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Non-uniform B-spline optimization method for generating swept surfaces

Xiaoyan Kui, Min Yang, Songpeng Yao, Hao Wang, Enya Shen, Qinsong Li, Beiji Zou

arXiv:2609.16042v1cs.GR

TL;DR

Swept surfaces are widely used in CAD, motivating work on high-precision approximation. The paper uses non-uniform B-splines with feature selection guided by surface irregularity; it reports fewer feature points at comparable accuracy and reduced average error versus Pagani et al.

  • Problem

    Swept surfaces are widely used in CAD, and the paper focuses on improving their high-precision approximation using surface shape characteristics to guide sampling.

    Source: §I · p. 1

  • Method

    A non-uniform B-spline method derives a surface-irregularity distribution from geometric features, adjusts feature weights across samples, selects feature points, and estimates approximation error to add points as needed.

    Source: B. Calculation of the distribution… · p. 4, §V · p. 9

  • Results

    At specified approximation accuracy 0.01, required feature points fall about 15.85% versus Pagani et al.; at multiplier 10, average error falls about 51.35% versus Pagani et al.

    Source: B. Analysis of the selected multiplier · p. 6, C. Analysis of specified accuracy · p. 7

  • Takeaways & Limitations

    The method can produce comparable approximation accuracy with fewer feature points and has better stability than the commonly used comparison methods.

    Source: §V · p. 9

Abstract

from arXiv · show

Swept surface construction is widely used in computer-aided design. We propose a novel optimization method using non-uniform B-splines to improve the approximate accuracy of swept surfaces. First, discrete points on the swept shape are computed, and geometric properties such as surface area, discrete curvature, first-order derivatives, and their rotation angles are used to derive a distribution function representing surface irregularity, with weights adjusted from samples. Then, feature points are selected based on the distribution function to determine control points for the approximate non-uniform B-spline surface via inverse calculation, producing an optimized approximation. Finally, the number of feature points is adjusted based on the estimated approximation error. Experiments on 969 randomly generated sweep samples and 1 pipe example show that the proposed algorithm achieves similar accuracy with fewer control points, reducing them by about 15.85% at a specified accuracy of 0.01. Moreover, with ample sampling points, it reduces the average error by approximately 51.35% when the feature point multiple is 10 times the path control points, outperforming comparable methods.

I. INTRODUCTION

Swept surfaces are formed by moving a cross-section curve along a path, but NURBS-based approximations can be inaccurate, especially for complex paths. The paper proposes surface-informed feature-point selection to improve approximation, contrasting it with knot-only refinement.

I. INTRODUCTION

  • NURBS approximations can struggle with complex paths, while earlier approaches either offset path control points or add points by subdividing path knots.Coquillart’s offset method may lack global fine control; Piegl et al. subdivide path knot intervals, making the result sensitive to parameterization and path-knot information.Source: §I · p. 1
  • The proposed method selects feature points from swept-surface shape information to improve non-uniform B-spline approximation accuracy.The paper motivates this approach by noting that surface shape can guide sampling.Source: §I · p. 1

II. FUNDAMENTALS OF SWEPT SURFACE APPROXIMATION

  • A swept-surface approximation places cross-section curves at selected path feature points, then solves for the approximate surface’s control points.The paper notes that the transformation matrix in a general sweep often cannot be accurately represented by NURBS [2].Source: §II · p. 2

A. Feature point selection.

  • The mid-knot method inserts the midpoint of the longest path-knot interval; enough subdivisions can achieve a specified accuracy, but the method ignores swept-surface information and may reduce error slowly.The method focuses on path-knot information when selecting feature points.Source: A. Feature point selection · p. 2

B. Reverse calculation of control points

The approximate surface is represented as a non-uniform B-spline, and its control points are recovered by inverse calculation from cross-section placements at path feature points.

B. Reverse calculation of control points

  • The approximate surface is expressed using B-spline basis functions in both parameter directions and surface control points.The basis functions correspond to the cross-section curve and path, respectively.Source: B. Reverse calculation of control points · p. 2
  • Cross-section control points are placed at each feature point to preserve accuracy along the surface’s u-direction isoparametric lines.The initial control points are obtained using the cross-section curve’s knot vector in the approximate surface’s u-direction.Source: B. Reverse calculation of control points · p. 2
  • Inverse calculation solves a linear system for the approximate surface control points, with parameter values chosen to keep its coefficient matrix non-singular.The passage identifies the solution as inverse calculation of the control points.Source: B. Reverse calculation of control points · p. 2

C. Local coordinate frame calculation

For general sweeps without additional cross-section operations, the method uses a rotation minimizing frame to orient the section along the path. It computes the frame with the double reflection method.

C. Local coordinate frame calculation

  • The rotation minimizing frame is used for sweeps without additional operations such as rotating the cross-section or keeping it parallel to a fixed direction.The frame is the transformation matrix for this general sweep setting.Source: C. Local coordinate frame calculation · pp. 2–3
  • The double reflection method advances the path frame from one point to the next through two reflections.For planar and spherical curves it gives an exact RMF; for spatial curves its global approximation error is fourth-order.Source: C. Local coordinate frame calculation · p. 3, Figure 2 · p. 3
  • The RMF adds no rotation relative to the path tangent and requires only the path’s first derivative.The paper treats frame-calculation error as negligible when the step size is chosen appropriately.Source: C. Local coordinate frame calculation · p. 3

D. Error calculation

The method samples swept surfaces and uses geometric features to measure their irregularity. To estimate approximation error more efficiently than a surface-to-surface Hausdorff calculation, it converts that error to path offset error.

D. Error calculation

  • Hausdorff distance requires sampling in both surface directions, and its calculation error depends on the sampling step size.The supplied passages note that high-accuracy use can require excessive time.Source: D. Error calculation · p. 3
  • The estimator converts swept-surface approximation error into path offset error and requires only path-direction sampling, with time complexity O(n_v).It avoids distance calculations between point sets.Source: D. Error calculation · p. 3

A. Calculation of swept surface information

  • Surface irregularity is measured from sampled area, curvature, first derivatives, and derivative rotation angles; only v-direction information is used for derivatives and rotation angles.The u-direction curvature is replaced by the cross-section curve curvature at the corresponding point.Source: A. Calculation of swept surface… · pp. 3–4

B. Calculation of the distribution function

The distribution function allocates more feature points to regions of greater surface irregularity. Its feature weights can be adjusted for each sweep using sampled approximation errors, and the authors report that this makes the distribution reflect sample characteristics more accurately.

B. Calculation of the distribution function

  • The distribution function allocates more feature points where surface irregularity is high and fewer where it is low.Its inputs include cumulative area, discrete curvature, first derivative, and derivative rotation angle in the v-direction.Source: B. Calculation of the distribution… · p. 4
  • Feature weights are dynamically adjusted because the influence of each surface feature can differ across sweep examples.The weights represent how much attention each feature receives when measuring irregularity.Source: B. Calculation of the distribution… · p. 4
  • The authors report that dynamically adjusted weights make the calculated distribution function more accurately reflect each sample's surface characteristics.The method's effectiveness was evaluated in Section IV.D.Source: B. Calculation of the distribution… · p. 5

C. Method of selecting feature points

Feature points are selected to distribute surface irregularity evenly across the chosen points. When a target accuracy is specified, the method increases the point count iteratively until the estimated error meets that target or an iteration limit is reached.

C. Method of selecting feature points

  • Selected feature points approximately equalize the v-direction distribution function across consecutive intervals.The feature-point parameters are drawn from the v-direction surface sampling points.Source: C. Method of selecting feature points · p. 5
  • After feature-point selection, the average method calculates the v-direction knot vector to keep the control-point inversion matrix invertible.The resulting control points yield a non-uniform B-spline approximation surface.Source: C. Method of selecting feature points · p. 5
  • For specified approximation-surface accuracy ε, the method initializes feature-point count to path control-point count, calculates approximation error against the ideal surface, and increases the count until accuracy is reached or the preset iteration limit.Source: C. Method of selecting feature points · p. 5

D. Overall methodology pipeline

The paper presents its swept-surface approximation method as a pipeline combining surface-information distribution, feature-point selection, and control-point inversion. Figure 3 summarizes this overall methodology.

D. Overall methodology pipeline

IV. EXPERIMENT

The experiments compare the proposed method with similar methods on randomly generated swept surfaces and separately examine dynamic weight adjustment. The setup uses ACIS-generated surfaces and uniform parameter-direction sampling.

IV. EXPERIMENT

  • ACIS screens randomly generated cross-sections and paths for valid swept surfaces.The RMF calculation step size is 0.01.Source: §IV · p. 6
  • Surface information is calculated using uniform sampling along the parameter direction.Source: §IV · p. 6
  • The experiments compare the proposed method with similar methods and separately analyze dynamic weight adjustment.The stated aims are to assess performance on randomly generated examples and the effectiveness of the weight-adjustment module.Source: §IV · p. 6

A. Random sample generation method

The study generates cubic cross-section curves and paths by randomizing their control points and knot vectors, then screens samples for valid swept surfaces. The resulting set contains 969 samples.

A. Random sample generation method

  • The cross-section curve lies in the xoy plane, has degree 3, and has 4–8 randomly generated control points.Source: A. Random sample generation method · p. 6
  • The path has degree 3, and its number of control points is randomly generated between 4 and 10; the corresponding knots obey U(0,1), except σ.U(a,b) denotes a uniform distribution on the interval (a,b).Source: A. Random sample generation method · p. 6
  • 969 samples passed legality checks, including checks for self-intersection and sharp points, and were used to test performance.The authors state that the random samples reflect characteristics of most swept-surface types.Source: A. Random sample generation method · p. 6

B. Analysis of the selected multiplier

On 969 random samples, the proposed method’s average error advantage over Pagani et al. grows at higher selection multiples. The pipe example shows a similar trend as feature-point counts increase.

B. Analysis of the selected multiplier

  • When few feature points are specified, the mid-knot method performs better overall; as the required count increases, the proposed method performs well against both alternatives.The authors attribute the small-count advantage to initializing feature points from the path knot vector, which captures v-direction features.Source: B. Analysis of the selected multiplier · p. 6
  • At selection multiples 7 and 10, the proposed method reduces average error versus Pagani et al. by about 13.63% and 51.35%, respectively.Table I reports average error across methods under different selection multiples.Source: B. Analysis of the selected multiplier · p. 6, Table I · p. 6
  • The practical example is a pipe surface swept along a path using a closed cross-section curve.Figure 5 presents the pipe model and its corresponding cross-section and path.Source: B. Analysis of the selected multiplier · p. 7, Figure 5 · p. 7
  • On the pipe example, mid-knot performs better at small feature-point counts, but its error reduction slows and may stagnate; the proposed method performs better as counts increase.The authors attribute this contrast to mid-knot's lack of swept-surface information during feature-point selection.Source: Figure 6 · p. 7, B. Analysis of the selected multiplier · p. 7
  • With 90 feature points, the proposed method places more points at two highly irregular surface segments, while the alternatives show larger errors at curved sections.The comparison reports lower error for the proposed method with the same number of feature points.Source: B. Analysis of the selected multiplier · p. 7

C. Analysis of specified accuracy

At target accuracies 0.1 and 0.01, the method used fewer feature points than both comparators while obtaining comparable approximation accuracy. The reported tests also indicate fewer non-converged samples and improved errors with dynamic weight adjustment.

C. Analysis of specified accuracy

  • At accuracy 0.1, feature points fell 22.60% vs mid-knot and 14.38% vs Pagani et al.[4]; at 0.01, reductions were 16.81% and 15.85%, respectively.At both targets, the method achieved comparable approximation accuracy with the fewest feature points, implying fewer control points because their count matches the feature-point count.Source: Table II · p. 7, C. Analysis of specified accuracy · p. 7
  • At selection multiple 10 on the pipe example, errors were 0.0033 for mid-knot, 0.0015 for Pagani et al.[4], and 0.0008 for ours.Source: Figure 7 · p. 8
  • At accuracy 0.01 on the pipe example, feature-point multiples were 8 for mid-knot, 6 for Pagani et al.[4], and 4 for ours, with respective errors 0.0075, 0.0079, and 0.0050.The paper describes these as comparable surface approximation accuracies, with fewer feature points used by the proposed method.Source: Figure 9 · p. 8, C. Analysis of specified accuracy · p. 9
  • The method had significantly fewer non-converged samples than mid-knot and Pagani et al.[4], indicating stronger stability.Table III reports non-converged sample counts under different accuracy requirements.Source: C. Analysis of specified accuracy · p. 8, Table III · p. 9

D. Effectiveness analysis of weight adjustment

V. CONLUSION

  • With many specified feature points, the proposed method achieved better accuracy than mid-knot and Pagani et al.[4].The conclusion also reports comparable accuracy with fewer feature points and better stability.Source: §V · p. 9
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