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Solving Variational Problems Using Nonlinear Rotation-invariant Coordinates

Josua Sassen, Behrend Heeren, Klaus Hildebrandt, Martin Rumpf
2019 Eurographics Symposium on Geometry Processing  
We consider Nonlinear Rotation-Invariant Coordinates (NRIC) representing triangle meshes with fixed combinatorics as a vector stacking all edge lengths and dihedral angles.  ...  We develop the machinery needed to use NRIC for solving geometric optimization problems.  ...  Introduction Variational problems are at the core of many applications in geometry processing. Here we consider Nonlinear Rotation-Invariant Coordinates (NRIC) for solving them on triangle meshes.  ... 
doi:10.2312/sgp.20191213 dblp:conf/sgp/SassenHHR19 fatcat:apf3wvuubnhvvb2r4j7s3w56qq

A Surface-Theoretic Approach for Statistical Shape Modeling [chapter]

Felix Ambellan, Stefan Zachow, Christoph von Tycowicz
2019 Lecture Notes in Computer Science  
We present a novel approach for nonlinear statistical shape modeling that is invariant under Euclidean motion and thus alignmentfree.  ...  We evaluate the performance of our model w.r.t. shape-based classification of pathological malformations of the human knee and show that it outperforms the standard Euclidean as well as a recent nonlinear  ...  [26] presented a physically motivated approach based on differential coordinates for which the inverse problem is well-known and can be solved at linear cost.  ... 
doi:10.1007/978-3-030-32251-9_3 fatcat:7263ok6mujbcpgq3zhi3vtiwdu

Geometric optimization using nonlinear rotation-invariant coordinates [article]

Josua Sassen, Behrend Heeren, Klaus Hildebrandt, Martin Rumpf
2019 arXiv   pre-print
We consider the Nonlinear Rotation-Invariant Coordinates (NRIC) that represent the nodal positions of a discrete triangular surface with fixed combinatorics as a vector that stacks all edge lengths and  ...  The goal of this paper is to develop the machinery needed to use the NRIC for solving geometric optimization problems.  ...  Acknowledgements The authors thank Laszlo Bardos from for the photo of Steffen's polyhedron used in Figure 1 .  ... 
arXiv:1908.11728v1 fatcat:tc6tar7svfa3hdynsaif6kuptq

Hamilton-Jacobi Modelling of Relative Motion for Formation Flying

2005 Annals of the New York Academy of Sciences  
Using Hamilton's equations, the variational equations for the new constants are found.  ...  This allows us to treat perturbations in an identical manner as in the classical Delaunay formulation of the two-body problem.  ...  We do this by solving the variational equations above using a Poincarè-Lindstedt procedure.  ... 
doi:10.1196/annals.1370.009 pmid:16510405 fatcat:klbqqh7wgnfgnp65ty7n5wljui

Page 3953 of Mathematical Reviews Vol. , Issue 93g [page]

1993 Mathematical Reviews  
Combined with the classical perturbation method, a method for solving the problem of nonlinear oscillations in a system with sev- eral degrees of freedom is presented.  ...  Previous studies of the nonlinear inverted double pendulum using Lyapunov’s di- rect method are not considered by the author. P.  ... 

The Parameters Optimizing Design of Double Suspension Arm Torsion Bar in the Electric Sight-Seeing Car by Random Vibration Analyzing Method

Shui-Ting Zhou, Yi-Jui Chiu, I-Hsiang Lin
2017 Shock and Vibration  
The suspension stiffness was calculated by using the virtual work principle, the vector algebra, and tensor of finite rotation methods and was verified by the ADAMS software.  ...  is about the impact of the performance and the sensitivity analysis for parameters of the torsion bar suspension in the electric sight-seeing car, which the authors' laboratory designed and which is used  ...  The first step is to solve the point coordinate of exterior ball joints of upper arms. point coordinate is solved according to geometric characteristics since the upper exterior point only rotates around  ... 
doi:10.1155/2017/8153756 fatcat:d4zfyih7irgsnieyjtze3672da

Page 2604 of Mathematical Reviews Vol. , Issue 97D [page]

1997 Mathematical Reviews  
[Morgan, Michael Ambrose] (1-SEAT-P; Seattle, WA) Geometric angle for rotated rotators, and the Hannay angle of the world. (English summary) Nonlinearity 9 (1996), no. 3, 787-799.  ...  The solution does not rely on coordinates used to describe the sys- tems and is computationally efficient. One illustrative example is presented.”  ... 

Geometric Collocation Method on SO(3) with Application to Optimal Attitude Control of a 3D Rotating Rigid Body

Xiaojia Xiang, Lizhen Wu, Lincheng Shen, Jie Li
2015 Mathematical Problems in Engineering  
In addition, the numerical method is applied to solve a constrained OAC problem on SO(3).  ...  The optimal control problem is transcribed into a nonlinear programming problem, in which the equivalent Lie algebra equation is being considered as the defect constraints instead of the configuration  ...  referred to as a Lie group variational integrator and then use Lagrange's method to derive the discrete necessary conditions (i.e., two-point boundary-value problem) which are solved by a standard nonlinear  ... 
doi:10.1155/2015/790409 fatcat:5chqlfmgrnf7he3uau4fmjp6pi

Page 5331 of Mathematical Reviews Vol. , Issue 82m [page]

1982 Mathematical Reviews  
Some functions of p which are invariant under rotation of the coordinate axes are obtained.” Babaev, M. A.; Salmanov, V.  ...  The author derives the basic variational principles of finite elasticity using the principle of virtual work.  ... 

Dynamic analysis of planar flexible beams with finite rotations by using inertial and rotating frames

M. Iura, S.N. Atluri
1995 Computers & structures  
Nonlinear effects appear only in the transformation of displacement components between global and local coordinates.  ...  Since the exact solutions for linear static theory of Timoshenko's beam are used to obtain the strain energy, the present stiffness operator is free from the locking problem without using any special technique  ...  There have been many methods to solve the nonlinear algebraic equations.  ... 
doi:10.1016/0045-7949(95)98871-m fatcat:oyixpbdrljbg7nxvmc76pqipae

Page 2966 of Mathematical Reviews Vol. , Issue 90E [page]

1990 Mathematical Reviews  
This study is of great interest to researchers using group-invari- ance to solve differential equations and to those interested in the nonlinear superposition principle. G. A.  ...  [Rogers, Colin] (3-WTRL); Ramgulam, U. (3-WTRL) A nonlinear superposition principle and Lie group invariance: application in rotating shallow water theory. Internat. J.  ... 

2D/3D rotation-invariant detection using equivariant filters and kernel weighted mapping

Kun Liu, Qing Wang, W. Driever, O. Ronneberger
2012 2012 IEEE Conference on Computer Vision and Pattern Recognition  
In many vision problems, rotation-invariant analysis is necessary or preferred.  ...  We demonstrate its performance on 2D object detection with in-plane rotations, and a 3D application on rotation-invariant landmark detection in microscopic volumetric data.  ...  Invariance and equivariance In detection tasks, the rotation invariance is defined w.r.t. to the object coordinate system.  ... 
doi:10.1109/cvpr.2012.6247766 dblp:conf/cvpr/LiuWDR12 fatcat:lx2nyy5w5nbkvg3yg4j3akctqm

Pose Space Surface Manipulation

Yusuke Yoshiyasu, Nobutoshi Yamazaki
2012 International Journal of Computer Games Technology  
We encode example deformations with a rotation-invariant mesh representation which handles large rotations in examples.  ...  To incorporate example deformations, we infer a pose from the handle translations/rotations and perform pose space interpolation, thereby avoiding involved nonlinear optimization.  ...  Storing rotation-invariant encodings requires a relatively large amount of memory. To solve this problem, we propose a multiresolution approach.  ... 
doi:10.1155/2012/596953 fatcat:to7rerasnjbi7fozyjp5ardr4u

A Second-Order Symplectic Integrator for Guiding-Center Equations [article]

John R. Cary
2018 arXiv   pre-print
Algorithm for finding a solution to a set of nonlinear equations Newton's method is commonly used for finding a solution to a set of nonlinear equations, F j (z=(z 1 ,...z 2n ))=0.  ...  The integrator, which requires a nonlinear equation solve at each step, is discussed in Sec. III. A fast method for performing this solve is described in Sec. IV.  ... 
arXiv:1809.05498v1 fatcat:s5pqo2mfirbk7jgcyrvnpfvkla

Page 2297 of Mathematical Reviews Vol. 52, Issue 6 [page]

1976 Mathematical Reviews  
Surveying in rotating systems. J. Phys. A 9 (1976), no. 1, 35-43. Authors’ summary: “The principles of radar are applied to the problem of measuring distances in rotating systems.  ...  Two applications using the above expansion of > are given: (i) If, in the coordinate system used in this paper, # is set equal to zero, the space has hypersurface-orthogonal geodesic rays with nonvanishing  ... 
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