Computer-Aided Design 39 (2007) 142–148
www.elsevier.com/locate/cad
Error analysis of reparametrization based approaches for curve offsetting
Hong-Yan Zhao, Guo-Jin Wang ∗
Department of Mathematics, State Key Laboratory of CAD&CG, Zhejiang University, Hangzhou 310027, China
Received 22 February 2005; accepted 6 November 2006
Abstract
This paper proposes an error analysis of reparametrization based approaches for planar curve offsetting. The approximation error in Hausdorff
distance is computed. The error is bounded by O(r sin2 β), where r is the offset radius and β is the angle deviation of a difference vector from the
normal vector. From the error bound an interesting geometric property of the approach is observed: when the original curve is offset in its convex
side, the approximate offset curve always lies in the concave side of the exact offset, that is, the approximate offset is contained within the region
bounded by the exact offset curve and the original curve. Our results improve the error estimation of the circle approximation approaches, as well
as the computation efficiency when the methods are applied iteratively for high precision approximation.
c 2006 Elsevier Ltd. All rights reserved.
Keywords: Offset; Convolution; Circle approximation approaches; Error bound; Angle deviation
1. Introduction
Offset curves/surfaces, also called parallel curves/surfaces,
are defined as a locus of the points which are at the constant
distance r along the normal from the original curves/surfaces.
Offsets are widely used in various CAD/CAM areas, such
as tool path generation [7,8], 3D NC machining [2,11], solid
modeling [17], graphics [18] and so on.
Given a planar parametric curve C(t) = (x(t), y(t))(t1 ≤
t ≤ t2 ), its offset curve with an offset radius r is
defined by Cr (t)
= C(t) + r N(t), where N(t) =
q 2
0
0
0
y (t), −x (t) / x (t) + y 0 2 (t) is the unit normal to the
original curve C(t). In general, the offset curve cannot be
represented in a polynomial or rational form because of the
square root term in the denominator of the unit normal N(t),
and so is hard to be applied in CAD systems. During the last
20 years, a lot of methods for the offset approximation [4–6,9,
10,15,16,19–22] have been proposed and developed.
In 1996, Lee et al. [12] presented an approximation method
of offset curves of given Bézier curves by rational Bézier curves
based on circle approximation. They approximated a unit circle
with piecewise quadratic polynomial curve segments Q j (s),
j = 0, . . . , n. The Hodograph curve Q0j (s) is piecewise linear;
therefore, the parallel constraint C0 (t)kQ0 (s(t)) provides the
reparametrization of s(t) as a rational polynomial of degree d,
where d is the degree of C(t). The method yields a very small
error bound as shown in tables in the literature [5,13].
In contrast, Lee et al. [13,14] approximated the
reparametrization s(t), while representing the circle Q(s) exactly by a rational quadratic curve. In their methods, the error stems only from the inaccurate reparametrization function s(t), which results in a mismatch in the parallel constraint of C0 (t)kQ0 (s(t)). Lee et al. [14] presented three such
reparametrization based methods, LRC, TMC and SRC. Unfortunately, the approximation error cannot be computed with the
distance function which was used in the quadratic curve approximation [12]. Neither can it be estimated by using the difference
function of ε(t) = kCra (t) − C(t)k2 − r 2 [4], because the term
ε(t) always equals zero. Lee et al. [13,14] measured the angular deviation of Q(s(t)) from the exact offset direction N(t) by
using the following error function:
δ(t) =
∗ Corresponding address: Department of Mathematics, State Key Laboratory
of CAD&CG, Zhejiang University, No. 38 Zheda Road, Hangzhou 310027,
Zhejiang Province, China. Tel.: + 86 571 87951609x8306.
E-mail address: wanggj@zju.edu.cn (G.-J. Wang).
c 2006 Elsevier Ltd. All rights reserved.
0010-4485/$ - see front matter doi:10.1016/j.cad.2006.11.004
hC0 (t), Q(s(t))i2
.
kC0 (t)k2
In this way, the accuracy of the algorithm is controlled by
the angle deviation. The distance between the exact offset point
Cr (t) and an approximated offset point Cra (t) is ε ≈ r sin β
H.-Y. Zhao, G.-J. Wang / Computer-Aided Design 39 (2007) 142–148
Fig. 1. The exact error is over-estimated by ε1 (t) = kCra (t) − Cr (t)k.
(see Fig. 1). However, this error measure considerably overestimates the approximation error.
Ahn et al. [1] computed the Hausdorff distance (see Fig. 1)
between the exact and approximated offset curves to estimate
the approximation error. However, the method can only be
used when the approximated offset is compatible [12,13] to the
original curve. Generally, distance sampling can be employed,
but it could not guarantee the maximum global error.
In this paper, we propose a different approach to compute the
Hausdorff distance to estimate the approximation error using
reparametrization based methods that were proposed in [13,14].
We first divide the original curve into several segments
at inflexion points. Secondly, we compute the relationships
between the angular deviations. Finally the error bound can
be expressed by those angles. The error bound is O(r sin2 β),
where β is the angle between N(t) and Q(s). From the error
bound, we observe that when the original curve is offset in its
convex side, the approximated offset curve is always bounded
by the exact offset curve and the original curve.
The paper is organized as follows. Section 2 reviews
the reparametrization based offsetting methods. Section 3
introduces the new error estimation method. In Section 4, we
give some comparisons using our method. Then comes the
conclusion of the paper.
2. Circle approximation approaches for curve offsetting
We briefly review the reparametrization based approaches
for curve offsetting in this section. The offset circle is
represented by piecewise rational curve segments. The exact
offset curve is then approximated by taking the vector sum of
the original curve with the circle segments.
Generally, a quadratic rational Bézier curve Q(s) is used to
represent a circular segment as the following
Q (s) =
(1 − s)2 ω0 P0 + 2 (1 − s) sω1 P1 + s 2 ω2 P2
(1 − s)2 ω0 + 2 (1 − s) sω1 + s 2 ω2
s1 ≤ s ≤ s2 ,
,
where P0 , P1 , and P2 are the control points of Q(s), ω0 , ω1 ,
and ω2 are weights with ω0 = ω2 = 1, ω1 = cos θ, and θ is the
angle of the circular segment, see Fig. 2.
143
Fig. 2. Circular arc with quadratic rational representation.
Fig. 3. Offsetting the curve segment in its convex direction.
Then the offset curve is approximated by taking the vector
sum of the original curve and quadratic rational Bézier curve
Q(s) which is defined as Cra (t) = C (t) + r Q (s (t)), where
s = s(t) is an approximated reparametrization. For instance,
a linear reparametrization method based on the tangent field
matching [3] is used in the MO (also called TMC in [14])
method [13] while Lee et al. [13] used two alternative different
linear reparametrization methods.
In the TMC method, the offset approximation error is
estimated by the angle deviation between the vector Q(s) and
the corresponding normal vector N(t). However, the error is
considerably over-estimated since the distance due to the angle
deviation does not faithfully reflect the Hausdorff distance from
the exact offset curve. In this paper, we present a more precise
error estimation in the Hausdorff distance. Using this result, the
conventional reparametrization based methods become more
practical because of the improved error estimation.
3. Error analysis
In the reparametrization based approach, the distance
between the original curve and the offset approximation curve
is equal to the offset radius r for any parameter value. But the
direction of the difference vector Cra (t) − C(t) is inconsistent
with the normal direction N(t) of C(t). Therefore, the error will
be introduced in the form of the associated angular deviation.
Denote β to be the angle between N(t) and Q(s(t)). Let
C(t + ∆t) be the point on the original curve whose normal
passes through the approximated offset point Cra (t) of the curve
C(t), see Figs. 3 and 5. That is, for any t ∈ [t1 , t2 ], there exists
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H.-Y. Zhao, G.-J. Wang / Computer-Aided Design 39 (2007) 142–148
Fig. 4. Angles γ , δ, and β in Case 1. (b) is obtained from (a) by appending an
auxiliary parallel line.
Thus the bound for λ can be derived from Eqs. (3.1)–(3.3) as
follows
a parameter value t + ∆t and a constant λ satisfying
C(t + ∆t) + λr N(t + ∆t) = C(t) + r Q(s(t)),
cos γ − sin β sin (β − γ ) ≤ λ ≤ cos γ ,
where N(t + ∆t) is the normal of the curve at the point
C(t + ∆t).
Thus the approximation error in Hausdorff distance is
represented by r (λ − 1). Let γ denote the angle between the
normal vector N(t +∆t) and the vector Q(s(t)), and δ the angle
between the difference vector C(t + ∆t) − C(t) and the vector
N(t + ∆t). The relationship between angles γ and δ is derived
as
C (t) − C (t + ∆t)
· N (t + ∆t)
λ = Q (s (t)) · N (t + ∆t) +
r
kC (t) − C (t + ∆t)k
= cos γ +
· cos δ.
(3.1)
r
Then the approximation error is obtained as follows
kC (t) − C (t + ∆t)k
r (λ − 1) = r cos γ − 1 +
· cos δ .
r
We will discuss the approximation error of the offset in more
detail along the convex direction and concave direction of the
original curve respectively.
Case 1: offset in the convex direction
The angles γ , δ, and β are respectively the angles between
the vector N(t + ∆t) and the vector Q(s(t)), the difference
vector C(t) − C(t + ∆t) and the vector N(t + ∆t), and the
normal vector N(t) and the vector Q(s(t)), see Fig. 4(a). A
dotted line that is parallel to the norm N(t) is drawn from the
point C(t + ∆t), see Fig. 4(b).
It is easily seen from Fig. 4(b) that
0 ≤ γ ≤ β,
cos β cos (β − γ ) ≤ λ ≤ cos γ .
Based on the following inequalities
cos β cos(β − γ ) − 1 ≥ cos2 β − 1 = −sin2 β,
cos γ − 1 ≤ 0,
the approximation error r (λ − 1) is obtained as
−r sin2 β ≤ r (λ − 1) ≤ 0.
tan 6 ABO =
(3.2)
(3.3)
(3.4)
Therefore, the approximation error is at least of O(r sin2 β)
and its upper bound is zero. It indicates that the approximate
offset always lies in the concave side of the exact offset curve,
i.e., lies in the region bounded by the exact offset curve and the
original curve.
Case 2: offset in the concave direction
Let ρ(t) be the curvature radius of the original curve C(t).
We suppose there is no self-intersection for both the exact offset
curve and the approximate offset curve, i.e., the offset radius
r < ρ(t) for all t, see Fig. 5. Denote η(t) = r/ρ(t).
Let A = C(t) and B = C(t) + r Q(s(t)), see Fig. 6. For each
parameter t, a neighborhood Oδ (t) = {u|t − δ ≤ u ≤ t + δ} can
be found so that the curve segment C (u) (u ∈ Oδ (t)) is entirely
contained inside the circle with radius ρ − ε(0 < ε < ρ).
Obviously, the angle 6 ABO > π/2 as the angle deviation β is
very small.
In 4AOB, by using laws of sins we have
It can be derived from (3.5) that
From Fig. 4(a) we have
kC (t) − C (t + ∆t)k ≤ r sin β.
or
sin (π − 6 ABO − β)
r
=
.
6
sin ABO
ρ−ε
and
π
π
≤ δ ≤ + (β − γ ) .
2
2
Therefore
−sin (β − γ ) ≤ cos δ ≤ 0.
Fig. 5. Offsetting the curve segment in its concave direction.
r
ρ−ε
sin β
.
− cos β
As γ < π − 6 ABO and 6 ABO > π/2, we have
tan γ < − tan 6 ABO =
sin β
sin β
,
r →
cos β − ρ−ε
cos β − η
(3.5)
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H.-Y. Zhao, G.-J. Wang / Computer-Aided Design 39 (2007) 142–148
and (3.6), it follows that
cos γ ≥ 1 −
sin2 β
(cos β − η)2 + sin2 β
,
and
λ − 1 ≥ cos γ − 1 ≥ −
sin2 β
(cos β − η)2 + sin2 β
,
or
cos β
cos β
−1
−1≤
sin2 β
cos γ
1−
2
2
(cos β−η) +sin β
cos β (cos β − η)2 + sin2 β
=
−1
(cos β − η)2
λ−1 ≤
=
Fig. 6. Angles γ , δ, and β in Case 2. (b) is obtained from (a) by appending an
auxiliary parallel line.
ε → 0.
γ ≥ β,
(cos β − η)2
β
β 2 cos β cos2 2 − (cos β − η)2
= 2 sin
2
(cos β − η)2
β cos β (cos β + 1) − (cos β − η)2
= 2 sin2
2
(cos β − η)2
β cos β (1 + 2η) − η2
= 2 sin2
.
2
(cos β − η)2
(3.6)
Then the relation among the angles γ , δ, and β can be
obtained using the concavity of the curve segment. The line
started from point B that is parallel to the normal direction of
the point C(t) divides the angle γ into two angles, one of which
is equal to β, see Fig. 6(b).
Then we have
(cos β − 1) (cos β − η)2 + cos β sin2 β
2
Therefore the bound for r (λ − 1) of Case 2 is derived as follows
−
r sin2 β
(cos β − η)2 + sin2 β
β
≤ 2r sin
2
2
cos β (1 + 2η) − η2
(cos β − η)2
.
(3.9)
π
π
− (γ − β) ≤ δ ≤ .
2
2
(3.7)
On the other hand, in the triangle determined by the three
vertices A, B and C(t + ∆t), we have
kC (t + ∆t) − C (t)k ≥ r sin γ ,
(3.8)
and
r sin γ
.
sin δ
Thus a bound for kC(t) − C(t + ∆t)k is derived based on the
above equation and the inequalities (3.7) and (3.8) as follows:
r sin γ ≤ kC (t + ∆t) − C (t)k ≤
r sin γ
.
cos (γ − β)
Then the bound for λ in Eq. (3.1) can be obtained as
cos γ ≤ λ ≤
≤ r (λ − 1)
π
(γ − β) + δ ≥ ,
2
or
kC (t + ∆t) − C (t)k =
cos β
cos β
.
≤
cos (γ − β)
cos γ
From
cos γ ≥ 1 − sin2 γ = 1 −
tan2 γ
,
1 + tan2 γ
It is seen from Eq. (3.9) that the bound for the approximation
error is at least O(r sin2 β). However, if the value of cos β is
close to η, the denominator of the error bound will approach to
zero, which will make the bound inexact or invalid. This occurs
at cusp points of the offset curve.
Similarly, we can compute the range of the value
Q(s(t))·N(t) by symbolic computation. Thus the error bounds
for the two cases can be obtained by (3.4) and (3.9) respectively.
It is worthwhile to mention that the error analysis becomes
much more complicated when there is self-intersection in
the curve segment C(t). The global error control cannot be
guaranteed as the assumption of “There exists a footpoint of
Cra (t) at an original curve point C(t + ∆t) and the offset point
Cra (t) will be on the normal line to the curve at the point
C(t + ∆t)” does not hold.
4. Comparisons
In this section, we present examples of the application
of our error estimation method. Figs. 7–9 show some
experimental results on the construction of approximated
offset curves. The input curve of Fig. 7 is a cubic
Bézier curve with 4 control points: (−0.785938, 0.891849),
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H.-Y. Zhao, G.-J. Wang / Computer-Aided Design 39 (2007) 142–148
Fig. 7. (a) Cubic Bézier curve and its offset approximation curve. (b) Offset approximation after subdivisions.
Fig. 8. (a) Cubic B-spline curve and its offset approximation curve. (b) Curve subdivision after knot insertion and self-intersection removal.
Fig. 9. (a) Quintic B-spline curve and its offset approximation curve. (b) Offset approximation after subdivision.
(−0.993306, −0.59695), (0.3, −2.5) and (0.9, −0.2). An
offset radius 1.0 is used. We compare the Hausdorff distance
between the offset curve and the approximant using LRC,
TMC, and SRC methods. Table 1 shows the error bound of each
approximation method. The Hausdorff distances are 0.3237,
0.2696, and 0.1278 respectively. Clearly, SRC outperforms
LRC and TMC. If the tolerance is given by 10−1 , then the curve
must be subdivided. We subdivide C(t) at the parameter value
with maximum error. After subdivision, the new subdivided
Bézier curve segments C1 (t) and C2 (t) are approximated by
LRC, TMC, and SRC (see Fig. 7(b)). The Hausdorff distance is
compared again.
The input curve of Fig. 8 is a uniform cubic Bspline curve with 7 control points: (−3.01619, 2.34143),
(−3.97193, 2.20842), (−1.07045, 0.0722807), (0.319568,
−2.77522), (−0.152767, 2.299), (2.92416, −0.939865) and
(2.8027, 3.02775). An offset radius 0.5 is used for the example.
Intersection occurs on the offset curve, so we first remove the
intersecting loop (see Fig. 8(b)). The curve is subdivided into
Table 1
The error bound of the Hausdorff distance between the offset curve and the
approximation using LRC, TMC, and SRC methods
Segment
LRC
TMC
SRC
C(t)
C1 (t)
C2 (t)
[−0.3237, 0]
[−0.0124, 0]
[−0.1006, 0]
[−0.2696, 0]
[−0.0110, 0]
[−0.1936, 0]
[−0.1278, 0]
[−0.0088, 0]
[−0.0274, 0]
four cubic Bézier curve segments. Each segment has an inflexion point on it. We subdivide the segment at the inflexion point,
and the original curve is subdivided finally into eight curve segments. Having introduced in Section 1, Lee et al. [13] proposed
to estimate the approximation error using ε ≈ r sin β. For the
curve segments Ci (t) (i = 1, 4, 5, 8), which are offset in the
convex direction, our error bound is [−r sin2 β, 0], better than
Lee’s error estimation. For those offset in the concave direction, Ci (t) (i = 2, 3, 6, 7), still, Table 2 shows our estimation
is more effective than Lee’s.
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H.-Y. Zhao, G.-J. Wang / Computer-Aided Design 39 (2007) 142–148
Table 2
Comparison of the error bounds using our estimation method and Lee’s method
Segment
C1 (t)
C2 (t)
C3 (t)
C4 (t)
C5 (t)
C6 (t)
C7 (t)
C8 (t)
LRC
Lee’s
Ours
TMC
Lee’s
Ours
SRC
Lee’s
Ours
0.1747
0.0193
0.0754
0.2762
0.0595
0.0647
0.0177
0.1166
[−0.0610, 0]
[−0.0011, 0.0007]
[−0.0231, 0.0185]
[−0.1526, 0]
[−0.0071, 0]
[−0.0123, 0.0087]
1.0e–3 × [−0.9428, 0.6481]
[−0.0272, 0]
0.1114
0.0130
0.0533
0.1386
0.0452
0.0473
0.0115
0.0814
[−0.0248, 0]
1.0e–3 × [−0.8215, 0.6526]
[−0.0094, 0.0075]
[−0.0384, 0]
[−0.0041, 0]
[−0.0231, 0.0218]
1.0e–3 × [−0.8443, 0.7123]
[−0.0133, 0]
0.1132
0.0126
0.0488
0.1382
0.0457
0.0467
0.0117
0.0804
[−0.0256, 0]
1.0e–3 × [−0.7648, 0.6075]
[−0.0080, 0.0064]
[−0.0382, 0]
[−0.0042, 0]
[−0.0225, 0.0212]
1.0e–3 × [−0.8697, 0.7338]
[−0.0129, 0]
Table 3
Comparisons between offset approximation methods
Segment
Lee’s method
Ahn’s method
LRC
TMC
SRC
C(t)
C1 (t)
C2 (t)
1.24
0.0443
0.0256
0.1030
0.0391
0.0430
[−0.2780, 0]
[−0.1133, 0]
[−0.1109, 0]
[−0.1235, 0]
[−0.0496, 0]
[−0.0636, 0]
[−0.0970, 0]
[−0.0435, 0]
[−0.0487, 0]
No comparison has ever been made between the
reparametrization based method with other offset approximation methods. Table 3 shows the comparison of the Hausdorff
distance between Lee et al. [12], Ahn et al. [1] and LRC,
TMC, and SRC methods. The input curve is a quintic Bézier
curve with 6 control points: (3, −1), (4, 2), (4.5, 2), (5.5, 1.5),
(6.5, 0) and (7, −1), shown in Fig. 9. The Hausdorff distance
is shown in the second row in Table 3. If the given tolerance
is given by 10−1 , only the SRC method can meet the need.
We subdivide C(t) at the farthest point from the line passing
through both end points of C(t). Table 3 shows Lee’s method,
Ahn’s method, and the SRC method yield similar approximation results.
5. Conclusion
In this paper, we present an error analysis for the
reparametrization based approaches for curve offsetting. The
offset curve is considered as an envelope curve of the swept
circle along the original curve. Our method measures the
Hausdorff distance between the exact and approximated offsets.
A tight error bound has been obtained in this paper, which also
reveals some interesting properties of the approaches. Our work
has improved the error estimation for the reparametrization
based approaches and makes the approaches more practical and
convenient.
Acknowledgements
We wish to thank Dr. Ligang Liu for his great help
in improving the exposition of the paper. We are also
grateful to Prof. Myung-Soo Kim for his valuable comments
and constructive suggestions. This work was supported
by the National Basic Research Program of China (No.
2004CB719400), the National Natural Science Foundation of
China (No. 60673031 & No. 60333010) and the National
Natural Science Foundation for Innovative Research Groups
(No. 60021201).
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