+ Department of Computer Science, Technion, IIT, Haifa 32000, Israel. Abstract ... degree d, the offset curve Cr(t) is approximated by planar rational curves, C a.
1
Planar Curve Offset Based on Circle Approximation
∗
In-Kwon Lee∗ , Myung-Soo Kim∗ , and Gershon Elber+ * Department of Computer Science, POSTECH, Pohang 790-784, South Korea + Department of Computer Science, Technion, IIT, Haifa 32000, Israel
Abstract An algorithm is presented to approximate planar offset curves within an arbitrary tolerance > 0. Given a planar parametric curve C(t) and an offset radius r, the circle of radius r is first approximated by piecewise quadratic B´ezier curve segments within the tolerance . The exact offset curve Cr (t) is then approximated by the convolution of C(t) with the quadratic B´ezier curve segments. For a polynomial curve C(t) of degree d, the offset curve Cr (t) is approximated by planar rational curves, Cra (t)’s, of degree 3d − 2. For a rational curve C(t) of degree d, the offset curve is approximated by rational curves of degree 5d − 4. When they have no self-intersections, the approximated offset curves, Cra (t)’s, are guaranteed to be within -distance from the exact offset curve Cr (t). The effectiveness of this approximation technique is demonstrated in the offset computation of planar curved objects bounded by polynomial/rational parametric curves.
Keywords: Offset, convolution, curve approximation, hodograph, NC machining, B´ezier/Bspline curves, rational curve.
1
Introduction
Constant radius offsetting for curves and surfaces is one of the most important geometric operations in CAD/CAM due to its immediate application to NC machining [6, 14]. At the same time, the offsetting is one of the most difficult geometric operations to implement. The ∗
Computer-Aided Design, Vol. 28, No. 8, pp. 617–630, August, 1996
exact offset curves and surfaces have very high algebraic degree compared with the original curves and surfaces. For example, the offset of a cubic B´ezier curve has algebraic degree 10 [9]. Given a planar regular parametric curve C(t) = (x(t), y(t)), the offset curve Cr (t) (with respect to a fixed radius r) is defined by: Cr (t) = C(t) + r · N (t), where
(y (t), −x (t)) N (t) = x (t)2 + y (t)2
is the unit normal of C(t). (In this paper, we assume the normal vectors of a planar curve are pointing to the right of the curve advancing direction.) Due to the square root function in the denominator of N (t), the exact offset curves are not rational, in general [9]. Farouki and Sakkalis [10] considered the special case of x (t)2 + y (t)2 = σ(t)2 for some polynomial σ(t). These curves are called “Pythagorean hodograph curves,” and their offset curves are rational. However, they have less degrees of freedom than other polynomial curves with the same degree [10]. For example, a cubic Pythagorean hodograph curve has only one degree of freedom; at least degree five is required for a Pythagorean hodograph curve to have an inflection point. The Pythagorean hodograph curves have many useful properties which need further investigation. Nevertheless, for the time being, approximation techniques seem to be the only feasible practical solution to the planar curve offsetting. Previous offset computation methods for parametric curves approximated either (i) the original curve C(t) by line segments and circular arcs [22], or (ii) the offset curve Cr (t) by low degree polynomial curves [3, 4, 17, 18, 19, 23, 24]. The offsets for line segments and circular arcs are also line segments and circular arcs [22]. However, many line segments and circular arcs are required to approximate a given planar curve. Although quadratic or cubic curves approximate the given curve with fewer curve segments, the offset even for a quadratic curve is not rational. The approximation method of [22] is thus limited only to line segments and circular arcs. The other methods approximate the offset curve by spline curve segments [3, 4, 17, 18, 19, 23, 24]. They estimate the approximation error only at finite sample points, which may inaccurately reflect the global error. Moreover, each curve subdivision is made at the midpoint of the curve, which may not be the optimal subdivision point. Elber and Cohen [6] estimated the offset approximation error by computing the maximum value of the difference function ψ(t) = Cra (t) − C(t)2 − r2 , where Cra (t) is an approximation of Cr (t). Curve refinement is made at the parameter value(s) which gives the maximum error. This technique reduces the size of curve data. Any of the previous methods [3, 4, 17, 18, 19, 2
23, 24] can be used to construct the approximated offset spline curve Cra (t). None of these methods, however, guarantees that the curve Cra (t) has similar speed to Cr (t). Even when the overall traces of the two curves are close to each other, the distance Cra (t) − Cr (t) may be taken to be larger than the actual distance, and the error measure ψ(t) = Cra (t)−C(t)2 −r2 may over-estimate the actual distance between Cra and Cr . In Reference [6], the error measure ψ(t) = Cra (t) − C(t)2 − r2 was used in conjunction with the assumption that the offset approximation converges to the actual offset. There are, however, some special cases in which this assumption is not valid. For example, consider the offset curve approximation Cra (t) = C(t) + (r, 0); that is, Cra (t) is a translation of C(t) along the x axis by an amount of r. Clearly, the error measure ψ(t) ≡ 0; but the curve Cra (t) is completely different from the exact offset curve Cr (t). In this way, the measure may also under-estimate the error. Elber and Cohen [6] provided a way to remedy this shortcoming; they proposed to measure the angle between the difference vector δ(t) = Cra (t) − C(t) and the unit normal vector N (t). As a result, the distance and angle functionals in [6] bound each approximated offset curve point within a small neighborhood of the exact offset point, and thus construct a very precise approximated offset curve. However, it is difficult to completely eliminate error over-estimation. All the previous methods have the same problem, the only exception being the simplest method of [22]. Taking a slightly different approach, this paper suggests a method that measures the actual approximation error more precisely. The planar curve offsetting problem is interpreted as a sweeping problem in which the center of a circle of radius r is moving along the given planar curve. The boundary of the sweep is obtained as an envelope curve of the swept circle, which is identical to the exact offset curve [2, 13]. This envelope curve is also known as a convolution curve. The convolution curve of two parametric curves C(t) and Q(s) is defined as the curve C(t)+Q(s(t)), where C (t) and Q (s) have the same direction at s = s(t) [2, 13]. This relationship between C (t) and Q (s) requires a proper reparametrization of s = s(t) as a function of t. To make the reparametrization easier, the moving circle is approximated with piecewise quadratic B´ezier curve segments within a tolerance > 0. Let C(t) = (x(t), y(t)), t1 ≤ t ≤ t2 , be a polynomial curve of degree d and Q(s) = (xQ (s), yQ (s)), s1 ≤ s ≤ s2 , be a quadratic curve. Then, Q(s) has a linear hodograph Q (s) = (xQ (s), yQ (s)) = (as + b, cs + d), for some constants a, b, c, d ∈ R. Assume the mapping, t ∈ [t1 , t2 ] → s(t) ∈ [s1 , s2 ], is a bijection, where C (t) and Q (s(t)) have the same direction. Then, the condition of C (t) and Q (s) having the same direction implies a linear equation of s: x (t)(cs + d) − y (t)(as + b) = 0, where x (t) and y (t) are polynomials of degree d − 1. The proper reparametrization of s = s(t) is obtained as a rational polynomial of degree d − 1 in t. Under the reparametrization, the approximated offset circle, Q(s(t)), is a planar rational curve of degree 2(d − 1), and the convolution curve, C(t) + Q(s(t)), is 3
Q(s)
C(t)
r r+
r− Cra (t)
Cr (t)
Figure 1: Bounding Offset Curve Approximation Error
a planar rational curve of degree 3d − 2. The Pythagorean hodograph curves of degree dP have exact offset curves of degree 2dP − 1, and other general polynomial curves of degree d have the same degrees of freedom for dP = 2d − 1 (see [10]). Consequently, we have the relation, 2dP − 1 = 4d − 3 ≥ 3d − 2, which means that our approximated offset curve has lower degree than the exact offset curve of a Pythagorean hodograph with the same degrees of freedom. Furthermore, the proposed method can also be applied to planar rational curves. For a rational curve C(t) of degree d, the exact offset curve can be approximated by rational curves of degree 5d − 4 (see Section 3.2 for more details). When the circle of radius r is approximated by quadratic B´ezier curves within an > 0 bound, the approximated curves are in between the two circles of radii r − and r + . The envelope curve for an approximated quadratic B´ezier curve is also in between the two envelope curves for the two bounding circles (see Figure 1). Here, for simplicity, we assume the envelope curves have no self-intersections and the base curve is a non-trimmed infinite or closed curve. Otherwise, there are some delicate issues; the details of which will be discussed in Section 3.2. The two bounding envelope curves are at distance from the envelope curve of the circle of radius r. Thus, it is guaranteed that the approximated offset curve is within distance from the exact offset curve. Note that the distance Cra (t) − Cr (t) for each instance of t is not guaranteed to be less than ; furthermore, neither is the angle between the difference vector Cra (t) − C(t) and the normal vector N (t) guaranteed to be within a certain error bound. Nevertheless, the exact and approximated offset curves, Cr and Cra , are guaranteed to be within distance of each other. Compared with the approximation of an arbitrary planar parametric curve, the circle approximation is easier to construct and the error bound can be estimated very precisely [5, 11, 21]. With the piecewise quadratic B´ezier approximation of the circle, one can obtain only G1 -continuity of the resulting offset curve segments. Nevertheless, our method generates 4
very precise offset curve approximation with a smaller number of curve subdivisions than the previous methods, especially when high precision is required for the offset approximation. (See Elber et al. [7] for more detailed comparisons of the previous methods.) Both the precision and data reduction of our offset approximation algorithm are essentially based on the simple error estimation, which is guaranteed to be the same as the circle approximation error. In many practical applications of the planar curve offsetting (e.g., NC tool path generation), the accuracy of offset curve approximation and the reduction of output data size are two important criteria which must be considered. This paper is organized as follows. Section 2 reviews circle approximation methods with piecewise quadratic B´ezier curve segments [5, 11, 21]. Section 3 describes an algorithm to approximate the offset curve with rational parametric curves. Section 4 describes an algorithm to eliminate the local and global self-intersecting loops in the offset of a planar curved object. Section 5 demonstrates some experimental results. Finally, we conclude the paper in Section 6.
2
Circle Approximation with Quadratic B´ ezier Curve Segments
In this section, we review five different methods to approximate a unit circular arc by using a quadratic B´ezier curve segment Q(s), 0 ≤ s ≤ 1. The approximation methods are derived in a straightforward manner from the cubic B´ezier curve approximations of a circle [5, 11]. Morken [21] also developed the same circle approximations with quadratic B´ezier curves. The main contribution of Morken [21] is in proving that each method is optimal under the given curve constraints; the optimality proof is not clear in the previous work [5, 11]. We present the curve construction methods in detail here (without optimality proofs) since they will be used for the construction of approximated offset curves, Cra (t)’s, in the next section.
2.1
General Formulas for Quadratic B´ ezier Curves
The quadratic B´ezier curve with three control points, P0 , P1 , and P2 , is given by: Q(s) = (1 − s)2 P0 + 2s(1 − s)P1 + s2 P2 ,
for 0 ≤ s ≤ 1.
For the circle approximation, we may assume that P0 = (1, 0) and P2 = (cos α, sin α) (see Figure 2). For the sake of simplicity, we consider only the case of α ≤ π2 . The quadratic B´ezier curve Q(s) = (x(s), y(s)), 0 ≤ s ≤ 1, has the coordinate equations: x(s) = (1 − s)2 + 2Ds(1 − s) cos 5
α + s2 cos α, 2
y P2 = (cos α, sin α) P1 = (D cos α2 , D sin α2 ) Q(s) D P0 = (1, 0)
α
x Figure 2: Quadratic B´ezier Curve
y(s) = 2Ds(1 − s) sin
α + s2 sin α, 2
where D is the distance between the origin and the control point P1 . The approximation error of the curve Q(s) from the unit circle can be formulated as follows: ε(s) = x2 (s) + y 2 (s) − 1, for 0 ≤ s ≤ 1. Instead of using ε(s), which has a square root term, we may use the following function φ(s) for the error estimation: φ(s) = x2 (s) + y 2 (s) − 1,
for 0 ≤ s ≤ 1.
One can provide a bound on ε(s), once a bound on φ(s) is established, from the relation: φ(s) = x2 (s) + y 2 (s) − 1 = (ε(s) + 1)2 − 1 = ε2 (s) + 2ε(s) ≈ 2ε(s), as the approximation converges to the exact offset. That is, in using the function φ(s) instead of ε(s), we double the actual estimated error. As a degree four polynomial, φ(s), 0 ≤ s ≤ 1, has five extreme values at most. The parameter values of s for the five extreme values are two boundary values and the values obtained by solving φ (s) = 0 (assume D > 1): s0 = 0 √ 1 4D2 + 2 cos α − 6 s1 = − 2 4(D − cos α2 ) 1 s2 = 2 6
s3 s4
1 = + 2 = 1.
√
4D2 + 2 cos α − 6 4(D − cos α2 )
The five extremal values of φ(s) are: φ(s0 ) = φ(s4 ) = 0
(D cos α2 − 1)2 (D − cos α2 )2 1 α 2 1 φ(s2 ) = φ = − 1. · D + cos 2 4 2 φ(s1 ) = φ(s3 ) = −
2.2
Method 1: Tangent to the Circle at Both Ends
Let Q1 (s) = (x1 (s), y1 (s)), 0 ≤ s ≤ 1, be the quadratic B´ezier curve which is tangent to the circle at the end points of the curve. When each quadratic B´ezier curve is tangent to the circle at the two end points, two adjacent curves meet with G1 -continuity. Then, the middle control point P1 should be of the form (1, tan α2 ), and we have D1 = 1/ cos α2 ; therefore, s1 = s0 = 0, s3 = s4 = 1, and φ1 (s0 ) = φ1 (s1 ) = φ1 (s3 ) = φ1 (s4 ) = 0, while the maximum error occurs at s = s2 = 12 with magnitude:
δ1 (α) = φ1
1 α 1 = · D1 + cos 2 4 2
2
1 sin4 α2 1 4 −1= · α + o(α6 ). α = 2 4 cos 2 64
This magnitude is the same as the squared distance between Q 12 and (cos α2 , sin α2 ). The 1 4 resulting curve Q1 (s), 0 ≤ s ≤ 1, always lies in the exterior of the circle. The value of 64 α is the first nonzero term in a power series expansion of the error around α = 0. Figure 3 shows the function φ1 (s) of this method. The approximation error can be further reduced by scaling the B´ezier control polygons or eliminating the condition of G1 -continuity as will be discussed in the later subsections.
2.3
Method 2: Uniform Scaling of Method 1
The curve Q1 (s) of Method 1 lies in the exterior of the unit circle. This means that the error function φ1 (s) takes only nonnegative values. A slightly better approximation can be obtained by scaling all the three control points simultaneously by a certain magnitude. That is, the curve Q2 (s) = (x2 (s), y2 (s)), 0 ≤ s ≤ 1, is defined by: x2 (s) = ρx1 (s) and y2 (s) = ρy1 (s),
for some ρ > 0.
Then, the error function φ2 (s) of this method is given by: φ2 (s) = x22 (s) + y22 (s) − 1 = ρ2 (x21 (s) + y12 (s)) − 1 = ρ2 φ1 (s) + ρ2 − 1. 7
φ(s) φ1 (s)
φ2 (s)
φ4 (s) φ5 (s) s=1
φ3 (s)
s
Figure 3: Error Functions φi (s)’s for Five Methods
8
Multiplying x1 (s) and y1 (s) by ρ has the same effect as multiplying each control point of Q1 (s) by ρ. Since φ2 (s) = ρ2 φ1 (s), the two functions φ1 (s) and φ2 (s) have their extreme values at the same parameter values of s. Furthermore, the extremal value δ2 (α) = φ2 ( 12 ) satisfies: δ2 (α) = ρ2 δ1 (α) + ρ2 − 1. Since φ1 (0) = φ1 (1) = 0, we have φ2 (0) = φ2 (1) = ρ2 −1. We can minimize the approximation error by taking ρ so that 1 φ2 (0) + φ2 = 0, 2 or equivalently, ρ2 − 1 + ρ2 δ1 (α) + ρ2 − 1 = 0. Thus, the value of ρ is given by:
ρ=
8 cos2 α2 2 = , 2 + δ1 (α) 8 cos2 α2 + sin4 α2
and the approximation error is given by the value: δ2 (α) =
δ1 (α) 1 4 1 ≈ δ1 (α) = α + o(α6 ), 2 + δ1 (α) 2 128
which means that the size of δ2 (α) is slightly less than half of δ1 (α). Figure 3 shows the function φ2 (s) of this method. In order for two consecutive quadratic B´ezier curve segments of Method 2 to meet with 0 G -continuity, they should have the same angle α. (Note that G1 -continuity is also guaranteed when the same angle α is used.) This is because the value of ρ depends on the value of δ1 (α), and thus on α, too.
2.4
Method 3: Interpolating Three Circle Points
When the quadratic B´ezier curve is not restricted to be tangent to the unit circle at the two end points of the curve, the approximation error can be further reduced by moving the middle control point P1 of Method 1 toward the origin along the diagonal line. A simple approximation is obtained by making the midpoint of the quadratic B´ezier curve Q3 (s) = (x3 (s), y3 (s)) pass through the midpoint of the circular arc; that is, we make 1 α α 1 Q3 2 = (cos 2 , sin 2 ) and φ3 2 = 0. The resulting curve Q3 (s), 0 ≤ s ≤ 1, always lies
in the interior of the circle. From the condition φ3 12 = 0, it follows that D3 = 2 − cos α2 . Figure 3 shows the function φ3 (s) of this method. Maximum errors are obtained at s = s1 and s3 with (negative) magnitude:
1 α δ3 (α) = φ3 (s1 ) = φ3 (s3 ) = − 1 − cos 4 2 9
2
= − sin4
1 4 α =− α + o(α6 ). 4 256
2.5
Method 4: Interpolation with Equi-oscillating Error
To further reduce the maximum error, the curve can be modified to have an equi-oscillating error. The error function φ4 (s) has the same magnitude of maximum and minimum values (see Figure 3). ¿From the following polynomial equation of degree four:
1 = 0, 2 the corresponding value of D4 can be computed as
√ √ α α D4 = − 2 cos + 2 + 2 2 + cos2 . 2 2 Figure 3 shows the function φ4 (s) of this method. Maximum errors are obtained at s = s1 , s2 , and s3 with magnitude: φ4 (s1 ) + φ4
√ 1 α 1 δ4 (α) = φ4 ( ) = (1 − 2) cos + 2 4 2
2.6
√ α 2 2 + 2 2 + cos2 − 1 = 0.00268α4 + o(α6 ). 2
Method 5: Uniform Scaling of Method 3
The curve Q3 (s) of Method 3 is contained in the interior of the unit circle. This means that the error function φ3 (s) takes only nonpositive values. By using a similar scaling technique to Method 2, a better approximation can be obtained. Let the curve Q5 (s) be defined by: Q5 (s) = (x5 (s), y5 (s)) = (ρx3 (s), ρy3 (s)), where
ρ=
for 0 ≤ s ≤ 1,
2 2 = 2 + δ3 (α) 2 − sin4
α 4
.
(Note that δ3 (α) < 0 and ρ > 1.) Then the approximation error is given as: δ5 (α) =
δ3 (α) 1 4 1 ≈ δ3 (α) = − α + o(α6 ). 2 + δ3 (α) 2 512
Since δ3 (α) < 0, the absolute value |δ5 (α)| is slightly larger than half of |δ3 (α)|. Figure 3 shows the function φ5 (s) of this method. For two adjacent quadratic B´ezier curve segments of Method 5 to meet with G0 -continuity, they should have the same angle α. The actual absolute maximum error values of i (α), i = 1, . . . , 5, for the five methods are given by:
i (α) = max |εi (s)| = | δi (α) + 1 − 1|. 0≤s≤1
The maximum error occurs at s2 = 12 in Method 1, at s1 and s3 in Method 3, at s1 , s2 = 12 , s3 in Methods 2, 4, and at s0 , s1 , s2 , s3 , and s4 in Method 5. Table 1 and Figure 4 show the maximum errors i (α), i = 1, . . . , 5, of the five methods with respect to various values of α. The error 5 is about half of 3 , 2 is about half of 1 , and 4 is about 0.7 3 . 10
α (degree)
1 90.0 6.07 × 10−2 70.0 2.00 × 10−2 50.0 4.84 × 10−3 30.0 6.01 × 10−4 10.0 7.27 × 10−6 5.0 4.53 × 10−7 1.0 7.25 × 10−10
2 2.90 × 10−2 9.83 × 10−3 2.41 × 10−3 3.00 × 10−4 3.63 × 10−6 2.27 × 10−7 3.62 × 10−10
3 1.07 × 10−2 4.08 × 10−3 1.10 × 10−3 1.45 × 10−4 1.81 × 10−6 1.13 × 10−7 1.81 × 10−10
4 7.72 × 10−3 2.89 × 10−3 7.65 × 10−4 1.00 × 10−4 1.24 × 10−6 7.77 × 10−8 1.24 × 10−10
5 5.29 × 10−3 2.03 × 10−3 5.48 × 10−4 7.26 × 10−5 9.05 × 10−7 5.66 × 10−8 9.06 × 10−11
Table 1: The Values of i (α) for Five Methods.
α (degree) 90
80
Method 1 Method 2 Method 3 Method 4 Method 5
70
60
50
40
30
20
10
0 1e-11
1e-10
1e-09
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
Figure 4: The Inverse Functions of i (α)’s
11
0.01
0.1
3
Approximating Planar Offset Curves
In this section, we consider a regular parametric polynomial/rational curve C(t) of degree d and its offset approximation with piecewise rational curves. The approximation is based on the hodographs of the curve C(t) and the quadratic B´ezier curve segments, Q(s)’s, which approximate the unit circle. We assume the parametric curve C(t) is C 1 -continuous.
3.1
Hodograph of a Planar Curve
The hodograph H(t) of a C 1 -continuous parametric curve C(t) is the locus of the first derivative C (t). For example, the hodograph of a quadratic curve is a straight line and a cubic curve has a quadratic hodograph curve. We summarize below some properties of the hodograph which will be used for the approximation of offset curves. Definition 3.1 Let C(t) = (x(t), y(t)), t1 ≤ t ≤ t2 , be a planar regular parametric curve. The mapping θC , y (t) ∈ [0, 2π), θC : t ∈ [t1 , t2 ] → θC (t) = arctan x (t) is called the tangential angular map of the curve C(t). Lemma 3.1 Let HQ (s) = (xQ (s), yQ (s)) be the hodograph of Q(s) = (xQ (s), yQ (s)), s1 ≤ s ≤ s2 . If the tangential angular map θQ (s) of Q(s) is one-to-one, any ray L (starting from the origin) intersects with HQ (s) at no more than one point.
Proof: Suppose L intersects with HQ (s) at two different points HQ (sa ) and HQ (sb ), s1 ≤ (sa )) and Q (sb ) = (xQ (sb ), yQ (sb )) have the same sa < sb ≤ s2 . Then, Q (sa ) = (xQ (sa ), yQ direction, which implies that θQ (sa ) = θQ (sb ). This is a contradiction to the fact that θQ is one-to-one. Lemma 3.2 Let HQ (s) = (xQ (s), yQ (s)) and H(t) = (x (t), y (t)) be the hodographs of Q(s) = (xQ (s), yQ (s)), s1 ≤ s ≤ s2 , and C(t) = (x(t), y(t)), t1 ≤ t ≤ t2 , respectively. Furthermore, let HQ (s0 ) and H(t0 ) be the intersection points of a ray L (starting from the origin) with HQ (s) and H(t), respectively. Then, Q(s) and C(t) have the same tangent direction at Q(s0 ) and C(t0 ), respectively. (s0 )) and H(t0 ) = (x (t0 ), y (t0 )). Proof: The direction of L is the same as HQ (s0 ) = (xQ (s0 ), yQ Therefore, Q (s0 ) = (xQ (s0 ), yQ (s0 )) and C (t0 ) = (x (t0 ), y (t0 )) have the same direction.
12
3.2
Approximated Offset Curve
Assume the tangential angular map θQ is one-to-one and θC ([t1 , t2 ]) ⊂ θQ ([s1 , s2 ]). For each t ∈ [t1 , t2 ], consider the ray Lt which starts from the origin and passes through H(t). By Lemma 3.1, the ray Lt intersects with the hodograph curve HQ at a unique point HQ (s(t)), where s(t) ∈ [s1 , s2 ]. This means that the mapping, t ∈ [t1 , t2 ] → s(t) ∈ [s1 , s2 ], is welldefined. By Lemma 3.2, Q (s(t)) and C (t) have the same direction. Therefore, the convolution curve C(t) + Q(s(t)) is also well-defined for t ∈ [t1 , t2 ]. A quadratic curve Q(s), s1 ≤ s ≤ s2 , has a linear one-to-one hodograph curve HQ (s) = (as+b, cs+d), where a, b, c, d ∈ R. If Q (s(t)) = (as(t)+b, cs(t)+d) and C (t) = (x (t), y (t)) have the same direction, we have: cs(t) + d y (t) = , as(t) + b x (t) and the parameter s(t) ∈ [s1 , s2 ] is uniquely determined by: s(t) =
by (t) − dx (t) . cx (t) − ay (t)
(1)
The approximated offset curve Cra (t), t1 ≤ t ≤ t2 , is defined by: Cra (t) = C(t) + r · Q(s(t)), where s(t) is given in Equation (1). The curve Cra (t) is the envelope curve of Q(s) along the parametric trajectory curve C(t) [2]. For a polynomial curve C(t) of degree d, the reparametrized curve Q(t) = Q(s(t)) is a planar rational curve of degree 2(d − 1). Thus, the approximated offset curve, Cra (t) = C(t) + r · Q(s(t)), is a planar rational curve of degree 3d − 2. For a rational polynomial curve C(t) of degree d, the function s(t) is a rational polynomial of degree 2d − 2. Therefore, Q(s(t)) is a rational curve of degree 2(2d − 2), and Cra (t) is a rational curve of degree 5d − 4. When the curve C(t) is a non-trimmed (infinite or closed) curve and there is no selfintersection in the exact and approximated offset curves, the global error between Cr (t) and Cra (t) is bounded by r · , where is the maximum error between the unit circle and the quadratic B´ezier curve approximations, Q(s)’s. In this special case, the sweeping of the circular disk of radius r generates a connected banded region (in the plane) which is bounded by two disjoint boundary curves. The exact offset curve Cr (t) is one of the two boundary curves. Similarly, the approximated offset curve Cra (t) is a boundary curve of the sweeping volume of a convex region (bounded by the quadratic B´ezier curve segments, r · Q(s)’s). Since this convex region is in between the two offset circles of radii r(1 − ) and r(1 + ), the boundary curve Cra (t) is also in between the two offset boundary curves Cr(1− ) (t) and 13
y
Cra (t)
Cr (t) S 1 : x2 + y 2 = r2 C(t) = (t, |t|) Q x
Figure 5: Counter Example
y
y
y x
(a)
x
(b)
x
(c)
Figure 6: Different Topology Between Two Similar Sweep Volumes: (a) S 1 and Q, (b) C ∗S 1 , and (c) C ∗ Q.
14
Cr(1+ ) (t), both of which are at distance r · from the exact offset curve Cr (t). Therefore, the distance between Cra (t) and Cr (t) is also bounded by r · . Remark: When the curve C(t) is a trimmed curve segment or there is self-intersection in the exact/approximated offset curve, the global error between Cr (t) and Cra (t) is not guaranteed to be bounded by r · . One of the referees of this paper pointed out this fact with the counter-example shown in Figure 5. The distance between Cr (t) and Cra (t) is more than r · near the curve ends and the self-intersections. The boundary of a 2D curved object is usually modeled with a finite sequence of trimmed parametric curve segments. Two consecutive curve segments may share a common vertex at which they have G1 -discontinuity. To eliminate this limitation of trimmed curve segments, we may treat each vertex as a circular arc of radius 0 which smoothly connects two consecutive curve segments with C 1 -continuity. In this way, the object boundary may be considered as a single C 1 -continuous closed curve. More serious is the limitation resulting from the self-intersections of offset curves. In Figure 5, the distance between Cr (t) and Cra (t) (in the middle portions of the two curves) can be bounded by r· after the self-intersections are eliminated. However, this special treatment may not work out in general. By sweeping an approximated convex object (instead of the exact offset disk), the approximated sweep volume may have a different topology from the exact sweep volume (see Figure 6). In practice, the error bound r · is usually taken to be a very small quantity. The topological changes in the sweep volumes are realized only when the sweep boundaries have almost tangential intersections within the distance 2r · . Therefore, this kind of degeneracy is predictable. The tangential intersections among offset curve segments introduce various difficulties in the determination of offset curve arrangement. In the current implementation, we first determine the topology of the offset boundary by using a polygonal approximation, and then compute the offset boundary curve approximation within an r · bound using the rational curve segments, Cra (t)’s. Therefore, the issue resulting from self-intersecting offset curves is intentionally eliminated. (See Section 4 for more details and the justification of this approach.)
3.3
Subdivision of the Curve C(t)
The input curve C(t), t1 ≤ t ≤ t2 , is subdivided into finer curve subsegments, Ci (t)’s, so that each subsegment Ci (t), ti1 ≤ t ≤ ti2 , satisfies the condition: θCi ([ti1 , ti2 ]) ⊂ θQj ([sj1 , sj2 ]) for some quadratic B´ezier curve segment Qj (s), sj1 ≤ s ≤ sj2 . (The notation Qj ’s here means the consecutive quadratic curve segments in the circle approximation. This is different from the 15
y
y
x
Method 1
y
x
Method 2
y
x
Method 3
y
x
Method 4
x
Method 5
Figure 7: Hodographs of Qj ’s of Five Circle Approximation Methods. notation Qi ’s of Section 2 which means the quadratic B´ezier curve segments constructed by Method i, for i = 1, . . . , 5.) Figure 7 shows the hodographs of Qj (s)’s which are constructed by the five circle approximation Methods 1–5 of Section 2. Each method generates quadratic B´ezier curve segments which form a convex object (approximating the unit circle). Methods 3–5 do not have G1 -continuity at the curve end points; they have only G0 -continuity. There is a small jump of tangent direction at each common vertex of two adjacent curve segments. We may consider the singular vertex as a circular arc of radius 0, which smoothly fills the gap in the discontinuity of tangent directions. These singular vertices (treated as circular arcs of radius 0) generate circular arcs in the hodograph which can smoothly fill the tiny gaps in the hodographs of Methods 3–5 (Figure 7). In Figure 8–(a), the hodograph H(t) of a curve C(t) is subdivided into finer subsegments according to the tangential angular regions of Qj (s)’s of Method 1. The corresponding subdivision of C(t) and the construction of approximated offset curves, Cra (t)’s, is shown in Figure 8–(b). Assume the curve C(t), t1 ≤ t ≤ t2 , is convoluted with two adjacent quadratic B´ezier curve segments Q1 (s), s1 ≤ s ≤ sa , and Q2 (s), sb ≤ s ≤ s2 , which are not G1 -continuous at the common vertex v = Q1 (sa ) = Q2 (sb ) (see Figure 9). The curve C(t) is subdivided into three subsegments: (i) C1 (t) ≡ C(t), t1 ≤ t ≤ ta , (ii) Cv (t) ≡ C(t), ta ≤ t ≤ tb , and (iii) C2 (t) ≡ C(t), tb ≤ t ≤ t2 , where ta and tb are such that C (ta ) and Q1 (sa ) (respectively, C (tb ) and Q2 (sb )) have the same tangent direction. Then, the overall convolution is obtained as a connected sequence of three consecutive convolution curve segments: (i) C1 (t) + r · Q1 (s(t)), t1 ≤ t ≤ ta , (ii) Cv (t) + r · v, ta ≤ t ≤ tb , and (iii) C2 (t) + r · Q2 (s(t)), tb ≤ t ≤ t2 . Since each offset curve segment and the given curve C(t) have the same tangent direction at the same parameter value t, the three offset curve segments are connected with G1 -continuity. Methods 3–5 have smaller approximation errors than Methods 1 and 2; however, they need a convolution curve segment, Cv (t) + r · v, to fill the gap between each pair of nearby convolution curve segments, Ci (t) + r · Qi (s(t)), i = 1, 2, so that the offset curve segments are connected with G1 -continuity. Table 2 shows the magnitude of tangential direction
16
(a)
(b)
Figure 8: (a) Subdivision of H(t), and (b) Corresponding Subdivision of C(t) and Construction of Cra (t).
y
C2 (t) + r · Q2 (s(t)) C2 (t)
Q2 (s) v Q1 (s)
∆θ
Cv (t)
Cv (t) + r · v
C1 (t) C1 (t) + r · Q1 (s(t))
Figure 9: Convolution Curve from Methods 3, 4, and 5
17
x
α (degree) Method 3, 5 90 1.8 × 10−1 70 9.6 × 10−2 50 3.7 × 10−2 30 8.6 × 10−3 10 3.3 × 10−4
Method 4 1.6 × 10−1 8.2 × 10−2 3.1 × 10−2 7.2 × 10−3 2.7 × 10−4
Table 2: Magnitude of Normal Angle Discontinuity ∆θ (radian) discontinuity in Methods 3–5. The tangential gap filling curve segment, Cv (t) + r · v, is a simple translation of Cv (t) by a vector r · v; therefore, they have the same size. The length of the curve segment Cv (t) is between ∆θ/κ1 and ∆θ/κ2 , where κ1 and κ2 are the maximum and minimum curvatures of Cv (t), respectively. The size of the curve segment, Cv (t) + r · v, is usually larger than the offset curve approximation error, r · , especially when the curve Cv (t) has low curvature (see Tables 1 and 2). When we use Methods 3 and 4, the quadratic B´ezier curve Q(s), 0 ≤ s ≤ 1, has its circle approximation error less than on a domain slightly larger than [0, 1]; that is, |φ(s)| < , −∆s ≤ s ≤ 1 + ∆s, for some ∆s > 0 (see Figure 3). Therefore, the two approximated offset curve segments Ci (t) + r · Qi (s(t)), i = 1, 2, can be extended into larger curve segments while staying within the approximation error bound, r · . If the two extended offset curve segments intersect, we can remove the gap filling curve segment, Cv (t) + r · v, by filling the gap with the extensions of the two offset curve segments up to their common intersection point. Otherwise, we can replace the curve segment, Cv (t) + r · v, by the line segment connecting the two curve end points (if the line approximation has error less than r · ). However, in this case, they have only G0 -continuity at the intersection point. For G1 continuity or for the case in which the line approximation has error larger than the tolerance r · , we can use a quadratic B´ezier curve approximation of the curve segment, Cv (t) + r · v, if the curve approximation error is less than r · .
3.4
Under and Over Estimations
It is sometimes desirable to have an offset approximation which completely overestimates. In this case, the offset approximation is never below the exact offset distance, only above or equal to it. Method 1 (respectively, Method 3) completely overestimates (respectively, underestimates) the exact offset. Clearly, for machining purposes, when the center of the ball end tool follows the offset approximation, a completely overestimating offset approxi18
mation scheme will prevent gauging. This important question has been ignored in the offset approximation research. In fact, the method of Cobb [3] completely underestimates the exact offset. In Cobb [3], an approximated offset curve is computed by translating each control point of the curve along the normal direction of C(t) at the corresponding node parameter value by a distance r, and the approximated offset curve Cra (t) of a B´ezier or B-spline curve C(t) is given by: Cra (t) =
n
i=0
(Pi + r · Ni ) Bi (t) =
n
i=0
Pi Bi (t) + r ·
n
i=0
Ni Bi (t) = C(t) + r · V (t),
where Bi (t) is the B´ezier or B-spline basis function and Ni is the unit normal vector at the node parameter value of the corresponding control point Pi of C(t), for each i. Each control point Ni of the vector field curve V (t) is on the unit circle S 1 . Hence, by the convex hull property of the B´ezier and B-spline representations, V (t) is contained in the unit ball. Consequently, we have Cra (t)
− C(t) = r · V (t) = r ·
n
i=0
Ni Bi (t) = r ·
n
i=0
Ni Bi (t) ≤ r,
which implies that the resulting offset approximation always underestimates the exact offset. (See [7] for more details.) Interpolatory schemes [19, 23] are, in general, piecewise overestimating and underestimating the exact offset. Here, if desired, we can alleviate the errors of Methods 1 and 3, making them piecewise overestimating and underestimating using scaling as is done in Methods 2 and 5. The piecewise overestimating and underestimating offset approximation may be sufficient when the offset is employed as a modeling tool, for instance. Hence, making the offset approximation more accurate (at the expense of sacrificing the overestimation property) might be profitable in this instance.
4
Eliminating Self-Intersecting Loops
In this section, we consider how to compute the offset of a planar curved object. The offsets of boundary curves are first computed; then, their local and global self-intersections are eliminated. Examples of local and global loops are shown in Figure 10–(c). The determination of self-intersecting loops is closely related with the arrangement of offset curve segments [13]. A robust implementation of curve arrangement is one of the most difficult open problems in geometric modeling [14]. The only reliable robust implementation today is to use polygonal approximations of the offset curve segments and determine the arrangement of resulting line segments. (See Reference [12] for the current state-of-the-art of robust line segments arrangement in the plane.) Ahn, Kim, and Lim [1] demonstrated the efficiency and robustness 19
Local Loops
Global Loop
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Figure 10: Elimination of Self-Intersection Loops
20
of this technique in approximating the boundary of a 2D general sweep. The general sweep is the most general form of sweep in which the moving object changes its shape dynamically while moving along a trajectory curve. The offset computation is a special case of general sweep computation. Therefore, the general technique of Reference [1] can be applied to the case of offsetting. There are also some computational shortcuts to be applied in the special case of offsetting [20]. The input curve is approximated by discrete points and their connecting piecewise line segments (Figure 10–(b)). The offset curve is then approximated by the offsets of the discrete points (Figure 10–(c)). (Lee and Kim [20] have more details on how to approximate the offset computation error in this discrete approximation method for planar curves.) The offset polygon may have redundancies which do not contribute to the final offset boundary. For example, the segment between each pair of corresponding cusps in the offset polygon is redundant since the segment belongs to the interior of the offset object. Each cusp corresponds to the curve point which has curvature equal to 1/r, where r is the offset radius. The input curve subsegment which has curvature higher than 1/r should be eliminated from consideration. Let a line segment p) = p1 p2 be offset into a line segment )o = o1 o2 . We can easily observe that, in a local redundant segment, the inner product of two vectors, p) · )o, has negative value [6]. Figure 10–(d) shows the offset polygon after eliminating these local redundant offset segments. In the next step, we use a plane sweep algorithm to detect all the intersections among the offset line segments (Figure 10–(e)), and construct a valid boundary topology for the offset polygon (Figure 10–(f)). Ahn, Kim, and Lim [1] suggested an efficient and robust plane sweep algorithm which constructs the boundary of a polygonal approximation of the general sweep. We use a similar algorithm in this paper. Once we have computed a polygonal approximation of the offset boundary, we can easily extract the valid parameter intervals of the boundary curve of the input object. These valid intervals correspond to the object boundary segments which contribute to the offset boundary segments (Figure 10–(g)). However, a direct computation of approximated offset curve segments from these valid parameter intervals may not lead to a correct result. The intersection points of polygonal approximations may not provide accurate parameter values for the intersection points of the offset curve segments. We need to refine the numerical precision of the intersection points among offset curve segments [16]. For this purpose, the intersections among offset line segments are used as initial solutions (Figure 10–(e)). In Figure 10–(h), the offset curve end points are refined from the small segments shown in thick curves. Figure 10–(i) shows the final offset boundary and the original input object boundary. The polygonal approximation may omit some valid loops in the offset. A simple way to resolve this problem might be to approximate the input curve with high precision. However, this method generates many tiny line segments which cause problems in robustness 21
as well as computational efficiency. The missing loops are due to the tangential and/or multiple intersections of the offset curve segments. These degeneracies correspond to the topological changes of the offset boundary as the offset radius r is slightly changed. For each degeneracy, the corresponding offset curve segments are intersected with high precision to determine a correct topology. However, it is not always possible to determine the correct topological arrangement while making them consistent with the numerical curve intersection data, especially when the curve segments have (almost) tangential/multiple intersections [15].
5
Experimental Results
Figures 11–13 show some experimental results on the construction of approximated offset curves. The input curve of Figure 11 is a uniform cubic B-spline 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. For each tolerance |δ|, approximated offset curves of (rational) degree 7 are generated by using the convolution with the quadratic B´ezier curve segments of Methods 1–5. They are compared with the offset curves generated by other methods, in terms of the number of control points generated. The input curve for Figure 12 is a cubic B´ezier curve with 4 control points: (−0.785938, 0.891849), (−0.993306, −0.59695), (0.3, −2.5), and (0.9, −0.2). An offset radius 1.0 is used and approximated offset curves of (rational) degree 7 are generated. In Figure 13, the input circle is represented by a rational quadratic B-spline curve with 4 curve segments and a total of 9 control points. An offset radius 0.6 is used and approximated offset curves of (rational) degree 6 are generated. Note that, in Methods 3–5, the convolution curve segments, Cv (t)+v, have the same degree as the base curve since they are simple translations of the original curve subsegments. The table in each figure shows the numbers of control points for the approximated offset curves generated by different methods. The columns under labels M1 , . . . , M5 provide the results of Methods 1,. . . ,5, respectively. In Methods 3–5, the convolution curve segments, Cv (t) + v, may be further approximated by points, line segments, and quadratic curve segments within the given error bound (see the discussion at the end of Section 3.3). The numbers in parentheses (of Methods 3–5) show the best case lower bounds in which all the convolution curve segments, Cv (t) + v, could be replaced by points. The tolerances are given by the value of |δ|, which is the absolute maximum value of the function: φ(t) ≈ 2ε(t) (see Section 2.1). The method Cob corresponds to the divide-and-conquer based method of Cobb [3], in 22
which the global error bounding method of [6] is adapted to measure the offset approximation error instead of measuring the distance between the midpoints of the exact and approximated offset curves. If the approximation error is above the given tolerance, the input curve is subdivided at the midpoint of the curve. Most of the previous methods belong to this class of midpoint subdivision. The method Elb corresponds to the adaptive subdivision method of Elber and Cohen [6] which is applied to the offset curve construction method of Cobb [3]. When the approximation error is above the given tolerance, a knot point is inserted at the parameter value which corresponds to the maximum error. The method Elb 2 is a variation of Elb which subdivides the input curve at the parameter value of maximum error. The method Til is an extension of Tiller and Hanson [24] while adapting the global error bounding technique of [6]. The control polygon of an input spline curve is offset to define the control polygon of the offset curve. In Figure 13, the input circle is represented by a rational quadratic B-spline curve for which the control polygon has a regular shape. Therefore, the offset control polygon also has a regular shape and represents the exact offset circle with no approximation error. As long as the control polygon of an input curve has no sharp corner, the method of Tiller and Hanson [24] performs very nicely. However, as shown in the examples of Figures 11 and 12, when the control polygon has sharp corners, the offset control polygon does not provide a good approximation to the exact offset curve. The method Lst is an extension of Hoschek and Wissel [18] while adapting the global error bounding technique of [6]. Instead of sampling only the midpoint, a finite number of sample points are generated and the approximated offset curve is constructed as a uniform B-spline curve. The B-spline control points are determined so that they generate a piecewise cubic B-spline curve with the least squares error from the discrete sample points. If the approximation error is above the given tolerance, instead of subdividing the input curve as in [18], more control points are used for the construction of an approximated offset Bspline curve. This modification reduces the total number of control points generated. The method Lst 2 is a variation of Lst which subdivides the input curve at the parameter value of maximum error. Except for some special cases such as the one shown in Figure 13, the optimization based method Lst generates a smaller number of control points than all the other methods. Our method ranks just next to the methods Lst and Lst 2 for high precision offset approximation. The main reason why our method generates more control points than Lst and Lst 2 is that our offset curves have relatively high degree. Because of this, our method performs worse than all the other methods up to the approximation error 10−2 in the examples shown in Figures 11–13. However, when we consider the number of curve subdivisions as the performance criterion, our method performs better. For example, in Figure 11, the number of offset curve segments generated by Methods 1–2 is about 7 times less than the number of control points, 23
which is even less than the number of cubic B-spline offset curve segments generated by the method Lst. The question is: what is the rationale to support this performance criterion ? As one might have realized in the offset examples of Figures 11–13, data explosion is a serious problem in offset computation, especially for high precision approximation. Therefore, it is necessary to store only a minimum amount of computation results and generate all the other geometric information from these minimum data on the fly when they are required. In the case of subdivision based methods [3, 4, 17, 19, 23, 24], we can store only the parameter values of the base curve at which the curve subdivisions have been made. It is quite straightforward to generate the control points of the offset curve segments on the fly from these knot points on the base curve. Storing the corresponding parameter values instead of the control points reduces the data storage significantly. In our method, assuming degree 7 rational curves are used for the offset curve approximation, this storage reduction would result in a significant saving, with almost 14 times less output data size. Unfortunately, it is difficult to apply this technique to the optimization based method Lst since the method Lst applies the B-spline curve approximation directly to the offset curve. Therefore, the relationship between the control point and the base curve parameter value is not clear. However, for this purpose, we can use the method Lst 2 instead of Lst since the base curve subdivision is used in Lst 2 . Figure 14 shows the result of this storage reduction technique applied to the examples shown in Figures 11–13. The numbers of base curve subdivisions are shown for the subdivision based methods. The method Lst 2 produces a smaller number of curve subdivisions when higher degree B-spline curve segments are used for the offset curve approximation. However, we have not used high degree curves in the experiment because of high computational requirement for the evaluation of global error bounds in the offset approximation, especially for high precision approximation. This fact also demonstrates the effectiveness of our methods in that the global error bounds are estimated automatically by measuring the circle approximation errors. Once the range of curve tangent directions has been computed, given any offset approximation error, the number of required curve subdivisions are estimated a priori without doing any curve subdivision at all. Figure 15 shows some practical examples which require this kind of output data size reduction. The first three examples (a)–(c) have applications in the generation of decorative fonts and shapes in desktop publishing. The last example (d) has applications in the generation of NC tool paths for pocket machining. In these applications, the data explosion is a serious problem. The examples are generated by offsetting 2D curved objects. The base objects are shown in thick curves and their offsets for various offset radii are shown in thin curves. They are constructed by eliminating all the self-intersecting loops of the offset curves (see Section 4). All the input curves are uniform cubic B-spline curves. 24
0.1 Cob Elb Elb_2 Til Lst Lst_2 Method_1
0.01
Offset Error
0.001
0.0001
1e-05
1e-06 10
100
1000
10000
Number of control points
|δ| 10−1 10−2 10−3 10−4 10−5
Cob 28 73 208 637 1846
Elb 19 57 174 417 1357
Elb2 22 55 190 550 1690
Til 25 67 202 640 1918
Lst 16 48 84 138 240
Lst2 31 49 94 166 277
M1 78 92 141 211 365
M2 78 92 120 176 302
M3 88 (64) 128 (92) 148 (106) 228 (162) 368 (260)
M4 88 (64) 108 (78) 128 (92) 228 (162) 368 (260)
M5 88 (64) 108 (78) 128 (92) 208 (148) 308 (218)
Figure 11: Offset for a Uniform Cubic B-spline Curve.
0.1 Cob Elb Elb_2 Til Lst Lst_2 Method_1
0.01
Offset Error
0.001
0.0001
1e-05
1e-06 1
10
100
1000
Number of control points
|δ| 10−1 10−2 10−3 10−4 10−5
Cob 10 31 94 316 865
Elb 11 24 74 216 974
Elb2 13 25 97 247 769
Til 10 31 97 322 886
Lst 7 13 19 31 50
Lst2 10 19 31 46 88
M1 29 36 50 85 141
M2 22 29 43 71 127
M3 18 (15) 38 (29) 48 (36) 88 (64) 138 (99)
M4 18 (15) 38 (29) 48 (36) 88 (64) 138 (99)
Figure 12: Offset for a Cubic B´ezier Curve.
25
M5 18 (15) 38 (29) 48 (36) 68 (50) 128 (92)
0.1 Cob Elb Elb_2 Til Lst Lst_2 Method_1
0.01
Offset Error
0.001
0.0001
1e-05
1e-06
1e-07
1e-08 1
|δ| 10−1 10−2 10−3 10−4 10−5
Cob 17 33 129 513 1025
Elb 13 53 69 261 524
Elb2 11 27 123 299 1019
Til 9 9 9 9 9
Lst 9 13 24 46 98
Lst2 9 17 33 65 177
10
M1 25 61 73 121 229
M2 25 49 73 121 199
100 Number of control points
M3 31 (25) 63 (49) 63 (49) 127 (97) 223 (169)
1000
M4 31 (25) 31 (25) 63 (49) 127 (97) 207 (157)
10000
M5 31 (25) 31 (25) 63 (49) 95 (73) 191 (145)
Figure 13: Offset for a Circle.
6
Conclusion
We presented a new offset curve approximation method which is based on the offset circle approximation. Given a unit circle approximation error , the approximated offset curve is guaranteed to be within an error bound r · from the exact offset curve as long as there are no self-intersections in the exact/approximated offset curves, where r is the offset radius. The error bound r · can also be guaranteed for the offset of a planar curved object when the offset boundary has no degeneracies such as (almost) tangential/multiple intersections. Since the circle approximation is easier to construct and the error bound can be estimated very precisely [5], our method provides an extremely simple error estimation to the offset approximation. The main disadvantage of our method is that the approximated offset curves are rational curves of relatively high degree. Nevertheless, our method generates very precise offset curve approximations with a smaller number of curve subdivisions than the previous methods. (See Reference [7] for detailed comparisons of the previous methods.) By storing only the parameter values of the curve subdivisions and generating the offset curve control points on the fly when they are needed, we can reduce the output data size significantly. As a result, the algorithm is very precise and efficient. In many practical applications
26
|δ| 10−1 10−2 10−3 10−4 10−5
Cob 7 22 67 210 613
Elb2 5 16 61 181 561
Til 6 20 65 211 637
Lst2 8 14 29 53 90
M1 10 12 19 29 51
M2 10 12 16 24 42
M3 8 12 14 22 36
M4 8 10 12 22 36
M5 8 10 12 20 30
M1 3 4 6 11 19
M2 2 3 5 9 17
M3 1 3 4 8 13
M4 1 3 4 8 13
M5 1 3 4 6 12
M1 0 6 8 16 34
M2 0 4 8 16 29
M3 0 4 4 12 24
M4 0 0 4 12 24
M5 0 0 4 8 20
(a) |δ| 10−1 10−2 10−3 10−4 10−5
Cob 2 9 30 104 287
Elb2 3 7 31 81 255
Til 2 9 31 106 294
Lst2 2 5 9 14 28 (b)
|δ| 10−1 10−2 10−3 10−4 10−5
Cob 4 12 60 252 508
Elb2 1 9 57 145 505
Til 0 0 0 0 0
Lst2 0 4 12 28 84 (c)
Figure 14: Number of Curve Subdivisions: (a) Figure 11, (b) Figure 12, and (c) Figure 13
27
(a)
(b)
(c)
(d)
Figure 15: (a) H Shape, (b) Human Shape, (c) Oryx Shape, and (d) Pocket Machining Paths
28
of planar curve offsetting (e.g., NC tool path generation), the accuracy of the offset curve approximation and the reduction of output data size are more important than any other criteria.
7
Acknowledgement
The authors thank the anonymous referees for their valuable comments and constructive suggestions. The counter-example shown in Figure 5 was provided by one of the referees.
References [1] Ahn, J.-W., Kim, M.-S., and Lim, S.-B., (1993), “Approximate General Sweep Boundary of a 2D Curved Object,” CVGIP: Graphical Models and Image Processing, Vol. 55, No. 2, pp. 98–128. [2] Bajaj, C., and Kim, M.-S., (1989), “Generation of Configuration Space Obstacles: The Case of Moving Algebraic Curves,” Algorithmica, Vol. 4, No. 2, pp. 157–172. [3] Cobb, B., (1984), Design of Sculptured Surfaces Using The B-spline Representation, Ph.D. thesis, University of Utah, Computer Science Department. [4] Coquillart, S., (1987), “Computing Offset of B-spline Curves,” Computer-Aided Design, Vol. 19, No. 6, pp. 305–309. [5] Dokken, T., Daehlen, M., Lyche, T., and Morken, K., (1990), “Good Approximation of Circles by Curvature-Continuous B´ezier Curves,” Computer Aided Geometric Design, Vol. 7, pp. 33–41. [6] Elber, G., and Cohen, E., (1991), “Error Bounded Variable Distance Offset Operator for Free Form Curves and Surfaces,” International Journal of Computational Geometry and Applications, Vol. 1, No. 1, pp. 67–78. [7] Elber, G., Lee, I.-K., and Kim, M.-S., (1995), “State of the Art in Offset Approximation of Freeform Curves,” Manuscript, September, 1995. [8] Farouki, R., and Neff, C., (1990), “Analytic Properties of Plane Offset Curves,” Computer Aided Geometric Design, Vol. 7, pp. 83–99. [9] Farouki, R., and Neff, C., (1990), “Algebraic Properties of Plane Offset Curves,” Computer Aided Geometric Design, Vol. 7, pp. 101–127. 29
[10] Farouki, R., and Sakkalis, T., (1990), “Pythagorean-hodographs,” IBM Journal of Research and Development, Vol. 34, pp. 736–752. [11] Goldapp, M., (1991), “Approximation of Circular Arcs by Cubic Polynomials,” Computer Aided Geometric Design, Vol. 8, pp. 227–238. [12] Guibas, L., and Marimont, D., (1995), “Rounding Arrangements Dynamically,” Proc. 11th Annual Symp. on Computational Geometry, pp. 190–199. [13] Guibas, L., Ramshaw, L., and Stolfi, J., (1983), “A Kinetic Framework for Computational Geometry,” Proc. 24th Annual Symp. on Foundations of Computer Science, pp. 100–111. [14] Held, M., (1991), On the Computational Geometry of Pocket Machining, Lecture Notes in Computer Science, Vol. 500, Springer-Verlag. [15] Hoffmann, C., (1989), Geometric & Solid Modeling: An Introduction, Morgan Kaufmann Pub., San Mateo, California. [16] Hoschek, J., (1985), “Offset Curves in the Plane,” Computer-Aided Design, Vol. 17, No. 2, pp. 77–82. [17] Hoschek, J., (1988), “Spline Approximation of Offset Curves,” Computer Aided Geometric Design, Vol. 5, pp. 33–40. [18] Hoschek, J., and Wissel, N., (1988), “Optimal Approximate Conversion of Spline Curves and Spline Approximation of Offset Curves,” Computer-Aided Design, Vol. 20, No. 8, pp. 475–483. [19] Klass, R., (1983), “An Offset Spline Approximation for Plane Cubic Splines,” ComputerAided Design, Vol. 15, No. 4, pp. 297–299. [20] Lee, I.-K., and Kim, M.-S., (1992), “Primitive Geometric Operations on Planar Algebraic Curves with Gaussian Approximation,” Visual Computing, T.L. Kunii (Ed.), Springer-Verlag, 1992, pp. 449–468. [21] Morken, K., (1991), “Best Approximation of Circle Segments by Quadratic B´ezier Curves,” Curves and Surfaces, P.J. Laurent, A. Le M´ehaut´e, and L.L. Schumaker (Eds.), Academic Press, Boston, pp. 331–336. [22] Persson, H., (1978), “NC Machining of Arbitrarily Shaped Pockets,” Computer-Aided Design, Vol. 10, No. 4, pp. 169–174. 30
[23] Pham, B., (1988), “Offset Approximation of Uniform B-splines,” Computer-Aided Design, Vol. 20, No. 8, pp. 471–474. [24] Tiller, W., and Hanson, E., (1984), “Offsets of Two Dimensional Profiles,” IEEE Computer Graphics & Application, Vol. 4, pp. 36–46.
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