Computing Offsets of NURBS Curve and Surface

2012 ◽  
Vol 542-543 ◽  
pp. 537-540
Author(s):  
Ying Yue ◽  
Jun Jia
Keyword(s):  
Curve Surface ◽  
Computational Time ◽  
Normal Vector ◽  
Unit Normal Vector ◽  
Nurbs Curve ◽  
Vector Direction ◽  
Normal Vectors ◽  
Offset Curve ◽  
Offset Error ◽  
Unit Normal

This paper presents an algorithm for the offsetting of NURBS curve/surface. First the unit normal vectors of the progenitor NURBS curve/surface is computed precisely, then the offset curve/surface can be obtained by offsetting the progenitor curve/surface in the normal vector direction with the required distance. Considerable extra computational time can be saved, especially when they are to be offset by several times. As the method successfully computes the unit normal vector of the progenitors, the offset error of this method is zero. The method can also be generalized to other degree NURBS curve/surface.

Algorithm for Tool Path Offsetting Based on NURBS Surface

2014 ◽  
Vol 945-949 ◽  
pp. 111-114
Author(s):  
Ying Yue
Keyword(s):  
Tool Path ◽  
Computational Time ◽  
Machining Accuracy ◽  
Normal Vector ◽  
Nurbs Surface ◽  
Unit Normal Vector ◽  
Vector Direction ◽  
Normal Vectors ◽  
Offset Surface ◽  
Unit Normal

This paper presents an algorithm for tool path offsetting based on NURBS surface. First the progenitor free surface is fitted with a bi-cubic NURBS surface and the unit normal vectors of the NURBS surface is computed precisely, then the offset surface can be obtained by offsetting the NURBS surface in the normal vector direction with the required distance. Considerable extra computational time can be saved, especially when they are to be offset by several times. As the method successfully computes the unit normal vector of the progenitors, the offset error of this method is zero. The method can also be generalized to other degree NURBS surface, and it can improve the machining accuracy of the surface.

Curvature Theory of the Envelop Curve / Surface for a Line in Planar Motion and a Plane in Spatial Motion

2010 ◽  
Author(s):  
Wei Wang ◽  
Delun Wang
Keyword(s):  
Curve Surface ◽  
Geodesic Curvature ◽  
Unit Vector ◽  
Planar Motion ◽  
Normal Vector ◽  
Developable Surface ◽  
Spatial Motion ◽  
Unit Normal Vector ◽  
Curvature Theory ◽  
Unit Normal

The curvature theories for the envelop curve of a line in planar motion and the envelop ruled surface of a plane in spatial motion are extensively researched in the differential geometry language. A line-envelop curve in planar motion is firstly derived by means of the adjoint approach. The higher order curvature theory of the envelop curve reveals a unified form in the infinitesimal and finitely separated positions for a line in planar motion. And then, a plane in spatial motion traces the envelop surface, which is a developable surface and whose invariants are concisely derived. The geodesic curvature of the spherical image curve for the generator’s unit vector is readily derived and compared with that of the unit normal vector of the envelop surface. As a result, the curvature theory for a plane-envelop surface in spatial motion are shown in terms of that of the spherical motion, corresponding to the generator’s unit vector and unit normal vector of the envelop surface. Meanwhile, the instantaneous cubic (cone) of stationary curvature and the direction “Burmester’s line” of the generator of the developable envelop surface are revealed. Therefore, a solid theoretical basis is provided for the synthesis of mechanisms and the machining of surface.

scholarly journals Design and Analysis of Posterior Semicircular Canal BPPV Diagnostic Test Based on Physical Simulation

2021 ◽  
Author(s):  
Xiaokai Yang ◽  
Qiancheng Yang ◽  
Zhaobang Liu
Keyword(s):  
Diagnostic Test ◽  
Semicircular Canal ◽  
Physical Simulation ◽  
Normal Vector ◽  
Forward Angle ◽  
Unit Normal Vector ◽  
Posterior Semicircular Canal ◽  
Average Value ◽  
Crista Ampullaris ◽  
Unit Normal

Abstract To discusses and analyzes how to realize the design of posterior semicircular canal BPPV diagnostic maneuver. First, measure the spatial attitude of the human semicircular canal, establish a BPPV virtual simulation platform, then analyze the key positions of the maneuver, and finally design a new diagnostic maneuver according to the demand, and perform physical simulation verification. The average value of the unit normal vector of the right posterior semicircular plane is [ 0.660, 0.702, 0.266], after rotate -46.8 ° around Z axis and 15.4 ° around Y axis, it parallel to the X axis. After that, when the tilt back angle reaches 70 °, the free otoconia in the left utricle will fall into the common crus; when bend forward 53.3°, the unit normal vector of the crista ampullaris plane of the posterior semicircular canal to the XY plane; when bend forward angle reaches 30°, the otoconia slides to the opening of the ampulla; when bend forward angle reaches 70°, the otoconia slides to the bottom of the crista ampullaris. The shallow pitching Yang maneuver is designed as turn head 45° to the one side, bend forward 45°, tilt back 90°, and bend forward 90°. The deep pitching Yang maneuver is designed as bend forward 90°, turn head 45° to one side, tilt back 135°, and bend forward 90°. A new posterior semicircular BPPV diagnostic test is designed to make the induced nystagmus have the characteristics of long latency, reversal, and repeatability, will not cause the inhibitory stimulation of the contralateral superior semicircular canal, and has good operation fault tolerance, which is of great value for clinical and scientific research.

Spaces and Operators

2012 ◽  
Author(s):  
G. F. Roach ◽  
I. G. Stratis ◽  
A. N. Yannacopoulos
Keyword(s):  
Normal Vector ◽  
Connected Component ◽  
Unit Normal Vector ◽  
Open Set ◽  
Outward Unit ◽  
Outward Unit Normal Vector ◽  
Unit Normal

This chapter consists mainly of definitions and various properties (without proofs) of spaces and operators used in this book. It defines O as an open set in Rᶰ such that it is locally on one side of its boundary Γ‎ := δ‎O, which is supposed to be bounded and Lipschitz. The chapter is mainly focused on the case of N = 3. Further, without loss of generality, the chapter supposes that Γ‎ is connected (for otherwise, one could work separately at each connected component). Such a set O is referred to as ‘regular’ in what follows. Let n denote the outward unit normal vector to Γ‎. In addition, let Oₑ := Rᶰ∖Ō: By N₀ we denote the set N ∪ {0}.

scholarly journals Constant angle spacelike surface in de Sitter space S₁³

2017 ◽  
Vol 35 (3) ◽  
pp. 79-93
Author(s):  
Tugba Mert ◽  
Baki Karlıga
Keyword(s):  
Vector Field ◽  
De Sitter Space ◽  
Normal Vector ◽  
Spacelike Surface ◽  
De Sitter ◽  
Unit Normal Vector ◽  
Normal Vector Field ◽  
Constant Angle ◽  
Unit Normal Vector Field ◽  
Unit Normal

In this paper; using the angle between unit normal vector field of surfaces and a fixed spacelike axis in R₁⁴, we develop two class of spacelike surface which are called constant timelike angle surfaces with timelike and spacelike axis in de Sitter space S₁³.

scholarly journals Extension of the unit normal vector field from a hypersurface

2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Roland Duduchava ◽  
Eugene Shargorodsky ◽  
George Tephnadze
Keyword(s):  
Vector Field ◽  
Existence And Uniqueness ◽  
Elementary Proof ◽  
Gradient Field ◽  
Normal Vector ◽  
Unit Normal Vector ◽  
Normal Vector Field ◽  
Unit Normal Vector Field ◽  
Outer Unit ◽  
Unit Normal

AbstractIn many applications it is important to be able to extend the (outer) unit normal vector field from a hypersurface to its neighborhood in such a way that the result is a unit gradient field. The aim of this paper is to provide an elementary proof of the existence and uniqueness of such an extension.

The Hodograph and the Balancer of any Frequency Term of the Shaking Force of Spatial Mechanisms

1996 ◽  
Vol 210 (2) ◽  
pp. 135-141 ◽  
Author(s):  
S-T Chiou ◽  
J-C Tzou
Keyword(s):  
Fourier Coefficients ◽  
Normal Vector ◽  
Unit Normal Vector ◽  
Cycle Frequency ◽  
Spatial Mechanisms ◽  
Hodograph Plane ◽  
Frequency Term ◽  
Unit Normal

It is proved in this paper that the hodograph of a frequency term (for example the kth frequency term) of the shaking force of spatial mechanisms is an ellipse. Furthermore, expressions are provided for the lengths and attitudes of the semi-axes of this ellipse in terms of Fourier coefficients of the shaking force. Accordingly, a pair of counterweights, contra-rotating at k times of cycle frequency with their axes parallel to the unit normal vector of the hodograph plane, can be installed for eliminating the frequency term of the shaking force of spatial mechanisms. An example of a seven-link 7-R spatial linkage is included.

Calculation of the unit normal vector for wall shear stress in the lattice Boltzmann model

2020 ◽  
Vol 199 ◽  
pp. 104422
Author(s):  
Li Min ◽  
Huang Jingcong ◽  
Zhang Yang ◽  
Wang Yuan ◽  
Wu Changsong ◽  
...  
Keyword(s):  
Shear Stress ◽  
Wall Shear Stress ◽  
Lattice Boltzmann ◽  
Lattice Boltzmann Model ◽  
Normal Vector ◽  
Unit Normal Vector ◽  
Wall Shear ◽  
Boltzmann Model ◽  
Unit Normal

scholarly journals Harmonic Evolute Surface of Tubular Surfaces via B -Darboux Frame in Euclidean 3-Space

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
E. M. Solouma ◽  
Ibrahim AL-Dayel
Keyword(s):  
Mean Curvature ◽  
Normal Vector ◽  
Unit Normal Vector ◽  
Geometric Characteristics ◽  
Darboux Frame ◽  
Tubular Surfaces ◽  
Unit Normal

In this article, we look at a surface associated with real-valued functions. The surface is known as a harmonic surface, and its unit normal vector and mean curvature have been used to characterize it. We use the Bishop-Darboux frame ( B -Darboux frame) in Euclidean 3-space E 3 to study and explain the geometric characteristics of the harmonic evolute surfaces of tubular surfaces. The characterizations of the harmonic evolute surface’s ϱ and ς parameter curves are evaluated, and then, they are compared. Finally, an example of a tubular surface’s harmonic evolute surface is presented, along with visuals of these surfaces.

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