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Uniform Interpolation Constrained Geodesic Learning on Data Manifold [article]

Cong Geng, Jia Wang, Li Chen, Wenbo Bao, Chu Chu, Zhiyong Gao
2020 arXiv   pre-print
Specifically, we use an autoencoder network to map data samples into latent space and perform interpolation via an interpolation network.  ...  We add prior geometric information to regularize our autoencoder for the convexity of representations so that for any given interpolation approach, the generated interpolations remain within the distribution  ...  Theorem 1 If a piece-wise differentiable curve γ : [a, b] → M [8] , with parameter proportional to arc length, has length less or equal to the length of any other piece-wise differentiable curve joining  ... 
arXiv:2002.04829v4 fatcat:k3kdwfhzunf4xjnpm5cqgxzinm

A Two-stage CNC Interpolation Algorithm for Corner Smoothing Trajectories with Geometric Error and Dynamics Constraints

Jixiang Yang, Qin Hu, Han Ding
2016 Procedia CIRP  
The on-line interpolation stage generates interpolated drive commands with the trapezoidal acceleration profile, where chord errors and centripetal accelerations at transition corners are also constrained  ...  The off-line pre-processing stage calculates the path lengths, and approximates the relation of spline parameters and spline lengths with 7 th order polynomial splines within preset tolerance.  ...  constrained, followed by substituting the are-length information into the spline or linear function to calculate the interpolated position coordinates.  ... 
doi:10.1016/j.procir.2016.10.022 fatcat:oossrmlhk5bmtp22ofmp2by5l4

Real-Time NURBS Interpolation under Multiple Constraints

Mingxing Nie, Yaping Wan, Aijun Zhou, Kapil Sharma
2022 Computational Intelligence and Neuroscience  
Due to the special form of NURBS curve, there is a nonlinear relationship between its curve and arc length; feed fluctuations and mechanical shocks which are caused during the interpolation process will  ...  However, with the increasing demands for high-accuracy and efficient machining, NURBS interpolation has faced a growing number of challenges.  ...  Required arc length is the minimum length for the complete S curve acc/dec pro le from U_start to U_end at the test curve.  ... 
doi:10.1155/2022/7492762 fatcat:howz3jrpifaerfuiogpxvaj76m

Spline interpolation for demographic variables: The monotonicity problem

Len Smith, Rob J. Hyndman, Simon N. Wood
2004 Journal of Population Research  
In demography, it is often necessary to obtain a monotonic interpolation of data. A solution to this problem is available using the Hyman filter for cubic splines.  ...  Figure 1 shows the cumulative deaths in Australia in 1914 with an unconstrained spline interpolation shown as a solid line.  ...  At upper ages he constrains the slope to be zero although elsewhere he concludes that there seems to be no reliable solution at older ages: the spline function often starts to decline with the open age  ... 
doi:10.1007/bf03032212 fatcat:uphjjcpzvnbmvegguw43k77hzq

Smooth trajectory generation for rotating extensible manipulators

Mihai Dupac
2016 Mathematical methods in the applied sciences  
Moreover, the manipulator equations of motion can be easily handled, and the constrained trajectory of the non-active end of the manipulator derived directly from the position of the end-effector.  ...  Possible trajectories are modelled using Cartesian and polar piecewise cubic interpolants expressed as polynomial Hermite-type functions.  ...  Among these, piecewise interpolating curves with slope continuity, geometrically continuous Catmull-Rom splines [4] , parametric and/or geometric continuous splines [9, 15] or uniform cubic B-Spline  ... 
doi:10.1002/mma.4210 fatcat:tuomskgpebd3ddgkqmgwv7ryqi

The feedrate scheduling of NURBS interpolator for CNC machine tools

An-Chen Lee, Ming-Tzong Lin, Yi-Ren Pan, Wen-Yu Lin
2011 Computer-Aided Design  
Then, the NURBS curve is divided into several NURBS sub-curves using curve splitting method which guarantees the convergence of predictor-corrector interpolation (PCI) algorithm.  ...  Finally, simulations and experiments with two free-form NURBS curves are conducted to verify the feasibility and applicability of the proposed method.  ...  The adaptive-feedrate with curvature-based feedrate interpolation algorithm [18] is utilized to scan a NURBS curve for determining critical points with large curvatures.  ... 
doi:10.1016/j.cad.2011.02.014 fatcat:pzmzcerd4jd5pngpntuhwrd6v4

Constrained variational refinement

Scott N. Kersey
2009 Journal of Computational and Applied Mathematics  
The results are applied to closed periodic curves, open curves with various boundary conditions, and (approximate) Hermite interpolation.  ...  A non-uniform, variational refinement scheme is presented for computing piecewise linear curves that minimize a certain discrete energy functional subject to convex constraints on the error from interpolation  ...  This is what one would expect for an inactive point, which corresponds with the slack condition. Now, consider the setup in Fig. 4.1(b) .  ... 
doi:10.1016/j.cam.2008.03.033 fatcat:raceuivlm5dd3g7axtlm25ptsa

Page 8110 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews  
Rabut, Shape-constrained smooth- ing with inf-convolution cubic splines (61-70); Paolo Costan- tini, Isabella Cravero and Carla Manni, Constrained interpolation by Frenet frame continuous quintics (71-  ...  Kaklis, Spatial geometric interpolation (375- 385); Hongling Wang, Joseph Kearney and Kendall Atkinson, Arc-length parameterized spline curves for real-time simulation (387-396); Hongling Wang, Joseph  ... 

Generating Commands for Parametric Curve Motion Using Buffered DDA

Yih-Fang CHANG
2005 JSME international journal. Series C, Mechanical systems, machine elements and manufacturing  
The DDA interpolates sequential position data, where the gaps between pairs of data are variable, using a fixed increment of a parameter constrained by the required accuracy of the chord of the motion.  ...  A parametric curve, determined by the non-uniform rational B-spline (NURBS), is employed to verify the improvements in curve motion.  ...  Therefore, DDA is normally applied for interpolating a curve with a speed of under 1BLU/∆t. The speed can be increased to a high value using high-performance hardware.  ... 
doi:10.1299/jsmec.48.785 fatcat:wtgytboiknfwbkazgtl5vmgkf4

Interactive design of constrained variational curves

Wieger Wesselink, Remco C. Veltkamp
1995 Computer Aided Geometric Design  
A constrained variational curve is a curve that minimizes some energy functional under certain interpolation constraints.  ...  Modeling curves using constrained variational principles is attractive. because the designer is not bothered with the precise representation of the curve (e.g. control points).  ...  In order to deal with this unwanted effect the length of the curve has to be restricted. A common way to do this is to combine the bend energy with another functional, the length of a curve.  ... 
doi:10.1016/0167-8396(94)00033-o fatcat:hait6ulvhbeqrb4wdlzj7txmvq

Isometric Piecewise Polynomial Curves

Eugene Fiume
1995 Computer graphics forum (Print)  
There has been relatively little work on interacting with direct spatial properties of curves and surfaces, such as their arc length or surface area.  ...  As a rst step, we derive families of parametric piecewise polynomial curves that satisfy various positional and tangential constraints together with arc-length constraints.  ...  For a wide-range of arc lengths, interacting with these constrained curves is quite similar to their unconstrained counterparts.  ... 
doi:10.1111/1467-8659.1410047 fatcat:3w4g6mqta5e5ff632rrgdaz734

Fast Constrained Surface Extraction by Minimal Paths

Roberto Ardon, Laurent D. Cohen
2006 International Journal of Computer Vision  
The initialization procedure is reduced to introducing 3D curves into the image. These curves are supposed to belong to the surface to extract and thus, also constitute user given information.  ...  Our paper describes a fast construction obtained by exploiting the Fast Marching algorithm and a fast analytical interpolation method.  ...  Figure 7 . 7 (a) View of different intersecting planes of a 3D volume with the two constraining curves drawn on it. (b) Network of paths obtained with our method. (c) Interpolated surface.  ... 
doi:10.1007/s11263-006-6850-z fatcat:t22kpq4jmrb2rk4yefwt3ipwc4

Page 912 of SMPTE Motion Imaging Journal Vol. 102, Issue 10 [page]

1993 SMPTE Motion Imaging Journal  
Shoemake, “Animating Rotations with Quaternion Curves,” Computer Graphics, 19:245-254, July 1985. . A.  ...  Constraining the surfaces to maintain contact with their frames and minimiz- ing surface area is a way to generate these often beautiful surfaces using constrained optimization.  ... 

Spline Curve Approximation and Design by Optimal Control Over the Knots [chapter]

Rony Goldenthal, Michel Bercovier
2004 Geometric Modelling  
An interesting aspect is that the interpolation or the approximation matrix might become singular due to invalid knot vector values with respect to the current parametrization (violation of Schoenberg-Whitney  ...  In [1] Optimal Control methods over re-parametrization for curve and surface design were introduced.  ...  Length estimation: Find the curve that approximates (or interpolates) the set of given points and has the shortest length.  ... 
doi:10.1007/978-3-7091-0587-0_5 fatcat:xwqlv23t4rgbdjlpcd2ig5ygfa

Spline Curve Approximation and Design by Optimal Control Over the Knots

Rony Goldenthal, Michel Bercovier
2004 Computing  
An interesting aspect is that the interpolation or the approximation matrix might become singular due to invalid knot vector values with respect to the current parametrization (violation of Schoenberg-Whitney  ...  In [1] Optimal Control methods over re-parametrization for curve and surface design were introduced.  ...  Length estimation: Find the curve that approximates (or interpolates) the set of given points and has the shortest length.  ... 
doi:10.1007/s00607-003-0046-y fatcat:7r4ruh7sfbdafbkzhr3edh4bly
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