B. Figure, 1: Neighborhood motion maps of G U 2 , as label maps

B. Figure, as label maps, for ? ? (? 1 , ? 2 ) that differ from those for ? ? (0, ? 1 ) Each label (p, q) corresponds to the frame f ? p,q . Neighborhood motion maps which correspond to non-injective zones are marked by brown dashed frames. The edges of the neighborhood motion maps graph are marked by color line segments which connect different neighborhood motions maps (see Chapter 3 of Part I). The elements which have not changed with respect to the set

B. Figure, ? 3 ) that differ from those for ? ? (? 1 , ? 2 ) Each label (p, q) corresponds to the frame f ? p,q . Neighborhood motion maps which correspond to non-injective zones are marked by brown dashed frames. The edges of the neighborhood motion maps graph are marked by color line segments which connect different neighborhood motions maps (see Chapter 3 of Part I). The elements which have not changed with respect to the set

B. Appendix, Neighborhood motion maps for G U 2 (8-neighborhood case) (-4,4) (-3,4) (-2,4) (-1,4) (0,4), pp.4-6

B. Figure, as label maps, for ? ? (? 3 , ? 4 ) that differ from those for ? ? (? 2 , ? 3 ) Each label (p, q) corresponds to the frame f ? p,q . Neighborhood motion maps which correspond to non-injective zones are marked by brown dashed frames. The edges of the neighborhood motion maps graph are marked by color line segments which connect different neighborhood motions maps (see Chapter 3 of Part I). The elements which have not changed with respect to the set

B. Appendix, Neighborhood motion maps for G U 2 (8-neighborhood case) (-4,4) (-3,4) (-2,4) (-1,4) (0,4), pp.4-6

C. Figure, 1: The set of neighborhood motion maps M 1 , for rotation angles ? ? (? 0 , ? 1 ), visualized by the label map L U 1 (see Chapter

C. Figure, The set of neighborhood motion maps M 1 , for rotation angles ? ? (? 1

C. Figure, The set of neighborhood motion maps M 1 , for rotation angles ? ? (? 2

A. Yilmaz, O. Javed, and M. Shah, Object tracking, ACM Computing Surveys, vol.38, issue.4, 2006.
DOI : 10.1145/1177352.1177355

V. Ostromoukhov and R. D. Hersch, Halftoning by Rotating Non-Bayer Dispersed Dither Arrays. Milestone Series, pp.238-255, 1999.
DOI : 10.1117/12.207537

URL : http://diwww.epfl.ch/w3lsp/pub/papers/colour/SPIE95_RotatedNonBayer.pdf

V. Ostromoukhov, R. D. Hersch, and I. Amidror, Rotated dispersed dither, Proceedings of the 21st annual conference on Computer graphics and interactive techniques , SIGGRAPH '94, pp.123-130, 1994.
DOI : 10.1145/192161.192188

T. Y. Kong and A. Rosenfeld, Digital topology: Introduction and survey, Computer Vision, Graphics, and Image Processing, vol.48, issue.3, pp.357-393, 1989.
DOI : 10.1016/0734-189X(89)90147-3

R. Klette and A. Rosenfeld, Digital Geometry: Geometric Methods for Digital Picture Analysis, 2004.

L. Middleton and J. Sivaswamy, Hexagonal Image Processing: A Practical Approach Advances in Pattern Recognition, 2005.

J. Serra, Image Analysis and Mathematical Morphology, 1982.

K. Pluta, P. Romon, Y. Kenmochi, and N. Passat, Bijective Digitized Rigid Motions on Subsets of the Plane, Journal of Mathematical Imaging and Vision, vol.38, issue.4, pp.84-105, 2017.
DOI : 10.1145/1177352.1177355

URL : https://hal.archives-ouvertes.fr/hal-01497610

K. Pluta, P. Romon, Y. Kenmochi, and N. Passat, Honeycomb Geometry: Rigid Motions on the Hexagonal Grid, DGCI, pp.33-45, 2017.
DOI : 10.1016/j.tcs.2007.03.032

URL : https://hal.archives-ouvertes.fr/hal-01497608

B. Nouvel and É. Rémila, Configurations induced by discrete rotations: periodicity and quasi-periodicity properties, Discrete Applied Mathematics, vol.147, issue.2-3, pp.325-343, 2005.
DOI : 10.1016/j.dam.2004.09.018

URL : https://doi.org/10.1016/j.dam.2004.09.018

B. Nouvel and É. Rémila, On Colorations Induced by Discrete Rotations, DGCI, volume 2886 of Lecture Notes in Computer Science, pp.174-183, 2003.
DOI : 10.1007/978-3-540-39966-7_16

A. W. Paeth, Graphics Gems. chapter A Fast Algorithm for General Raster Rotation, pp.179-195, 1990.

É. Andres, The Quasi-Shear rotation, DGCI, pp.307-314, 1996.
DOI : 10.1007/3-540-62005-2_26

URL : https://link.springer.com/content/pdf/10.1007%2F3-540-62005-2_26.pdf

Y. Thibault, Rotations in 2D and 3D Discrete Spaces, 2010.
URL : https://hal.archives-ouvertes.fr/tel-00596947

W. S. Anglin, Using Pythagorean Triangles to Approximate Angles, The American Mathematical Monthly, vol.95, issue.6, pp.540-541, 1988.
DOI : 10.1080/00029890.1988.11972043

V. Ostromoukhov, Reproduction Couleur par Trames Irrégulières et Semi-régulières, 1995.

J. Reveillès, Géométrie Discrète, Calcul en Nombres Entiers et Algorithmique

M. Jacob and É. Andres, On Discrete Rotations, 5th International Workshop on Discrete Geometry for Computer Imagery, pp.161-174, 1995.

É. Andres, Cercles Discrets et Rotations Discrètes, 1992.

É. Andres, Modélisation Analytique Discrète d'Objets Géométriques. Habilitation à diriger des recherches, 2000.

T. Roussillon and D. Coeurjolly, Characterization of Bijective Discretized Rotations by Gaussian Integers
URL : https://hal.archives-ouvertes.fr/hal-01259826

B. Nouvel and É. Rémila, Incremental and Transitive Discrete Rotations, IW- CIA, pp.199-213393, 2006.
DOI : 10.1007/11774938_16

URL : https://hal.archives-ouvertes.fr/hal-00016037

P. Ngo, Y. Kenmochi, N. Passat, and H. Talbot, Topology-Preserving Conditions for 2D Digital Images Under Rigid Transformations, Journal of Mathematical Imaging and Vision, vol.8, issue.2, pp.418-433, 2014.
DOI : 10.1023/A:1008273227913

URL : https://hal.archives-ouvertes.fr/hal-00838183

P. Ngo, Y. Kenmochi, A. Sugimoto, H. Talbot, and N. Passat, Discrete rigid registration: A local graph-search approach, Discrete Applied Mathematics, vol.216, pp.461-481, 2017.
DOI : 10.1016/j.dam.2016.05.005

URL : https://hal.archives-ouvertes.fr/hal-01306035

P. Ngo, Y. Kenmochi, N. Passat, and H. Talbot, On 2D constrained discrete rigid transformations, Annals of Mathematics and Artificial Intelligence, vol.21, issue.11, pp.163-193, 2014.
DOI : 10.1016/S0262-8856(03)00137-9

URL : https://hal.archives-ouvertes.fr/hal-00838184

P. Ngo, N. Passat, Y. Kenmochi, and H. Talbot, Topology-Preserving Rigid Transformation of 2D Digital Images, IEEE Transactions on Image Processing, vol.23, issue.2, pp.885-897
DOI : 10.1109/TIP.2013.2295751

URL : https://hal.archives-ouvertes.fr/hal-00795054

I. Her, Geometric transformations on the hexagonal grid, IEEE Transactions on Image Processing, vol.4, issue.9, pp.1213-1222, 1995.
DOI : 10.1109/83.413166

Y. Thibault, Y. Kenmochi, and A. Sugimoto, Computing upper and lower bounds of rotation angles from digital images, Pattern Recognition, vol.42, issue.8, pp.1708-1717, 2009.
DOI : 10.1016/j.patcog.2008.12.027

URL : https://hal.archives-ouvertes.fr/hal-00622416

K. Pluta, T. Roussillon, D. Coeurjolly, P. Romon, Y. Kenmochi et al., Characterization of Bijective Digitized Rotations on the Hexagonal Grid, Journal of Mathematical Imaging and Vision, vol.38, issue.4, 2018.
DOI : 10.1145/1177352.1177355

URL : https://hal.archives-ouvertes.fr/hal-01540772

R. A. Gordon, Properties of Eisenstein Triples, Mathematics Magazine, vol.13, issue.1, pp.12-25, 2012.
DOI : 10.2307/3595782

J. Gilder, Integer-sided Triangles with an Angle of 60 ? . The Mathematical Gazette, pp.261-266, 1982.
DOI : 10.2307/3615511

A. I. Galarza and J. Seade, Introduction to Classical Geometries. Birkhäuser, 2007.

V. Berthé and B. Nouvel, Discrete rotations and symbolic dynamics, Theoretical Computer Science, vol.380, issue.3, pp.276-285, 2007.
DOI : 10.1016/j.tcs.2007.03.032

B. Zitova and J. Flusser, Image registration methods: a survey, Image and Vision Computing, vol.21, issue.11, pp.977-1000, 2003.
DOI : 10.1016/S0262-8856(03)00137-9

S. Basu, R. Pollack, and M. Roy, Algorithms in Real Algebraic Geometry, 2005.
DOI : 10.1007/978-3-662-05355-3

URL : https://hal.archives-ouvertes.fr/hal-01083587

G. E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decompostion, ATFL, pp.134-183, 1975.
DOI : 10.1007/3-540-07407-4_17

J. Renegar, On the computational complexity and geometry of the first-order theory of the reals. Part I: Introduction. Preliminaries. The geometry of semi-algebraic sets. The decision problem for the existential theory of the reals, Journal of Symbolic Computation, vol.13, issue.3, pp.255-299, 1992.
DOI : 10.1016/S0747-7171(10)80003-3

K. Kurdyka, P. Orro, and S. Simon, Semialgebraic Sard Theorem for Generalized Critical Values, Journal of Differential Geometry, vol.56, issue.1, pp.67-92, 2000.
DOI : 10.4310/jdg/1090347525

URL : https://doi.org/10.4310/jdg/1090347525

T. Toffoli and J. Quick, Three-Dimensional Rotations by Three Shears, Graphical Models and Image Processing, vol.59, issue.2, pp.89-95, 1997.
DOI : 10.1006/gmip.1997.0420

URL : http://pm1.bu.edu/~tt/publ/rot-gmip.ps

B. Chen and A. Kaufman, 3D Volume Rotation Using Shear Transformations, Graphical Models, vol.62, issue.4, pp.308-322, 2000.
DOI : 10.1006/gmod.2000.0525

URL : http://www-users.cs.umn.edu/~baoquan/papers/rot.pdf

Y. Thibault, A. Sugimoto, and Y. Kenmochi, 3D discrete rotations using hinge angles, Theoretical Computer Science, vol.412, issue.15, pp.1378-1391, 2011.
DOI : 10.1016/j.tcs.2010.10.031

URL : https://hal.archives-ouvertes.fr/hal-00734881

K. Pluta, G. Moroz, Y. Kenmochi, and P. Romon, Quadric Arrangement in Classifying Rigid Motions of a 3D Digital Image, CASC, pp.426-443, 2016.
DOI : 10.1016/S0262-8856(03)00137-9

URL : https://hal.archives-ouvertes.fr/hal-01334257

K. Pluta, P. Romon, Y. Kenmochi, and N. Passat, Bijectivity Certification of 3D Digitized Rotations, CTIC, pp.30-41, 2016.
DOI : 10.1007/978-0-85729-760-0

URL : https://hal.archives-ouvertes.fr/hal-01315226

R. Murray, Z. Li, and S. Sastry, A Mathematical Introduction to Robotic Manipulation, 52] K. Kanatani. Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics, 1994.

A. Cayley and A. R. Forsyth, The Collected Mathematical Papers of Arthur Cayley, 1898.

P. Singla and J. L. Junkins, Multi-resolution Methods for Modeling and Control of Dynamical Systems, 2008.
DOI : 10.1201/9781584887706

N. André, Largest Triangle with Vertices in the Unit Cube Mathematics Stack Exchange . URL https://math.stackexchange.com/q/44499, pp.2011-2017

H. Croft, K. Falconer, and R. Guy, Unsolved Problems in Geometry, 1994.
DOI : 10.1007/978-1-4612-0963-8

A. Schrijver, Theory of Linear and Integer Programming, 1998.

J. Cremona, Letter to the Editor, American Mathematical Monthly, vol.94, issue.8, pp.757-758, 1987.

D. Micciancio and B. Warinschi, A linear space algorithm for computing the hermite normal form, Proceedings of the 2001 international symposium on Symbolic and algebraic computation , ISSAC '01, pp.231-236, 2001.
DOI : 10.1145/384101.384133

URL : http://www-cse.ucsd.edu/~daniele/papers/HNFalg.ps

C. Pernet and W. Stein, Fast computation of Hermite normal forms of random integer matrices, Journal of Number Theory, vol.130, issue.7, pp.1675-1683, 2010.
DOI : 10.1016/j.jnt.2010.01.017

URL : https://hal.archives-ouvertes.fr/hal-00798442

K. Pluta, Y. Kenmochi, N. Passat, H. Talbot, and P. Romon, Topological Alterations of 3D Digital Images Under Rigid Transformations, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01333586

A. Amir, O. Kapah, and D. Tsur, Faster two-dimensional pattern matching with rotations, Theoretical Computer Science, vol.368, issue.3, pp.196-204, 2006.
DOI : 10.1016/j.tcs.2006.09.012

URL : https://doi.org/10.1016/j.tcs.2006.09.012

B. Mourrain, J. P. Tecourt, and M. Teillaud, On the computation of an arrangement of quadrics in 3D, Computational Geometry, vol.30, issue.2, pp.145-164, 2005.
DOI : 10.1016/j.comgeo.2004.05.003

URL : https://hal.archives-ouvertes.fr/inria-00350858

P. Bazin, L. M. Ellingsen, and D. L. Pham, Digital Homeomorphisms in Deformable Registration, IPMI, pp.211-222, 2007.
DOI : 10.1007/978-3-540-73273-0_18

P. J. Rabier, Ehresmann Fibrations and Palais-Smale Conditions for Morphisms of Finsler Manifolds, The Annals of Mathematics, vol.146, issue.3, pp.647-691, 1997.
DOI : 10.2307/2952457

Z. Jelonek and K. Kurdyka, Quantitative Generalized Bertini-Sard Theorem for Smooth Affine Varieties, Discrete & Computational Geometry, vol.34, issue.4, pp.659-678, 2005.
DOI : 10.1007/s00454-005-1203-1

URL : https://hal.archives-ouvertes.fr/hal-00389073

Z. Jelonek and K. Kurdyka, On asymptotic critical values of a complex polynomial, Journal f??r die reine und angewandte Mathematik (Crelles Journal), vol.36, issue.565, pp.1-11, 2003.
DOI : 10.1515/crll.2003.101

D. Cox, J. Little, and D. Shea, Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 1996.

Z. Jelonek, Topological Characterization of Finite Mappings, Bulletin of the Polish Academy of Sciences ? Mathematics, vol.49, issue.3, pp.279-283, 2001.

M. Safey-el-din and É. Schost, Properness Defects of Projections and Computation of at Least One Point in Each Connected Component of a Real Algebraic Set
URL : https://hal.archives-ouvertes.fr/inria-00099962

G. Moroz, Properness defects of projection and minimal discriminant variety, Journal of Symbolic Computation, vol.46, issue.10, pp.1139-1157, 2011.
DOI : 10.1016/j.jsc.2011.05.013

URL : https://hal.archives-ouvertes.fr/hal-01148309

F. Rouillier and P. Zimmermann, Efficient isolation of polynomial's real roots, Journal of Computational and Applied Mathematics, vol.162, issue.1, pp.33-50, 2004.
DOI : 10.1016/j.cam.2003.08.015

URL : https://doi.org/10.1016/j.cam.2003.08.015

E. Hansen, Global optimization using interval analysis ? the multi-dimensional case, Numerische Mathematik, vol.14, issue.3, pp.247-270, 1980.
DOI : 10.1007/BF01396702

A. Neumaier, Interval Methods for Systems of Equations. Encyclopedia of Mathematics and its Applications, 1991.

J. Abbott, Quadratic interval refinement for real roots, ACM Communications in Computer Algebra, vol.48, issue.1/2, pp.3-12, 2014.
DOI : 10.1145/2644288.2644291

URL : http://arxiv.org/pdf/1203.1227.pdf