) 117 – 133.
[8] D.S. Meek, D.J. Walton, Approximating quadratic NURBS curves by arc splines, Comput.-Aided Des. 25 (1993)
371– 376.
[9] A.W. Nutbourne, R.R. Martin, Di erential Geometry Applied to Curve and Surface Design, Ellis Horwood,
Chichester, UK, 1988.
[10] A.W. Nutbourne, P.M. McLellan, R.M.L. Kensit, Curvature pro les for plane curves, Comput.-Aided Des.
4 (1972) 176 –184.
[11] T.K. Pal, Mean tangent rotational angles and curvature integration, Comput.-Aided Des. 10 (1978) 30 – 34.
[12] T.K. Pal, A.W. Nutbourne, Two-dimensional curve synthesis using linear curvature elements, Comput.-Aided Des.
9 (1977) 121 – 134.
[13] A. Schechter, Synthesis of 2D curves by blending piecewise linear curvature pro les, Comput.-Aided Des. 10 (1978)
8 – 18.
[14] D.J. Struik, Lectures on Classical Di erential Geometry, Addison-Wesley, Reading, MA, 1950.
[15] A.E. Taylor, R.W. Mann, Advanced Calculus, Wiley, New York, 1983.
The generalised Cornu spiral and its application
to span generation
J.M. Ali a , R.M. Tookey b , J.V. Ball b , A.A. Ball b; ∗
b
a
School of Mathematical Sciences, The University of Science Malaysia, 11800 Minden, Penang, Malaysia
School of Manufacturing and Mechanical Engineering, The University of Birmingham, Edgbaston, West Midlands,
B152TT, United Kingdom
Received 12 January 1998; received in revised form 23 June 1998
Abstract
A Cornu spiral is a plane curve having a linear curvature pro le. This paper considers plane curves having rational linear
curvature pro les. These curves are de ned to be generalised Cornu spirals (GCS) and are quality curves in the sense
that they are continuous and smooth, can contain one in
ection at most, and have a bounded and monotonic curvature
pro le. In addition, the GCS has an extra degree of freedom over the Cornu spiral that is available for shape control.
Starting from the intrinsic equation of the GCS, the technique of curve synthesis is used to design a quality curve that
c 1999 Elsevier Science B.V. All rights reserved.
can be applied to a wide range of span generation problems.
Keywords: Generalised Cornu spiral; Curve synthesis; Span generation
1. Introduction
Over the last 25 years, there has been an increasing interest in the synthesis of curves and
surfaces for Computer-Aided Design. This has stemmed from engineers wanting to design surfaces
with guaranteed geometric properties, rather than the more customary, and time-consuming, practice
of constructing a surface and then interrogating and re ning it until it has the necessary characteristics
[9]. In other words, engineers want to ensure quality by design.
Much of the current research was initiated by Nutbourne et al. in 1972 [10]. They developed a
technique for constructing plane curves by integrating their curvature pro les. The work is based
upon the Fundamental Theorem for Space Curves which states that if two single-valued continuous
functions (s) and (s), s ¿ 0, are given, then there exists a unique space curve r(s), determined up
to a rigid-body motion, for which s is the arc-length, the curvature and the torsion [14]. Setting
∗
Corresponding author. Tel: 121 414 4150; fax: 121 414 3515.
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 2 0 7 - 6
38
J.M. Ali et al. / Journal of Computational and Applied Mathematics 102 (1999) 37–47
(s) = 0, the theorem can be applied to plane curves. Various curvature pro les were investigated in
[10] and applied to several span generation problems.
There has been particular interest in linear curvature segments which correspond to arcs of Cornu
spirals. Pal and Nutbourne [12] proved that at least two linear curvature segments are required to
interpolate end points, tangents and curvatures, and discussed quality issues for solutions with two
and three segments. Schechter [13] presented an interactive approach requiring up to ve segments
for a satisfactory span synthesis. More recently Meek and Walton [7] have presented an alternative
solution with conditions for existence and uniqueness.
This paper presents a practical engineering solution to the span generation problem, using a single
arc from a generalised Cornu spiral (GCS) [2]. The GCS has a rational linear curvature pro le and,
by de nition, is more versatile than the Cornu spiral in shape description. It is a quality curve in the
sense that it has a monotonic curvature pro le [3, 6]. However it is not in general possible for a GCS
(nor any curve with a monotonic curvature pro le) to interpolate speci ed end points, tangents and
curvatures. For example if the two end curvatures are equal then the blend curve must be a circular
arc which is unlikely to be geometrically consistent with the endpoints and tangents. In engineering
applications it is often preferable to resolve such inconsistencies by introducing small discontinuities
in curvature [4] rather than comprising the monotonic variation of curvature. Consequently, the
GCS construction in this paper matches the end points and tangents but only approximates the end
curvatures.
2. Intrinsic equation of a curve
Without any loss of generality, this paper considers arc-length s parametrised curves lying in the
xy plane, i.e., r(s) = [x(s) y(s)], 0 6 s 6 S, where S corresponds to the total arc-length. For plane
curves, it is straightforward to de ne the unit tangent vector t(s), the unit normal vector n(s) and
the (signed) curvature (s) directly from r(s) using
dx(s) dy(s)
t(s) =
;
ds
ds
dy(s) dx(s)
n(s) = −
ds
ds
and
dx(s) d 2y(s) dy(s) d 2 x(s)
−
;
ds
ds2
ds ds2
where a positive (s) corresponds to a counter clockwise rotation of the tangent.
Conversely, for a given curvature pro le (s), 0 6 s 6 S, it is possible by successive integrations
to determine t(s) and r(s) up to a rigid-body motion [14]. This process is referred to as curve
synthesis.
Let (s), 0 6 s 6 S, be de ned as the (signed) angle (in radians) from the positive x axis to t(s)
such that
(s) =
t(s) = [cos (s) sin (s)];
where
(s) = (0) +
Z
0
s
(t) dt:
(1)
39
J.M. Ali et al. / Journal of Computational and Applied Mathematics 102 (1999) 37–47
Integrating again gives
r(s) = x(0) +
Z
s
cos (0) +
0
Z
t
(u) du dt y(0) +
0
Z
0
s
sin (0) +
Z
0
t
(u) du
dt :
(2)
Eq. (2) thus provides the equation of the curve from its curvature pro le. The integrals contained in
the above equation, called Fresnel integrals, cannot be integrated in general and numerical techniques
are required to generate points on the curve (cf. [10, 1]).
3. Generalised Cornu spiral
A GCS is de ned to be a curve having a rational linear curvature pro le [2]. In its normalised
form, (s), 0 6 s 6 S, is given by
(s) =
p + qs
;
S + rs
(3)
where p, q, r (¿ − 1) and S (¿ 0) are the free parameters of the arc. The above form degenerates
to Logarithmic and Cornu spirals when q = 0 and r =0, respectively. Note that r ¿ − 1 ensures (s)
is well behaved and continuous over 0 6 s 6 S. The GCS can in
ect at most once, at s =−p=q.
Di erentiating Eq. (3) with respect to s gives
d(s)
Sq − pr
=
:
ds
(S + rs)2
Since d(s)=ds, 0 6 s 6 S, does not change sign then (s) is monotonic with respect to arc-length
and hence bounded by its end values p=S and (p + qS)=(1 + r)S, respectively. In addition, (s) is
di erentiable (smooth) since d(s)=ds is continuous.
As noted at the end of the previous section, there is no explicit curve equation for the GCS unless
the curvature pro le is linear [10], i.e., r = 0. Suppose r 6= 0, then Eq. (3) can be integrated to give
(pr − Sq)
rs
(s) = (0) +
ln 1 +
2
r
S
+
qs
;
r
(4)
and Eq. (2) has to be integrated numerically. The four available degrees of freedom, namely p, q,
r and S, together with the predictable nature of the in
ection, should provide sucient
exibility
for a wide range of shapes. In the next section, the span generation problem is framed with respect
to the GCS.
4. Span generation
The GCS has proven shape qualities and is therefore an ideal candidate as a blend or transition
curve [6]. This section formulates the span generation problem with respect to the GCS.
Consider two distinct points r0 and r1 with speci ed unit tangent vectors t0 and t1 and speci ed
curvatures 0 and 1 . Suppose 0 and 1 are associated with t0 and t1 respectively. The ideal solution
to the span generation problem is a single curve segment r(s), 0 6 s 6 S, that matches the data at its
40
J.M. Ali et al. / Journal of Computational and Applied Mathematics 102 (1999) 37–47
Fig. 1. Family of GCSs with various values of the shape factor r.
ends. Note that, as yet, the length of the span S has not been speci ed. The problem is approached
from an intrinsic perspective, hence the curvature constraints are rst considered, followed by the
tangent and then the position constraints.
For the GCS to match the end curvatures, (s) must satisfy (0) = 0 and (S) = 1 . Substituting
into Eq. (3) gives
(s) =
(S − s)0 + (1 + r)s1
S + rs
(5)
which eliminates p and q and leaves only r and S as free parameters. Since r controls the curvature
pro le and hence the shape of the GCS, it is referred to as the shape-factor.
Fig. 1 shows a family of GCSs with 0 = 0:010, 1 = 0:005, S = 100 and r =−0:9999; −0:9; −0:5;
0:0; 1:0; 2:0; 10:0 and 10 000. The most highly curved segment corresponds to r =−0:9999 and the
attest to r = 10 000. The gure demonstrates the extra versatility of shape control with a GCS
compared to the Cornu spiral (r = 0:0) whose ends are marked with small circles.
Integrating Eq. (5) (cf. Eq. (4)) gives
(s) = (0) + Sm(s)0 + (s − Sm(s))1 ;
(6)
where
S(1 + r)ln (1 + (rs=S)) − rs
(7)
Sr 2
is a continuous function of s provided r 6= 0. Expanding ln (1 + rs=S) as a Maclaurin series and then
substituting into the above expression gives
m(s) =
m(s) =
s
S
1−
s
2S
−
rs2
S2
1
s
−
2 3S
+
r 2 s3
S3
1
s
−
3 4S
+ o(r 2 )
for |rs=S| ¡ 1. It is clear that the limit of m(s) is well de ned as r approaches zero, hence m(s) is
a continuous function.
41
J.M. Ali et al. / Journal of Computational and Applied Mathematics 102 (1999) 37–47
Provided 0 6= 1 , then Eqs. (1) and (6) can be equated to give an alternative expression for m(s),
i.e.,
m(s) =
Rs
0
(t) dt − s1
:
S(0 − 1 )
Now both (s) and m(s) can be considered to be continuous functions of r as well as s, i.e., (r; s)
and m(r; s). Partial di erentiation of Eq. (5) with respect to r gives
@(r; s)
s(S − s)(0 − 1 )
;
=−
@r
(S + rs)2
which is also a continuous function of r and s, hence Leibniz’s rule [15] can be applied to give
@m(r; s)
=−
@r
Z
s
0
t(S − t)
dt;
S(S + rt)2
i.e., @m(r; s)=@r ¡ 0, r ¿ − 1 and 06s6S. This implies m(r; s) is strictly single valued (monotonic)
with respect to r and since the limits of m(r; s) as r approaches −1 and ∞ are s=S and 0 respectively,
then 0 ¡ m(r; s) ¡ s=S.
For the GCS to match the end tangents, (s) must satisfy (0) = 0 and (S) = 1 . Substituting
into Eq. (6) gives
1 = 0 + Sm(S)0 + S(1 − m(S))1 :
(8)
This expression constrains the arc-length S. When 0 = 1 ,
S = (1 − 0 )=1 ;
which corresponds to a circular solution, i.e., a GCS with r = 0; when 0 6= 1 , S must satisfy
0 −
1 − 0
S
1 − 0
− 1
S
¿ 0;
(9)
since 0 ¡ m(S) ¡ 1, and the GCS matching the end tangents and curvatures has the unique shapefactor r satisfying
(1 + r) ln(1 + r) − r 1 − 0 − S1
=
:
r2
S(0 − 1 )
(10)
Integrating Eq. (6) (cf. Eq. (2)) gives
r(s) =
x(0) +
y(0) +
Rs
cos(0 + Sm(t)0 + (t − Sm(t))1 ) dt
Rs
sin(0 + Sm(t)0 + (t − Sm(t))1 ) dt
0
0
:
(11)
Finally, for the GCS to match the end points, r(s) must satisfy r(0) = [x0 y0 ] and r(S) = [x1 y1 ].
Substituting into Eq. (11) gives
x1 = x0 +
Z
0
S
cos(0 + Sm(t)0 + (t − Sm(t))1 ) dt
(12)
42
J.M. Ali et al. / Journal of Computational and Applied Mathematics 102 (1999) 37–47
and
y1 = y0 +
S
Z
sin(0 + Sm(t)0 + (t − Sm(t))1 ) dt:
(13)
0
Eqs. (12) and (13) can both be rewritten as nonlinear equations in r, using Eqs. (7) and (10).
They are unlikely to have a common solution, since x1 and y1 can be varied independently. This
con rms that a single GCS cannot in general match the speci ed end points, tangents and curvatures.
However, a practical solution is presented in the next section in which the end curvatures are only
approximated. In curvature approximation it is more meaningful to consider percentage errors rather
than absolute errors. As a rule of thumb, a 5% discrepancy in curvature is just visible irrespective
of the curvature value. Consequently, the general aim of the GCS construction is to minimise the
percentage discrepancies in the end curvatures, and the expectation is that for well-conditioned data
the discrepancies will be small.
0 and 1 were introduced as the speci ed end curvatures but throughout this section they have
also denoted the end curvatures of the GCS. Now that GCSs are to be constructed which do not
necessarily match the speci ed end curvatures, there is a need to clarify the notation. For consistency
with all the numbered equations in this section, 0 and 1 are de ned as the end curvatures of the
GCS. Indeed, Eq. (5) can be taken as the de nition of a GCS; with 0 , 1 , r ( ¿ − 1) and S ( ¿ 0)
being the free parameters of the arc. Consequently, in the next section, the values of 0 and 1 are
initially assigned to match the speci ed end curvatures but are subsequently adjusted by a minimal
amount so that the GCS matches the end points and tangents.
5. A practical solution to span generation
Without any loss of generality, the problem is approached via the special case where r0 is the
origin and t0 is the unit vector in the direction of the positive x axis, i.e., x0 = y0 = 0 = 0. It is
noted that the span data can be translated and rotated so that these conditions are satis ed and, after
construction, the GCS can be transformed back to the original datum. For convenience, it is assumed
that the span data satis es (r1 − r0 ) · t0 ¿ 0 and t0 · t1 6= − 1 and the GCS does not turn through
more than radians, hence x1 ¿ 0 and − ¡ 1 ¡ .
Let , −=2 ¡ ¡ =2, be the (signed) angle (in radians) from the start tangent vector t0 to the
chord vector r(S) − r0 on the GCS. Then, using x0 = y0 = 0 = 0,
= tan
−1
y(S)
:
x(S)
(14)
Suppose
tan−1
y(S)
x(S)
= tan−1
y1
x1
(15)
and x(S) ¿ 0, then the arc-length S and the curvature pro le (s) can be scaled to ensure Eqs. (12)
and (13) are satis ed. Let
=
s
x(S)2 + y(S)2
;
x12 + y12
J.M. Ali et al. / Journal of Computational and Applied Mathematics 102 (1999) 37–47
43
then the scaled values are given by S= and (s) [10]. The shape-factor r is invariant under
scaling [2].
Consider the constant curvature pro le corresponding to 0 = 1 = . From Eqs. (11) and (14),
this curvature pro le corresponds to
= tan−1
1 − cos 1
sin 1
=
1
2
and is not dependent on . This suggests the following algorithm for span generation. If 0 = 1
and 1 =2 = tan−1 (y1 =x1 ), i.e., the end points, tangents and curvatures are consistent with a circular
solution, then the GCS can be constructed. Now suppose 0 = 1 but 1 =2 6= tan−1 (y1 =x1 ), then either
0 or 1 needs adjusting to ensure there is a solution since it can be assumed both the end points
and tangents are xed. When 0 6= 1 , ranges for S and r can be determined from Eqs. (9) and (10),
respectively. The m¿0 pairs of values of r and S satisfying Eq. (15) could possibly be determined
using a search routine. If m ¿ 0, construct the GCS with nearest to unity. This corresponds to the
one that minimises the percentage discrepancies in end curvatures. If m = 0, adjust the end curvatures
by the minimum amount until m ¿ 0.
Although the above algorithm is conceptually simple, there are computational aspects that make
it dicult to implement. These usually result from the span data being ill-conditioned. This has led
to the development of a practical two-part solution. It rst attempts to nd an exact solution up to
a constant scaling of S and (s) by xing 0 and 1 and calculating a value for S. If unsuccessful,
it then nds an approximate solution by assigning S and calculating new values for 0 and 1 .
The algorithm rst checks for a circular solution, that is, if 0 = 1 and 1 =2 = tan−1 (y1 =x1 ), then
the GCS is constructed. If there is no circular solution and 0 6= 1 , then a range for the arc-length
Sa 6S6Sb is calculated using the bounds in [8]. This range is based on the two circular arcs that
interpolate {r0 ; r1 ; t0 } and {r0 ; r1 ; t1 } respectively and gives reasonable bounds on S. For stability
reasons, the algorithm calculates 0 = 0:90 + 0:11 and 1 = 0:10 + 0:91 . Provided 0 and 1 span
1 =S, i.e., (0 −1 =S)(1 =S −1 )¿0, for all values of S in Sa 6S6Sb , then a range for the shape-factor
ra 6r6rb can be calculated using Eq. (10) and the values of corresponding to ra and rb , i.e., a
and b , can be calculated. If ( a −tan−1 (y1 =x1 ))(tan−1 (y1 =x1 )− b )¿0, a successive bisection routine
is used to nd the value of ra 6r6rb satisfying Eq. (15) and then a GCS can be constructed.
If an exact solution has not been found, then an iterative routine is used to adjust 0 and 1
until a solution is found. Appropriate values for S = (Sa + Sb )=2, 0 and 1 are rst assigned before
the iteration can begin. Since there is no circular solution, then either 0 or 1 needs adjusting if
(0 − 1 =S)(1 =S − 1 ) ¡ 0 or 0 = 1 = 1 =S. If 0 61 ¡ 1 =S or 1 =S ¡ 1 ¡ 0 , then 0 and 1 are
adjusted to 0′ and 1′ respectively using
0′ = (8:2S0 + 1:8S1 − 21 )=8S;
1′ = (181 − 1:8S0 − 8:2S1 )=8S:
These are the necessary values of 0′ and 1′ to give 0′ = 0 and 1′ = 21 =S − 1 , and hence (0′ −
1 =S)(1 =S −1′ ) ¿ 0. Similarly, if 1 ¡ 0 ¡ 1 =S or 1 =S ¡ 0 61 , then 0 and 1 are adjusted using
0′ = (181 − 8:2S0 − 1:8S1 )=8S;
1′ = (1:8S0 + 8:2S1 − 21 )=8S:
44
J.M. Ali et al. / Journal of Computational and Applied Mathematics 102 (1999) 37–47
Finally, if 0 = 1 = 1 =S, then 0 and 1 are adjusted using
0′ = 1 =S − ;
1′ = 1 =S + ;
where ¿ 0 is some user-set curvature tolerance. Experiments have shown that = 0:000005
gives reasonable results. When 0 = 1 , the adjustments have arbitrarily ensured 0′ ¡ 1 =S ¡ 1′ . It
is easy to prove that these adjustments give (0′ − 1 =S)(1 =S − 1′ ) ¿ 0.
It is possible that 0 and 1 are still inappropriate for span generation. If ( − 1 =2)(1 =2−
tan−1 (y1 =x1 )) ¿ 0, then the curvature pro le needs re
ecting about 1 =S, i.e., 0 and 1 need adjusting
to 0′ and 1′ using
0 = 21 =S − 0 ;
1′ = 21 =S − 1 :
In essence, the condition checks whether the current chord angle and the desired chord angle
tan−1 (y1 =x1 ) span the circular chord angle 1 =2. If this is the case, then re
ection of the curvature
pro le ensures and tan−1 (y1 =x1 ) are to the same side of 1 =2. It is proved in the appendix that
this re
ection gives ( − 1 =2)(1 =2 − tan−1 (y1 =x1 )) ¡ 0.
The iteration loop is now described for nding the GCS. It aims to balance the changes to 0 and
1 so that they are both adjusted by proportionally the same amount. 0 is rst xed and a value 1′
found using the method of false position to construct a GCS whose chord angle lies nearer the
desired chord angle tan−1 (y1 =x1 ). The method returns a value for 1′ in the range (S1 + 0 )=2S and
(2S1 − 1 )=S which ensures 0 and 1′ span 1 =S. Similarly, 1 is xed and 0′ determined. Provided
either 0 6= 0′ or 1 6= 1′ , then ! is calculated using
v1 =(v0 + v1 );
0;
!=
1;
1=2;
0 6= 0; 1 6= 0;
0 = 0′ = 0; 1′ 6= 0 or 1 = 0′ = 1′ = 0;
1 = 1′ = 0; ′0 6= 0 or 0 = 0′ = 1′ = 0;
otherwise;
where v0 = |(0′ − 0 )=0 | and v1 = |(1′ − 1 )=1 |, and new values for 0 and 1 determined using
0′′ = 0 + !(0′ − 0 );
1′′ = 1 + (1 − !)(1′ − 1 )
and the iterative process repeated. The loop terminates if 0 = 0′ and 1 = 1′ : this suggests that
either the point or tangent data is inappropriate for the problem. Note that at any stage the loop can
be exited if Eq. (15) is satis ed and the GCS constructed.
6. Example
Table 1 gives seven examples of the GCS construction algorithm using r1 = [90:1879 38:0816],
t1 = [0:7542 0:6567]. The table gives the initial curvatures for each GCS, together with the nal
curvatures, the shape-factor and the overall arc-length. GCS 1 is an exact solution, while the other
six are approximate solutions. As a sequence, GCSs 2, 1 and 3 illustrate that a proportional change
to both initial curvatures results in an approximately proportional change to both nal curvatures.
This is to be expected since the algorithm attempts to balance the proportional changes to the
curvatures at each stage. As a sequence, GCSs 3, 4 and 5 have the same initial 0 , but decreasing
J.M. Ali et al. / Journal of Computational and Applied Mathematics 102 (1999) 37–47
45
Table 1
Examples of constructed GCSs
GCS
Initial curvatures (mm−1 )
0
1
Final curvatures (mm−1 )
0
1
Shape-factor
Arc-length
(mm)
1
2
3
4
5
6
7
0:01000
0:01050
0.00950
0.00950
0.00950
0.00500
−0.01000
0.01000
0.01046
0.00962
0.00945
0.00933
0.00932
0.00867
0:5000
1:2083
−0:0082
−0:1903
−0:3294
−0:3368
−0:8449
100.0000
99.9628
100.0410
100.0609
100.0798
100.0807
100.2146
0.00500
0.00525
0.00475
0.00450
0.00425
0.01000
0.00500
0.00500
0.00523
0.00469
0.00452
0.00432
0.00432
0.00179
values for 1 . Again, the algorithm behaves in a predictable manner with the nal curvatures varying
almost proportionally. It can be deduced therefore that the algorithm is stable with respect to small
changes to the initial curvatures provided the curvature estimates are reasonable. The last two GCSs
demonstrate that the algorithm is robust and can cope with ill-conditioned span data. Ideally, the
curvature pro le needs re
ecting for GCS 6 and the sign of 0 needs changing for GCS 7.
Finally it should be remarked that for all the well-conditioned sets of span data, the extra shape
control of the GCS enables the speci ed end curvatures to be matched within 2%, and the discrepancies are not detectable by eye. However the discrepancies using a Cornu spiral would be much
larger and visible in GCSs 4 and 5.
7. Conclusions
This paper has introduced the generalised Cornu spiral. In particular, the curvature pro le of a
Cornu spiral has been generalised from a linear to a rational linear function of arc-length. This
gives one additional degree of freedom that can be used to control the shape of the curve. It has
been demonstrated that the GCS provides a one-parameter family of solutions to the simpli ed span
generation problem when the curvature constraint is relaxed.
The algorithm described in this paper has been extended to three dimensions by considering two
planar views of the span data and constructing GCSs in each of these planes. When combined using
parallel projections, the resulting space curve is expected to be a quality curve since the two planar
GCSs are themselves quality curves. The algorithm works well in practice provided the span data is
nearly planar, that is, the torsion pro le is insigni cant. However, no consideration has been given to
the resulting curvature pro le on the three-dimensional GCS due to the construction based approach
rather than an analytic one.
The Fundamental Theorem for Space Curves states there is a unique space curve, determined up
to a rigid body motion, corresponding to given curvature and torsion pro les [14]. In essence, this
curve is the solution to the system of three simultaneous rst-order di erential equations known
as the Serret–Frenet equations. A more direct approach to the three-dimensional problem therefore
would be to solve the so-called Ricatti equation that is the general solution to the Serret–Frenet
equations. This was established over a hundred years ago by Darboux [5] and seems worthy of
further consideration.
46
J.M. Ali et al. / Journal of Computational and Applied Mathematics 102 (1999) 37–47
Acknowledgements
The authors are pleased to acknowledge the support of the UK’s Engineering and Physical Sciences Research Council (GR/K 31268). The rst author would like to express his gratitude to The
University of Science Malaysia for a study grant under the Academic Sta Higher Education Scheme
that allowed him to complete his Ph.D. at The University of Birmingham.
Appendix. Re
ection of the curvature pro le
It is proved that if ( −1 =2)(1 =2−tan−1 (y1 =x1 )) ¿ 0, then re
ection of the curvature pro le about
1 =S, i.e., ′ (s) = 21 =S − (s), gives ( ′ − 1 =2)(1 =2 − tan−1 (y1 =x1 )) ¡ 0, where ′ corresponds to
the GCS with the re
ected curvature pro le. In essence, it is proved that ( − 1 =2)(1 =2 − ′ ) ¿ 0,
and hence the desired result follows immediately.
Consider the three curvature pro les a (s) = (s), b (s) = 1 =S and c (s) = 21 =S − (s) each
de ned over 06s6S, where S ¿ 0. a (s) is the original and c (s) the re
ected curvature pro le.
b (s) is a constant curvature pro le corresponding to the mean tangent rotational angle [11]. Recall
the de nition of , −=2 ¡ ¡ =2, from Eq. (14), and let a , b and c correspond to the curves
with a (s), b (s) and c (s) respectively. Since a = , b = 1 =2 and c = ′ , then ( −1 =2)(1 =2−
′ ) ¿ 0 if, and only if, ( a − Rb )( b − c ) ¿ 0.
Rt
Rt
t
For convenience, let fa (t) = 0 a () d, fb (t) = 0 b () d and fc (t) = 0 c () d, then there
are four cases to consider to complete the proof of ( a − b )( b − c ) ¿ 0, namely
(i) − =2 ¡ fa (t) ¡ fb (t) ¡ fc (t)60;
(ii) − =2 ¡ fa (t) ¡ fb (t)60 ¡ fc (t) ¡ =2;
(iii) − =2 ¡ fa (t) ¡ 0 ¡ fb (t) ¡ fc (t) ¡ =2;
(iv) 06fa (t) ¡ fb (t) ¡ fc (t) ¡ =2:
The rst two cases are proven here and then the other two follow mutatis mutandis.
(i) − =2 ¡ fa (t) ¡ fb (t) ¡ fc (t)60:
Let t ¿ 0, then 0 ¡ cos fa (t) ¡ cos fb (t) ¡ cos fc (t) and sin fa (t) ¡ sin fb (t) ¡ sin fc (t)60, since
the cosine function is positive, the sine function is negative and both functions are strictly increasing
in (−=2; 0]. Hence, 0 ¡ cos fa (t) ¡ cos fb (t) ¡ cos fc (t) and sin fa (t) ¡ sin fb (t) ¡ sin fc (t)60
for all t ¿ 0. This implies, since S ¿ 0, that
0¡
Z
S
cos fa (t) dt ¡
Z
S
cos fb (t) dt ¡
S
cos fc (t) dt
0
0
0
Z
and
Z
0
S
sin fa (t) dt ¡
Z
0
S
sin fb (t) dt ¡
Z
0
S
sin fc (t) dt60;
J.M. Ali et al. / Journal of Computational and Applied Mathematics 102 (1999) 37–47
47
and hence
RS
0
RS
0
sin fa (t) dt
cos fa (t) dt
¡
Rs
0
RS
0
sin fb (t) dt
cos fb (t) dt
RS
¡ R 0S
0
sin fc (t) dt
60;
cos fc (t) dt
i.e., tan a ¡ tan b ¡ tan c 60. Since tan is strictly increasing in (−=2; 0], then a ¡ b ¡ c
and ( a − b )( b − c ) ¿ 0.
(ii) − =2 ¡ fa (t) ¡ fb (t)60 ¡ fc (t) ¡ =2:
Let t ¿ 0, then 0 ¡
R S cos fc (t) and 0 ¡ sinRfSc (t). Hence, 0 ¡ cos fc (t) and
R S 0 ¡ sin fc (t)R Sfor all t ¿ 0.
This implies 0 ¡ 0 cos fc (t) dt and 0 ¡ 0 sin fc (t) dt, and hence 0 ¡ 0 sin fc (t) dt= 0 cos fc (t) dt,
i.e., 0 ¡ tan c . Now, tan a ¡ tan b 60 from case (i), and therefore it follows that tan a ¡ tan b
¡ tan c . Since tan is strictly increasing in (−=2; =2), then a ¡ b ¡ c and ( a − b )( b −
c ) ¿ 0.
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