Mitered Offsets and Skeletons for Circular Arc Polygons

2020 ◽  
Vol 30 (03n04) ◽  
pp. 235-256
Author(s):  
Bastian Weiß ◽  
Bert Jüttler ◽  
Franz Aurenhammer
Keyword(s):  
Monotonicity Property ◽  
Free Form ◽  
Shape Preserving ◽  
Circular Arc ◽  
Spline Curves ◽  
Circular Arcs ◽  
Event Driven ◽  
Planar Shapes ◽  
Practical Performance

The offsetting process that defines straight skeletons of polygons is generalized to arc polygons, i.e., to planar shapes with piecewise circular boundaries. The offsets are obtained by shrinking or expanding the circular arcs on the boundary in a co-circular manner, and tracing the paths of their endpoints. These paths define the associated shape-preserving skeleton, which decomposes the input object into patches. While the skeleton forms a forest of trees, the patches of the decomposition have a radial monotonicity property. Analyzing the events that occur during the offsetting process is non-trivial; the boundary of the offsetting object may get into self-contact and may even splice. This leads us to an event-driven algorithm for offset and skeleton computation. Several examples (both manually created ones and approximations of planar free-form shapes by arc spline curves) are analyzed to study the practical performance of our algorithm.

scholarly journals Measurement point sampling method for inspection of parts with free-form surfaces

2018 ◽  
Vol 10 (11) ◽  
pp. 168781401880957 ◽  
Author(s):  
Dezhong Zhao ◽  
Wenhu Wang ◽  
Jinhua Zhou ◽  
Ruisong Jiang ◽  
Kang Cui ◽  
...  
Keyword(s):  
Measurement Accuracy ◽  
Sampling Method ◽  
Free Form ◽  
Surface Inspection ◽  
B Spline ◽  
Spline Curves ◽  
Data Points ◽  
Manufacturing Accuracy ◽  
Method Of Measurement ◽  
Free Form Surfaces

Parts must be measured to evaluate the manufacturing accuracy in order to check whether their dimension is in expected tolerance. In engineering, parts with free-form surfaces are generally measured by high-precision coordinate-measuring machines. The measurement accuracy is usually improved by increasing the density of measurement points, which is time-consuming and costly. In this article, a novel sampling method of measurement points for free-form surface inspection is proposed. First, surface inspection is simplified into the inspection of a number of section curves of the surface. Second, B-spline curves constructed with an iterative method are employed to approximate these section curves. Subsequently, data points necessary to construct the B-spline curves are taken as the measurement points. Finally, the proposed method is compared with other two sampling methods. The results indicate that the proposed method greatly reduced the number of measurement points without decreasing the precision of surface modeling.

scholarly journals Exploring Benefits of Using Blending Splines as Transition Curves

2020 ◽  
Vol 10 (12) ◽  
pp. 4226
Author(s):  
Tanita Fossli Brustad ◽  
Rune Dalmo
Keyword(s):  
High Speed ◽  
Transition Curve ◽  
End Points ◽  
Circular Arc ◽  
Track Geometry ◽  
Railway Construction ◽  
Spline Curves ◽  
High Speed Trains ◽  
Transition Curves

Track geometry is a fundamental subject in railway construction. With the demand for increased capacity in terms of load and speed, the need for suitable transitions between consecutive track sections is highly relevant. Properly constructed transition curves lead to improved travel comfort, increased safety, and reduced wear. The well known clothoid curve is widely used as a transition curve; however, the linear curvature is not sufficiently smooth to meet the requirements for railways carrying high speed trains or heavy hauls. Blending spline curves are flexible spline constructions possessing favourable smoothness properties at the end points, which makes them considerable for use as transition curves. This paper demonstrates some selected blending splines applied as transition curves between two existing circular arc segments selected from the Ofotbanen railway. The main results in this paper are related to the smoothness at the end points and the behaviour of the curvature of the curves, where the new transition curves were shown to be smoother than the original clothoid. Another new result is the observation that the proposed method allows for the improvement of existing railways without forcing extensive changes to the original track. Some representative examples are included to highlight the flexibility of this first instance of blending splines as transition curves.

Shape-Preserving Positive Trigonometric Spline Curves

2016 ◽  
Vol 42 (2) ◽  
pp. 763-775
Author(s):  
Malik Zawwar Hussain ◽  
Farsia Hussain ◽  
Muhammad Sarfraz
Keyword(s):  
Shape Preserving ◽  
Spline Curves ◽  
Trigonometric Spline

scholarly journals On the generation of spiral-like paths within planar shapes

2017 ◽  
Vol 5 (3) ◽  
pp. 348-357 ◽  
Author(s):  
Martin Held ◽  
Stefan de Lorenzo
Keyword(s):  
Simple Algorithm ◽  
Basic Algorithm ◽  
Line Segments ◽  
Spiral Path ◽  
Curvature Variation ◽  
Circular Arcs ◽  
Straight Line ◽  
Planar Shapes ◽  
Double Spiral ◽  
Step Over

Abstract We simplify and extend prior work by Held and Spielberger [CAD 2009, CAD&A 2014] to obtain spiral-like paths inside of planar shapes bounded by straight-line segments and circular arcs: We use a linearization to derive a simple algorithm that computes a continuous spiral-like path which (1) consists of straight-line segments, (2) has no self-intersections, (3) respects a user-specified maximum step-over distance, and (4) starts in the interior and ends at the boundary of the shape. Then we extend this basic algorithm to double-spiral paths that start and end at the boundary, and show how these double spirals can be used to cover complicated planar shapes by composite spiral paths. We also discuss how to improve the smoothness and reduce the curvature variation of our paths, and how to boost them to higher levels of continuity. Highlights The algorithm computes a spiral path within planar shapes with and without islands. It respects a user-specified maximum step-over distance. Double spirals and composite spiral paths can be computed. Heuristics for smoothing the spirals are discussed. The algorithm is simple and easy to implement, and suitable for various applications.

Design constraints of five-arc Roots vacuum pumps

2002 ◽  
Vol 216 (2) ◽  
pp. 225-234 ◽  
Author(s):  
P-Y Wang ◽  
Z-H Fong ◽  
H S Fang
Keyword(s):  
Design Procedure ◽  
Vacuum Pump ◽  
Tooth Profile ◽  
Design Constraints ◽  
Circular Arc ◽  
Ultimate Pressure ◽  
Rotor Profile ◽  
Circular Arcs ◽  
Centre Of Rotation ◽  
Inlet Opening

The design constraints for the tooth profile of the five-arc Roots vacuum pump are derived and discussed in this paper. The addendum portion of the five-arc tooth profile comprises five smoothly connected circular arcs, while the dedendum portion consists of conjugate curves of the addendum portion of the mating rotor. The top land of the proposed rotor profile is a circular arc with its centre coinciding with the centre of rotation. Therefore, the gap between the top land of the rotor and the wall of the chamber turns into a long and narrow path, which provides better gas sealing and wider inlet opening. The design constraints of the rotor profile are quite complex owing to the limitations of zero carryover and the condition of non-undercutting. A design procedure is proposed for determining the feasible design region by considering the geometry constraints, zero carryover and non-undercutting. By using the proposed procedure, wider inlet opening and better gas sealing are expected, while the characteristic of zero carryover is maintained. The results of experiment show that the ultimate pressure of the prototype of the five-arc Roots vacuum pump is 2,5 × 10-3 torr, and the maximum pumping speed is 1600L/min. The performance of the prototype is excellent compared with commercially available mechanical dry vacuum pumps.

scholarly journals From 2D to 3D: Construction of a 3D Parametric Model for Detection of Dental Roots Shape and Position from a Panoramic Radiograph—A Preliminary Report

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Laura Mazzotta ◽  
Mauro Cozzani ◽  
Armando Razionale ◽  
Sabrina Mutinelli ◽  
Attilio Castaldo ◽  
...  
Keyword(s):  
Panoramic Radiograph ◽  
Maximum Error ◽  
Parametric Model ◽  
Parametric Models ◽  
Free Form ◽  
Spline Curves ◽  
Accuracy Level ◽  
Root Movement ◽  
Free Form Surfaces ◽  
Movement Simulation

Objectives. To build a 3D parametric model to detect shape and volume of dental roots, from a panoramic radiograph (PAN) of the patient.Materials and Methods. A PAN and a cone beam computed tomography (CBCT) of a patient were acquired. For each tooth, various parameters were considered (coronal and root lengths and widths): these were measured from the CBCT and from the PAN. Measures were compared to evaluate the accuracy level of PAN measurements. By using a CAD software, parametric models of an incisor and of a molar were constructed employing B-spline curves and free-form surfaces. PAN measures of teeth 2.1 and 3.6 were assigned to the parametric models; the same two teeth were segmented from CBCT. The two models were superimposed to assess the accuracy of the parametric model.Results. PAN measures resulted to be accurate and comparable with all other measurements. From model superimposition the maximum error resulted was 1.1 mm on the incisor crown and 2 mm on the molar furcation.Conclusion. This study shows that it is possible to build a 3D parametric model starting from 2D information with a clinically valid accuracy level. This can ultimately lead to a crown-root movement simulation.

Statistical evaluation of the influence of the uncertainty budget on B-spline curve approximation

2017 ◽  
Vol 11 (4) ◽  
Author(s):  
Xin Zhao ◽  
Hamza Alkhatib ◽  
Boris Kargoll ◽  
Ingo Neumann
Keyword(s):  
Laser Scanning ◽  
Statistical Significance ◽  
Model Misspecification ◽  
Uncertainty Budget ◽  
Free Form ◽  
Data Set ◽  
Nested Model ◽  
B Spline ◽  
Uncertainty In Measurement ◽  
Spline Curves

AbstractIn the field of engineering geodesy, terrestrial laser scanning (TLS) has become a popular method for detecting deformations. This paper analyzes the influence of the uncertainty budget on free-form curves modeled by B-splines. Usually, free-form estimation is based on scanning points assumed to have equal accuracies, which is not realistic. Previous findings demonstrate that the residuals still contain random and systematic uncertainties caused by instrumental, object-related and atmospheric influences. In order to guarantee the quality of derived estimates, it is essential to be aware of all uncertainties and their impact on the estimation.In this paper, a more detailed uncertainty budget is considered, in the context of the “Guide to the Expression of Uncertainty in Measurement” (GUM), which leads to a refined, heteroskedastic variance covariance matrix (VCM) of TLS measurements. Furthermore, the control points of B-spline curves approximating a measured bridge are estimated. Comparisons are made between the estimated B-spline curves using on the one hand a homoskedastic VCM and on the other hand the refined VCM. To assess the statistical significance of the differences displayed by the estimates for the two stochastic models, a nested model misspecification test and a non-nested model selection test are described and applied. The test decisions indicate that the homoskedastic VCM should be replaced by a heteroskedastic VCM in the direction of the suggested VCM. However, the tests also indicate that the considered VCM is still inadequate in light of the given data set and should therefore be improved.

B-SPLINE BASED STEREO FOR 3D RECONSTRUCTION OF LINE-LIKE OBJECTS USING AFFINE CAMERA MODEL

2001 ◽  
Vol 15 (02) ◽  
pp. 347-358 ◽  
Author(s):  
YIJUN XIAO ◽  
MINGYUE DING ◽  
JIAXIONG PENG
Keyword(s):  
Stereo Vision ◽  
Free Form ◽  
Control Points ◽  
Invariant Property ◽  
Affine Invariant ◽  
Camera Model ◽  
B Spline ◽  
Space Curves ◽  
Spline Curves ◽  
Affine Camera

This paper presents a novel curve based algorithm of stereo vision to reconstruct 3D line-like objects. B-spline approximations of 2D edge curves are selected as primitives for the reconstruction of their corresponding space curves so that, under the assumption of affine camera model, a 3D curve can be derived from reconstructing its control points according to the affine invariant property of B-Spline curves. The superiority of B-spline model in representing free-form curves gives good geometric properties of reconstruction results. Both theoretical analysis and experimental results demonstrate the validity of our approach.

Error Estimate of Approximate Double Circular Arc Interpolated Method for Curve Smoothing

2004 ◽  
Vol 471-472 ◽  
pp. 92-95 ◽  
Author(s):  
Zi Qiang Zhang ◽  
Qiu Sheng Yan ◽  
Zhi Dan Zheng ◽  
Shao Bo Chen
Keyword(s):  
Error Estimate ◽  
Reverse Engineering ◽  
Nc Machining ◽  
Intersection Point ◽  
Circular Arc ◽  
Original Curve ◽  
Circular Arcs ◽  
Curve Smoothing ◽  
Estimate Error

Approximate Double Circular Arc Interpolated Method which was put forward by author, is different from other circular arc interpolated methods in demanding only the corner between normal directions of each circular arcs at intersection point are less than designated allowed value but not demanding contiguous circular arcs are tangent, and makes the calculating be predigested. In order to estimate error of the method, emulated calculating is carried out, namely the course of curve being obtained by reverse engineering is simulated in this paper. The results show: if space between measure points is about 0.1mm in curve being obtained by reverse engineering, then, the most departure of smoothing results from original curve is 0.552μm for the stated example. Influence of the error on NC machining is quite small, so it can meet the needs of NC machining.

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