Supplementary material for this book is in progress; the current version is available HERE
Java applet demonstrating a special case Envelopes of Zags
There is an elementary paper on the latter, which appeared in
There is a description of the polygons and a Java applet demonstrating their construction HERE
C.M.(Henry) Hui, Plane Quartic Curves, PhD thesis, 1979. Scanned pdf file, 7.5MB HERE
J.W.Bruce and P.J.Giblin, Curves and Singularities, 2nd edition, Cambridge University Press, 1992.
P.J.Giblin and T.F.Banchoff Symmetry sets of piecewise circular curves, Proc. Royal Soc. Edinburgh 123A (1993), 181-194.
Primes and Programming, Cambridge University Press, 1993.
P.J.Giblin, F.E.Pollick and J.E.Rycroft Recovery of an unknown axis of rotation from the profiles of a rotating surface, J. Optical Soc. America 11A (1994), 1976-1984.
P.J.Giblin and F.Tari, Perpendicular bisectors, duals and local symmetry, Proc. Royal Soc. Edinburgh 125A (1995), 181-194
Gordon Fletcher, Geometrical Problems in Computer Vision, PhD thesis, 1996. PDF file HERE (8MB)
J.W.Bruce, P.J.Giblin and F.Tari, Parabolic curves of evolving surfaces, Int. J. Computer Vision. 17 (1996), 291-306.
J.W.Bruce, P.J.Giblin and F.Tari, Ridges, crests and sub-parabolic lines of evolving surfaces, Int. J. Computer Vision. 18 (1996), 195-210.
K.Astrom, R.Cipolla and P.J.Giblin, Generalised epipolar constraints, Proc. European Conference on Computer Vision 1996, Lecture Notes in Computer Science 1065, Springer-Verlag, 97--108.
R.Cipolla, G.J.Fletcher and P.J.Giblin, Following cusps, Int. J. Computer Vision 23 (1997), 115-129.
P.J.Giblin and G.Sapiro, Affine invariant distances, envelopes and symmetry sets. Geom. Dedicata. 71 (1998), 237-261.
J.W.Bruce, P.J.Giblin and F.Tari, Families of surfaces: focal sets, ridges and umbilics, Math. Proc. Camb. Phil. Soc. 125 (1999), 243-268.
P.J.Giblin and P.A.Holtom, `The centre symmetry set', Geometry and Topology of Caustics, Banach Center Publications, Vol 50, ed. S.Janeczko and V.M.Zakalyukin, Warsaw, 1999, pp.91-105. PDF file HERE
P.L.Hallinan, G.G.Gordon, A.L.Yuille, P.Giblin and D.Mumford, Two- and Three-Dimensional Patterns of the Face, A.K.Peters 1999.
Roberto Cipolla and Peter Giblin, Visual Motion of Curves and Surfaces, Cambridge University Press 2000.
Peter Giblin and Paul Holtom, Affine-distance symmetry sets,
Paul Holtom, PhD Thesis, Affine-invariant symmetry sets. PDF file (2.2MB) HERE
Peter Giblin and Vladimir Zakalyukin, Singularities of centre symmetry sets. Proceedings of London Mathematical Society 2005. PDF file HERE
Andre Diatta and Peter Giblin, Vertices and inflexions of sections of surfaces in R^3. Trends in Mathematics, Real and Complex Singularities, Birkhauser 2006, pp. 71-97. PDF file HERE
Andre Diatta and Peter Giblin, Geometry of isophote curves. Scale-space 2005 proceedings. PDF file HERE
Andre Diatta, Peter Giblin, Brendan Guilfoyle and Wilhelm Klingenberg, Level sets of functions and symmetry sets of surface sections, Mathematics of Surfaces 2005, PDF file HERE
Anthony Pollitt, PhD thesis `Euclidean and Affine Symmetry Sets and Medial Axes', 2004, PDF file HERE
Daniel Littlestone, MSc thesis,`Affine Area Parallels and Symmetry Sets', 2004, PDF file HERE
Declan Davis `Generic affine differential geometry of curves in R^n', 2005, PDF file HERE
Andrew Irving, MSc mini-dissertation, `Curves of constant width and centre symmetry sets', 2006, PDF file HERE
Andrew Irving, MSc Thesis, `Surfaces of constant width',2006, PDF file HERE
Andre Diatta, Peter Giblin, `Pre-symmetry sets of 3D shapes', DSSCV Workshop Proceedings, Lecture Notes in Computer Science Vol 3753 (2005), PDF file HERE
Peter Giblin and Vladimir Zakalyukin, `Recognition of centre symmetry set singularities', Geometriae Dedicata 130 (2007), 43-58. PDF file HERE
Peter Giblin, `Affinely invariant symmetry sets'. Geometry and Topology of Caustics -- Caustics '06. Banach Center Publications Vol 82 (2008), 71-84. PDF file HERE
P.J.Giblin and B.B.Kimia, `Local Forms and Transitions of the Medial Axis', in Siddiqi, K., and Pizer, S., eds., Medial Representations: Mathematics, Algorithms and Applications. Springer, 2008. Link here
Declan Davis, `Affine Differential Geometry and Singularity Theory', Ph.D.Thesis, University of Liverpool, 2008, PDF file HERE
Graham Reeve, `Curves of constant width, envelopes and duals of plane and space curves', MMath dissertation, University of Liverpool, 2008, PDF file HERE
Ben Kimia, Peter Giblin and Anthony Pollitt, `Transitions of the 3D medial axis under a one-parameter family of deformations'. IEEE Transactions in Pattern Analysis and Machine Intelligence 31 (2009), 900-918. PDF file of preprint version HERE
J.Damon, P.Giblin and G.Haslinger, `Local image features resulting from 3-dimensional image features, illumination and movement, I', Internat. J. Computer Vision 82 (2009), 25-47, PDF file HERE
Declan Davis, `Affine normal curvature of hypersurfaces from the point of view of singularity theory', Geom. Dedicata 141 (2009), 137--145, PDF file of preprint version HERE
J. Paul Warder, `Symmetries of curves and surfaces', Ph.D. Thesis, University of Liverpool, 2009, PDF file HERE
P.J.Giblin, J.P.Warder and V.M.Zakalyukin, `Bifurcations of affine equidistants', Proceedings of the Steklov Institute of Mathematics 267 (2009), 57--75. PDF file of preprint version HERE
P.J.Giblin and J.P.Warder, `Reconstruction from medial representations',American Mathematical Monthly 118 (2011), 712-725, preprint version HERE
J.Damon, P.Giblin and G.Haslinger, `Local image features resulting from 3-dimensional image features, illumination and movement, II', SIAM Journal of Imaging Sciences 4 (2011), 386-412 PDF file HERE. Supplementary material on illuminated corners PDF file HERE
Yu She, `Universal cycles', Nuffield Bursary Project, Summer 2011, PDF file HERE
P.J.Giblin and S.Janeczko, `Geometry of curves and surfaces through the contact map', Topology and its Applications 159 (2012), 466-475; Preprint version HERE
Ricardo Fabbri, Benjamin B. Kimia and Peter J.Giblin, `Camera Pose Estimation Using First-Order Curve Differential Geometry', in A. Fitzgibbon et al. (Eds.): ECCV 2012, Part IV, Lecture Notes in Computer Science Vol. 7575, pp. 229--242. Springer, Heidelberg (2012). Earlier version,with supplementary notes HERE
Matthew Temple, `Oscillating Möbius sequences', part of a Nuffield Bursary Project 2012, HERE
P.J.Giblin and J.P.Warder, `Evolving Evolutoids', American Math. Monthly 121 (2014), 871--889. Formerly called `Fixed Angle Envelopes'. Preprint November 2013, HERE
Peter Giblin, `Ancient approximation to the sine function', Math. Gazette. 98 (2014), 488--490; Preprint version HERE
P.J. Giblin and G.M.Reeve, `Centre symmetry sets of families of plane curves', Demonstratio Mathematica 48 (2015), 167--192; preprint version HERE
Alexander Gheorghiu, `An analysis on the iteration of four interesting functions', Nuffield Bursary Project 2014, HERE
Alexander Wettig, `Embroidery from chords of one or two circles', Preprint, based on a Nuffield project, HERE
Azeb Alghanemi and Peter Giblin, `On the geometry of the midlocus associated to a smooth curve in the plane and space', preprint December 2015. HERE
P.J.Giblin, `Applications of singularity theory and 3D modelling in arts and retail', pp. 265-271 in `UK Success Strories in Industrial Mathematics', ed. Philip Aston, Anthony Mulholland and Katherine Tant, Springer 2016. See HERE
P.J.Giblin, S.Janeczko and M.A.S.Ruas, `Reflexion maps and geometry of surfaces in R4.' Preprint August 2016 HERE
James Damon, Peter Giblin and Gareth Haslinger. Local Features in Natural Images via Singularity Theory,, x + 255pp. Springer Lecture Notes in Mathematics, Vol 2165 (2016). See HERE
P.J. Giblin and S.Janeczko, `Bifurcation sets of families of reflexions on surfaces in R3', Proceedings of the Royal Society of Edinburgh 147A (2017), 337-352 HERE
Peter Giblin and Graham Reeve, `Equidistants for families of surfaces', March 2018, preliminary version HERE
Peter Giblin and Graham Reeve, `Equidistants and their duals for families of plane curves', Advanced Studies in Pure Mathematics (Singularities in Generic Geometry) 78 (2018), 251--272. Preprint version HERE