TE2PM.html010066400054310000001000000023750616626676500126300ustar00dkpother00002540002445 Tightly Embedded 2-dimensional Polyhedral Manifolds.

Tightly Embedded 2-dimensional Polyhedral Manifolds
American Journal Mathematical, 1965, pp. 462-472.

This paper introduces the word "tight" to describe the geometric condition that corresponds in the smooth case to minimal total absolute curvature. For any two-dimensional surface, an embedding is tight if almost any height function has exaclty one maximum when restricted to the surface. This is equivalent to the fact that every local support hyperplane is a global support hyperplane. Although any tightly embedded smooth surface must be contained in some five-dimensional subspace, there is not such restriction for polyhedral surfaces:

Theorem A: For any n, there is a two-dimensional polyhedral surface M(n) embedded in R^n and not lying in any affine (n-1)-dimensional subspace.

In order to find examples for laarge n, it is necessary to use surfaces with high genus.

If M is a polyhedral surface tightly embedded in R^n, then n < (7 - 24 Euler characteristic of (M))/2.

about.html010066400054310000001000000005370617571051000130670ustar00dkpother00002540002445 Articles about Thomas Banchoff

Articles about Thomas Banchoff

  1. Computer Recreations, by A. K. Dewdney. Scientific American, April 1986, 254 No. 4. pp. 14-23. biblio.html010066400054310000001000000431570617542405300132260ustar00dkpother00002540002445 Thomas F. Banchoff's list of publications. List of Publications of Thomas F. Banchoff
    1. Tightly embedded 2-dimensional polyhedral manifolds, Amer. J. Math., 87 (1965), 462-472. (abstract) ( AMS review #31#2729)

    2. Critical points and curvature for embedded polyhedra, J. Differential Geometry, 1 (1967), 257-268. (abstract) ( AMS review #85b:53061)

    3. Total central curvature of curves, Duke Math. J., 37 (1970), 281-289. (abstract) ( AMS review #41#4447)

    4. Periodic points of Anosov diffeomorphisms (with Michael I. Rosen), Proceedings of Symposia in Pure Mathematics, Global Analysis, Vol. XIV (1970), 17-21.(abstract) ( AMS review #43#2738)

    5. The spherical two piece property and tight surfaces in spheres, J. Differential Geometry, 4 (1970), 193-205. (abstract) ( AMS review #42#3720)

    6. Critical points and curvature for embedded polyhedral surfaces, Amer. Math. Monthly, 77 (1970), 475-485. ( AMS review #41#4444)

    7. Non-rigidity theorems for tight polyhedral tori, Archiv der Mathematik, 21 (1970), 416-423. (abstract) ( AMS review #42#8426)

    8. The two-piece property and tight n-manifolds-with-boundary in E^n , Trans. Amer. Math. Soc. 161 (1971), 259-267. (abstract) ( AMS review #44#4760)

    9. [9] On a generalization of the isoperimetric inequality (with William Pohl), J. Differential Geometry, 6 (1971), 175-192. ( AMS review #46#4449)

    10. High codimensional 0-tight mappings on spheres, Proc. Amer. Math. Soc., 29 (1971), 133-137. (abstract) ( AMS review #43#5541)

    11. Applications of elementary calculus, Eight lectures in an NSF Sponsored Conference for College Teachers of Mathematics, Summer 1971, published in the Proceedings of the Conference.

    12. Polyhedral Catastrophe I: Maps of the Line to the Line, Dynamical Systems, Academic Press (1973), 7-22. ( AMS review #49#6268)

    13. Foliations of Knot Complements in the Bicylinder Boundary, Separata do Boletim da Sociedade Brasileira de Matematica Vol. 5, No. 1 (1974), pp. 31-43. (abstract)

    14. Global Geometry of Polygons I: The Theorem of Fabricius-Bjerre, Proc. A.M.S. 45 (1974), 237-241. (abstract) ( AMS review #51#6826)

    15. Real Time Computer Graphics Techniques in Geometry (with Charles Strauss), Proceedings of Symposia in Applied Mathematics Vol. 20, The Influence of Computing on Mathematical Research and Education, Amer. Math. Soc. (1974), 105- 111. (abstract) ( AMS review #49#7058)

    16. Tight Polyhedral Klein Bottles, Projective Planes, and Mobius Bands, Math. Ann. 207 (1974), pp. 233-243. (abstract)

    17. Triple Points and Surgery for Immersed Surfaces, Proc. A.M.S. 46 (1974), 407-413.

    18. Stiefel-Whitney Homology Classes and Singularities of Projections for Polyhedral Manifolds, Amer. Math. Soc. Proc. Sympos. Pure Math., Vol. XXVII, Stanford University 1973, Part I (1975), 333-347. (abstract) ( AMS review #51#14090)

    19. The Behavior of the Total Twist and Self-Linking Number of a Closed Space Curve under Inversions (with James E. White), Mathematica Scandinavica 36 (1975), 254- 262. (abstract) ( AMS review #52#4312)

    20. Height functions with three critical points (with Floris Takens), Illinois J. Mathematics, 76 (1975), 325-335. (abstract) ( AMS review #51#14079)

    21. Minimal submanifolds of the Bicylinder Boundary ,Boletim da Sociedade Brasileira de Matematica, 7 (1976), 37-57. ( AMS review #58#12844)

    22. Self-Linking Numbers of Space Polygons, Indiana University Mathematics Journal 25 (1976), 1171-1188. (abstract) ( AMS review #55#4227)

    23. Immersions and Mod 2 Quadratic Forms (with Lou Kauffman), American Mathematical Monthly 84 (1977), 168-185. (Awarded Lester Ford Award for exposition, Summer 1978). (abstract) ( AMS review #55#4210)

    24. Whitney Duality and Singularities of Projections (with Clint McCrory), Proceedings of Escuela Latino-americana de Mathematica, Rio de Janeiro, Springer-Verlag Lecture Notes in Mathematics 5997 (1977), 68-81. ( AMS review #56#6678)

    25. Computer Animation and the Geometry of Surfaces in 3- and 4-Space, Proceedings of the International Congress of Mathematicians, Helsinki (1978), (Invited 45 minute address), 1005-1013. ( AMS review #82b:53002)

    26. Real-Time Computer Graphics Analysis of Figures in Four-Space (with Charles Strauss),American Association of the Advancement of Science Selected Symposium 24 (1978), Westview Press, Colo., pp. 159-168.

    27. Combinatorial Formula for Normal Stiefel-Whitney Classes (with Clint McCrory), Proceedings of the A.M.S. 76 (1979), 171-177. (abstract)

    28. Selected Papers in Geometry (Edited by Ann Stehney, Tilla Milnor, Joseph D'Atri and Thomas Banchoff), Mathematical Association of America (1979). ( AMS review #82c:01041)

    29. Sur les points paraboliques des surfaces: erratum et complements (with Rene Thom), C. R. Acad. Sc. Paris, 291 (27 Octobre 1980), 503-505.

    30. Every Sphere Eversion has a Quadruple Point (with Nelson Max), Am. J. Math., Supplementary Issue (1981), 191-209. (abstract) ( AMS review #83g:57020)

    31. Cusps of Gauss Mappings (with T. Gaffney and C. McCrory), Pitman Advanced Publishing Program, 55 (1982), London, pp. 1-88.

    32. Frenet Frames and Theorem of Jacobi and Milnor for Space Polygons, Jugoslavenska Akademija Znanosti I Umjetnosti , iz Rada 396 (1982), pp. 101-108. (abstract)

    33. Double Tangency Theorems for Pairs of Submanifolds, Geometry Symposium Utrecht (1980), Springer-Verlag Lecture Notes 894, pp. 26-48. ( AMS review #83h:53005)

    34. Geometrical Class and Degree for Surfaces in Three-Space (N. Kuiper), J. Diff. Geom. 16 (1981), 559-576. (abstract) ( AMS review #83j:53056)

    35. Circular and Countercircular Images of Plane Curves (with E. Beckenbach), General Inequalities 3 (1983), ISNM64, Birkhäuser-Verlag, 321-337. (abstract) ( AMS review #86h:53003)

    36. The Nine-Vertex Complex Projective Plane (with W. Kühnel), Mathematical Intelligencer 5 (1983), 11-22. (abstract) ( AMS review #85d:51016)

    37. Linear Algebra Through Geometry (with J. Wermer), Springer-Verlag (1983). ( AMS review #84f:15001)

    38. DIAL: A Diagrammatic Animation Language (with S. Feiner and D. Salesin), IEEE Computer Graphics and Applications, 2, No. 7 (1982), 43-54.

    39. Computer Graphics in Geometric Research, Recent Trends in Mathematics, Teubner- Texte 50 (1983), 316-327. ( AMS review #85c:00001)

    40. Critical points and curvature for embedded polyhedra II, Differential Geometry, Proc. Special Year, Maryland, Progress in Math 32, Birkhuser (1983), 34-55. (abstract)

    41. La Quarta Dimensio i Salvador Dali, Breu Viatge al mon de la Mathematica, 1 (1984), pp. 19-24.

    42. Normal Curvatures and Euler Classes for Polyhedral Surfaces in 4-Space, Proc. A.M.S.,92 (1984), 593-596. (abstract) ( AMS review #86c:57019)

    43. Differential Geometry and Computer Graphics, Perspective in Mathematics, Anniversary of Oberwolfach (1984) Birkhuser-Verlag, Basel, pp. 43-60. (abstract) ( AMS review #86f:53001)

    44. Counting Tritangent Planes of Space Curves (with T. Gaffney and C. McCrory), Topology 34 (1985), 15-24. (abstract) ( AMS review #86m:58028a)

    45. Visualizing Two-Dimensional Phenomena in Four-Dimensional Space, Statistical Image Processing, Marcel Dekker (1986), 187-202. (abstract) ( AMS review #89a:58001)

    46. Topology and Mechanics with Computer Graphics: Linear Hamiltonian Systems in Four Dimensions (with H. Koçla;ak, F. Bisshopp and D. Laidlaw), Advances in Applied Mathematics (1986), 282-308. (abstract) ( AMS review #88a:58069)

    47. Computer Graphics and Differential Geometry: Because the Light is Better over Here. The Merging of Disciplines: New Directions in Pure, Applied, and Combinatorial Mathematics, Springer-Verlag(1986) 1-11. (abstract)

    48. EDGE: The Educational Geometry Environment (with Richard Schwartz) Pro. of the International Congress on Educational Computing in Mathematics 87 (1987) 11-29. (abstract)

    49. Global Theorems for Symmetry Sets of Smooth Curves and Polygons in the Plane (with Peter Giblin) Proc. Royal Soc. Edinburgh 106A (1987) 221-231. ( AMS review #88j:53003)

    50. Torus Decompositions of Regular Polytopes in 4-Space. Shaping Space (1988), Birkhäuser-Verlag, 221-230. ( AMS review #937 089)

    51. From Flatland to Hypergraphics: Interacting with Higher Dimensions Interdisciplinary Science Reviews,15, No. 4 (1990) 364-372.

    52. Beyond the Third Dimension, (1990) New York: W. H.Freeman & Co., Scientific American Library, 1-210. ( AMS review #91h:00001)

    53. Student Generated Interactive Software for Calculus of Surfaces in a Workstation Laboratory, (with student associates) UME Trends 1, No. 3, (1990) 7-8.

    54. Geometry of the Hopf Mapping and Pinkall's Tori of Given Conformal Type Computers in Geometry, Marcel Dekker, New York (1990) 57-62. (abstract) ( AMS review #90c:53035)

    55. Computer Graphics Tools for Rendering Algebraic Surfaces and the Geometry of Order Geometric Analysis and Computer Graphics, Springer-Verlag, New York (1991) 31-37. (abstract) ( AMS review #1:081:327)

    56. Computer Laboratory Magnification of Idiosyncrasies MAA Notes 20 (1991) 1-8.

    57. Investigating Volumes: The Air France Cup Geometry's Future, COMAP (1991) 87- 93.

    58. Dimensions On the Shoulders of Giants, National Academy of Sciences Press (1991) 11-59.

    59. Flatland: A New Introduction Princeton Science Series, Princeton University Press (1991) xv-xxxi.

    60. Linear Algebra Through Geometry (with J. Wermer), revised and expanded second edition, Springer-Verlag (1991).

    61. Illustrating Beyond the Third Dimension, (with Davide Cervone) Leonardo, special issue on mathematics and computer graphics, 25 3/4 (1992) 273-280.

    62. Euler Numbers, Complex Points and Singularities of Projections for Oriented Surfaces in Four-Space (with F. Farris) Pacific Journal of Mathematics 161 no. 1 (1993) 1, 24..

    63. Equilibrium Triangulations of the Complex Projective Plane (with Wolfgang Kühnel), Geometriae Dedicata. 44(1992)313-333. (abstract) ( AMS review #94a:52035)

    64. With Coxeter at the International Congress on Mathematics Education VII, Focus 12 No. 5 (1992) 4, 16.

    65. Symmetry Sets of Piecewise Circular Curves (with Peter Giblin), Proc. Royal Soc. Edinburgh 123A (1993) 1-15. (abstract) ( AMS review #95a:58007)

    66. Tangential and Normal Euler Numbers, Complex Points, and Singularities of Projections for Oriented Surfaces in Four-Space (with Frank Farris), Pacific Journal of Mathematics Vol. 161 No. 1. (1993) pp. 1-24. (abstract)

    67. Interactive Computer Graphics, Higher Dimensional Geometry, and Electronic Publication: From Flatland to Hypertext, The Serials Librarian 24 No. 3-4 (1994) 9-15. simultaneously in New Scholarship, New Serials: POroceedings of the North American Serials Group The Haworth Press (1994) 9-15.

    68. ODE on a Grecian Urn, American Mathematical Monthly 100 No. 9 (1993) cover.

    69. On the Geometry of Piecewise Circular Curves (with Peter Giblin), American Mathematical Monthly 101 No. 5 (1994) 403-416. (abstract) ( AMS review #95k:51030a)
    171-177. (abstract)

  2. Selected Papers in Geometry (Edited by Ann Stehney, Tilla Milnor, Joseph D'Atri and Thomas Banchoff), Mathematical Association of America (1979). ( AMS review #82c:01041)

  3. Sur les points paraboliques des surfaces: ebiblio.html.bak010066400054310000001000000410560617150320500137470ustar00dkpother00002540002445 Thomas F. Banchoff's list of publications. List of Publications of Thomas F. Banchoff
    1. Tightly embedded 2-dimensional polyhedral manifolds, Amer. J. Math., 87 (1965), 462-472. (abstract) ( AMS review #31#2729)

    2. Critical points and curvature for embedded polyhedra, J. Differential Geometry, 1 (1967), 257-268. (abstract) ( AMS review #85b:53061)

    3. Total central curvature of curves, Duke Math. J., 37 (1970), 281-289. (abstract) ( AMS review #41#4447)

    4. Periodic points of Anosov diffeomorphisms (with Michael I. Rosen), Proceedings of Symposia in Pure Mathematics, Global Analysis, Vol. XIV (1970), 17-21.(abstract) ( AMS review #43#2738)

    5. The spherical two piece property and tight surfaces in spheres, J. Differential Geometry, 4 (1970), 193-205. (abstract) ( AMS review #42#3720)

    6. Critical points and curvature for embedded polyhedral surfaces, Amer. Math. Monthly, 77 (1970), 475-485. ( AMS review #41#4444)

    7. Non-rigidity theorems for tight polyhedral tori, Archiv der Mathematik, 21 (1970), 416-423. (abstract) ( AMS review #42#8426)

    8. The two-piece property and tight n-manifolds-with-boundary in E^n , Trans. Amer. Math. Soc. 161 (1971), 259-267. (abstract) ( AMS review #44#4760)

    9. [9] On a generalization of the isoperimetric inequality (with William Pohl), J. Differential Geometry, 6 (1971), 175-192. ( AMS review #46#4449)

    10. High codimensional 0-tight mappings on spheres, Proc. Amer. Math. Soc., 29 (1971), 133-137. (abstract) ( AMS review #43#5541)

    11. Applications of elementary calculus, Eight lectures in an NSF Sponsored Conference for College Teachers of Mathematics, Summer 1971, published in the Proceedings of the Conference.

    12. Polyhedral Catastrophe I: Maps of the Line to the Line, Dynamical Systems, Academic Press (1973), 7-22. ( AMS review #49#6268)

    13. Global Geometry of Polygons I: The Theorem of Fabricius-Bjerre, Proc. A.M.S. 45 (1974), 237-241. (abstract) ( AMS review #51#6826)

    14. Real Time Computer Graphics Techniques in Geometry (with Charles Strauss), Proceedings of Symposia in Applied Mathematics Vol. 20, The Influence of Computing on Mathematical Research and Education, Amer. Math. Soc. (1974), 105- 111. ( AMS review #49#7058)

    15. Tight Polyhedral Klein Bottles, Projective Planes, and Mobius Bands, Math. Ann. 207 (1974), pp. 233-243. (abstract)

    16. Triple Points and Surgery for Immersed Surfaces, Proc. A.M.S. 46 (1974), 407-413.

    17. Stiefel-Whitney Homology Classes and Singularities of Projections for Polyhedral Manifolds, Amer. Math. Soc. Proc. Sympos. Pure Math., Vol. XXVII, Stanford University 1973, Part I (1975), 333-347. (abstract) ( AMS review #51#14090)

    18. The Behavior of the Total Twist and Self-Linking Number of a Closed Space Curve under Inversions (with James E. White), Mathematica Scandinavica 36 (1975), 254- 262. (abstract) ( AMS review #52#4312)

    19. Height functions with three critical points (with Floris Takens), Illinois J. Mathematics, 76 (1975), 325-335. (abstract) ( AMS review #51#14079)

    20. Minimal submanifolds of the Bicylinder Boundary ,Boletim da Sociedade Brasileira de Matematica, 7 (1976), 37-57. ( AMS review #58#12844)

    21. Self-Linking Numbers of Space Polygons, Indiana University Mathematics Journal 25 (1976), 1171-1188. (abstract) ( AMS review #55#4227)

    22. Immersions and Mod 2 Quadratic Forms (with Lou Kauffman), American Mathematical Monthly 84 (1977), 168-185. (Awarded Lester Ford Award for exposition, Summer 1978). (abstract) ( AMS review #55#4210)

    23. Whitney Duality and Singularities of Projections (with Clint McCrory), Proceedings of Escuela Latino-americana de Mathematica, Rio de Janeiro, Springer-Verlag Lecture Notes in Mathematics 5997 (1977), 68-81. ( AMS review #56#6678)

    24. Computer Animation and the Geometry of Surfaces in 3- and 4-Space, Proceedings of the International Congress of Mathematicians, Helsinki (1978), (Invited 45 minute address), 1005-1013. ( AMS review #82b:53002)

    25. Real-Time Computer Graphics Analysis of Figures in Four-Space (with Charles Strauss),American Association of the Advancement of Science Selected Symposium 24 (1978), Westview Press, Colo., pp. 159-168.

    26. Combinatorial Formula for Normal Whitney Classes (with Clint McCrory), Proceedings of the A.M.S. 76 (1979), 171-177.

    27. Selected Papers in Geometry (Edited by Ann Stehney, Tilla Milnor, Joseph D'Atri and Thomas Banchoff), Mathematical Association of America (1979). ( AMS review #82c:01041)

    28. Sur les points paraboliques des surfaces: erratum et complements (with Rene Thom), C. R. Acad. Sc. Paris, 291 (27 Octobre 1980), 503-505.

    29. Every Sphere Eversion has a Quadruple Point (with Nelson Max), Am. J. Math., Supplementary Issue (1981), 191-209. ( AMS review #83g:57020)

    30. Cusps of Gauss Mappings (with T. Gaffney and C. McCrory), Pitman Advanced Publishing Program, 55 (1982), London, pp. 1-88.

    31. Frenet Frames and Theorem of Jacobi and Milnor for Space Polygons, Jugoslavenska Akademija Znanosti I Umjetnosti , iz Rada 396 (1982), pp. 101-108. (abstract)

    32. Double Tangency Theorems for Pairs of Submanifolds, Geometry Symposium Utrecht (1980), Springer-Verlag Lecture Notes 894, pp. 26-48. ( AMS review #83h:53005)

    33. Geometrical Class and Degree for Surfaces in Three-Space (N. Kuiper), J. Diff. Geom. 16 (1981), 559-576. ( AMS review #83j:53056)

    34. Circular and Countercircular Images of Plane Curves (with E. Beckenbach), General Inequalities 3 (1983), ISNM64, Birkhäuser-Verlag, 321-337. ( AMS review #86h:53003)

    35. The Nine-Vertex Complex Projective Plane (with W. Kühnel), Mathematical Intelligencer 5 (1983), 11-22. ( AMS review #85d:51016)

    36. Linear Algebra Through Geometry (with J. Wermer), Springer-Verlag (1983). ( AMS review #84f:15001)

    37. DIAL: A Diagrammatic Animation Language (with S. Feiner and D. Salesin), IEEE Computer Graphics and Applications, 2, No. 7 (1982), 43-54.

    38. Computer Graphics in Geometric Research, Recent Trends in Mathematics, Teubner- Texte 50 (1983), 316-327. ( AMS review #85c:00001)

    39. Critical points and curvature for embedded polyhedra II, Differential Geometry, Proc. Special Year, Maryland, Progress in Math 32, Birkhuser (1983), 34-55. (abstract)

    40. Normal Curvatures and Euler Classes for Polyhedral Surfaces in 4-Space, Proc. A.M.S.,92 (1984), 593-596. ( AMS review #86c:57019)

    41. Differential Geometry and Computer Graphics, Perspective in Mathematics, Anniversary of Oberwolfach (1984) Birkhuser-Verlag, Basel, pp. 43-60. ( AMS review #86f:53001)

    42. Counting Tritangent Planes of Space Curves (with T. Gaffney and C. McCrory), Topology 34 (1985), 15-24. ( AMS review #86m:58028a)

    43. Visualizing Two-Dimensional Phenomena in Four-Dimensional Space, Statistical Image Processing, Marcel Dekker (1986), 187-202. (abstract) ( AMS review #89a:58001)

    44. Topology and Mechanics with Computer Graphics: Linear Hamiltonian Systems in Four Dimensions (with H. Koçla;ak, F. Bisshopp and D. Laidlaw), Advances in Applied Mathematics (1986), 282-308. ( AMS review #88a:58069)

    45. Computer Graphics and Differential Geometry: Because the Light is Better over Here. The Merging of Disciplines: New Directions in Pure, Applied, and Combinatorial Mathematics, Springer-Verlag(1986) 1-11.

    46. Global Theorems for Symmetry Sets of Smooth Curves and Polygons in the Plane (with Peter Giblin) Proc. Royal Soc. Edinburgh 106A (1987) 221-231. ( AMS review #88j:53003)

    47. Torus Decompositions of Regular Polytopes in 4-Space. Shaping Space (1988), Birkhäuser-Verlag, 221-230. ( AMS review #937 089)

    48. From Flatland to Hypergraphics: Interacting with Higher Dimensions Interdisciplinary Science Reviews,15, No. 4 (1990) 364-372.

    49. Beyond the Third Dimension, (1990) New York: W. H.Freeman & Co., Scientific American Library, 1-210. ( AMS review #91h:00001)

    50. Student Generated Interactive Software for Calculus of Surfaces in a Workstation Laboratory, (with student associates) UME Trends 1, No. 3, (1990) 7-8.

    51. Geometry of the Hopf Mapping and Pinkall's Tori of Given Conformal Type Computers in Geometry, Marcel Dekker, New York (1990) 57-62. ( AMS review #90c:53035)

    52. Computer Graphics Tools for Rendering Algebraic Surfaces and the Geometry of Order Geometric Analysis and Computer Graphics, Springer-Verlag, New York (1991) 31-37. ( AMS review #1:081:327)

    53. Computer Laboratory Magnification of Idiosyncrasies MAA Notes 20 (1991) 1-8.

    54. Investigating Volumes: The Air France Cup Geometry's Future, COMAP (1991) 87- 93.

    55. Dimensions On the Shoulders of Giants, National Academy of Sciences Press (1991) 11-59.

    56. Flatland: A New Introduction Princeton Science Series, Princeton University Press (1991) xv-xxxi.

    57. Linear Algebra Through Geometry (with J. Wermer), revised and expanded second edition, Springer-Verlag (1991).

    58. Illustrating Beyond the Third Dimension, (with Davide Cervone) Leonardo, special issue on mathematics and computer graphics, 25 3/4 (1992) 273-280.

    59. Euler Numbers, Complex Points and Singularities of Projections for Oriented Surfaces in Four-Space (with F. Farris) Pacific Journal of Mathematics 161 no. 1 (1993) 1, 24..

    60. Equilibrium Triangulations of the Complex Projective Plane (with Wolfgang Kühnel), Geometriae Dedicata. 44(1992)313-333. (abstract) ( AMS review #94a:52035)

    61. With Coxeter at the International Congress on Mathematics Education VII, Focus 12 No. 5 (1992) 4, 16.

    62. Symmetry Sets of Piecewise Circular Curves (with Peter Giblin), Proc. Royal Soc. Edinburgh 123A (1993) 1-15. (abstract) ( AMS review #95k:58007)

    63. Tangential and Normal Euler Numbers, Complex Points, and Singularities of Projections for Oriented Surfaces in Four-Space (with Frank Farris), Pacific Journal of Mathematics Vol. 161 No. 1. (1993) pp. 1-24. (abstract)

    64. Interactive Computer Graphics, Higher Dimensional Geometry, and Electronic Publication: From Flatland to Hypertext, The Serials Librarian 24 No. 3-4 (1994) 9-15. simultaneously in New Scholarship, New Serials: POroceedings of the North American Serials Group The Haworth Press (1994) 9-15.

    65. ODE on a Grecian Urn, American Mathematical Monthly 100 No. 9 (1993) cover.

    66. On the Geometry of Piecewise Circular Curves (with Peter Giblin), American Mathematical Monthly 101 No. 5 (1994) 403-416. ( AMS review #95k:51030a)
    btt.html010066400054310000001000000042050617074646300125550ustar00dkpother00002540002445 The Behavior of the Total Twist and Self-Linking Number of a Closed Space Curve under Inversions

    The Behavior of the Total Twist and Self-Linking Number of a Closed Space Curve under Inversions
    by Thomas F. Banchoff and James H. White

    Given that x:C -> E^3 is a smooth imbedding of a closed space curve, the total twist of a a unit normal vector field is a measurement of how much the normal plane turns as it moves along the curve. It can be shown that although the total twist is dependent on the particular vector field, its reduction mod Z, denoted Tw(x)~ is independent of the field. The first part of this paper proves that if x is an imbedded space curve and Ix is its image under an inversion through a sphere, the Tw(x)~ + Tw(Ix)~ = 0.

    It is then shown as a corollary that if x and Ix both have nowhere vanishing curvatures, then the normalized total torsion of x mod Z is identical to the negative of the normalized total torsion of Ix mod Z. Similar results hold for conformal transformations of E^3.

    The remainder of the article discussed the self-linking number of x, ie. the integer SL(x) which describes the linking number of x moved a small distance in the direction of its principal normal vector field. We prove the the self-linking number is the sum of the normalized total torsion and the Gauss integral, G(x), of x. In the second section, a deformation argument is used to prove the under an inversion, G(x) + G(Ix) = 0, which implies that SL(x) + SL(Ic) is the sum of the respective normalized torsions of x and Ix.

    The main theorem, presented in section 3, is:

    Theorem 4. If I is an inversion such that x and Ix have nowhere vanishing curvature, then SL(x) + SL(Ix) is equal to the winding numbers of the locus of osculating circles to x about the center of the sphere of inversion.

    In the last section, the behavior of plane curves under inversion is examined, and the we give the construction of a curve, C, an inversion, I, such that SL(C)=0 and SL(IC)=a for any integer a. cacc.html010066400054310000001000000011000617276510300126360ustar00dkpother00002540002445 Circular and Countercircular Images of Plane Curves

    Circular and Countercircular Images of Plane Curves
    by Thomas F. Banchoff and E. F. Beckenbach

    We extend Gauss's notion of the circular image of a plane curve, and we introduce the notion of a countercircular image of a plane curve from a point on the plane. The relationship between these two images are discussed, and we demonstrate examples of its application in global differential geometry of plane curves. cc1.gif010066400054310000001000005322570620143154100122270ustar00dkpother00002540002445GIF89abCU1IRűʄUAmG7vw':;%`x|߁齾7C wl,EQ~N` Vdԧ\n,fy|āv_[ey SwU>Bzr%7fnPpR!FSO"{ZBIc֤ۡ۫ESk6 z$1D-$7f24֦7BmP8UxP۰]M^_}`&_F|ܜdӞ {.bdKy!D0/u%)Kڒ󑹠́x~}[HqTH, jͤ, zGeImHJ~LB܌nvOTt-R{TB,V>./(֐rBFxb~g]a cSc?|L}3bUB4UO{'y0;I~5}wc1 gJu D*F~Ztz~k̏˯Ve[yRwrFEJN5KR QCC2}d!o(Rٝ}v|_rZt[Q3}U}FJJB|H[OPZ-LX Y .[P~f~_g rr+b~r>[qGd{fFI14H/@j6"/Lth<~b`^1gֈhEcG8!ojʜZI;9sJZʹ09Jk,bYNgU$ WaK9bz.Y̸ "QBإb,ؿy(WeNw8 ŽweKXH8Ѣ:ʸ#Xͪ饩FbRSGlƵJGmUmׯFIj1K\P9Z8hŅEE X+dyb@=sEEV^YhаWg^=:,vn׽{N-Zsu\w5-tPΛWoyؿo:=e:2{飝S]~Gl8eҁ >UW|`QFxETDA4UQd1P@!ǛQ) S \! 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