65
Basics Program Hyperbolic Ornaments Drawing in Non-Euclidean Crystallographic Groups Martin von Gagern joint work with Jürgen Richter-Gebert Technische Universität München Second International Congress on Mathematical Software, September 1 2006 Martin von Gagern Hyperbolic Ornaments

Hyperbolic Ornaments - Drawing in Non-Euclidean Crystallographic Groups · 2009. 6. 10. · Hyperbolic Geometry Anatomy of the Hyperbolic Plane Definition (Hyperbolic Axiom of Parallels)

  • Upload
    others

  • View
    11

  • Download
    0

Embed Size (px)

Citation preview

  • BasicsProgram

    Hyperbolic OrnamentsDrawing in Non-Euclidean Crystallographic Groups

    Martin von Gagernjoint work with Jürgen Richter-Gebert

    Technische Universität München

    Second International Congress on Mathematical Software,September 1 2006

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Educational Value

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Escher

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Hyperbolic Escher

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Hyperbolic Escher

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    SymmetriesHyperbolic Geometry

    Outline

    1 BasicsSymmetriesHyperbolic Geometry

    2 ProgramIntuitive InputGroup CalculationsFast Drawing

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    SymmetriesHyperbolic Geometry

    Rigid Motions

    Reflection Rotation Translation Glide Reflection

    Definition (Rigid Motion)

    Rigid Motions ( = Isometries) are the length-preserving mappingsof the plane onto itself.

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    SymmetriesHyperbolic Geometry

    Rigid Motions

    Reflection Rotation Translation Glide Reflection

    Definition (Rigid Motion)

    Rigid Motions ( = Isometries) are the length-preserving mappingsof the plane onto itself.

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    SymmetriesHyperbolic Geometry

    Groups of Rigid Motions

    • Group E(2): all euclidean planar isometries• Discrete Subgroups

    Definition (Discreteness)

    A group G is discrete if around every point P of the plane there is aneighborhood devoid of any images of P under the group operations.

    The discrete groups of rigid motions in the euclidean plane:

    • 17 Wallpaper Groups• 7 Frieze Groups• 2 kinds of Rosette Groups

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    SymmetriesHyperbolic Geometry

    Groups of Rigid Motions

    • Group E(2): all euclidean planar isometries• Discrete Subgroups

    Definition (Discreteness)

    A group G is discrete if around every point P of the plane there is aneighborhood devoid of any images of P under the group operations.

    The discrete groups of rigid motions in the euclidean plane:

    • 17 Wallpaper Groups• 7 Frieze Groups• 2 kinds of Rosette Groups

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    SymmetriesHyperbolic Geometry

    Groups of Rigid Motions

    • Group E(2): all euclidean planar isometries• Discrete Subgroups

    Definition (Discreteness)

    A group G is discrete if around every point P of the plane there is aneighborhood devoid of any images of P under the group operations.

    The discrete groups of rigid motions in the euclidean plane:

    • 17 Wallpaper Groups• 7 Frieze Groups• 2 kinds of Rosette Groups

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    SymmetriesHyperbolic Geometry

    Anatomy of the Hyperbolic Plane

    Definition (Hyperbolic Axiom of Parallels)

    Given a point P outside a line `there exist at least two lines through P that do not intersect `.

    • Many facts of euclidean geometry don’t rely on the Axiom ofParallels and are true in hyperbolic geometry as well.

    • The sum of angles in a triangle is less than π.• Lengths are absolute, scaling is not an automorphism.• Geometry of constant negative curvature.

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    SymmetriesHyperbolic Geometry

    Poincaré Disc Model

    • hyperbolic points:inside of the unit circle

    • hyperbolic lines:lines and circlesperpendicular to the unit circle

    • hyperbolic angle:identical to euclidean angle

    • hyperbolic distance:changes withdistance from center

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    SymmetriesHyperbolic Geometry

    Poincaré Disc Model

    • hyperbolic points:inside of the unit circle

    • hyperbolic lines:lines and circlesperpendicular to the unit circle

    • hyperbolic angle:identical to euclidean angle

    • hyperbolic distance:changes withdistance from center

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    SymmetriesHyperbolic Geometry

    Poincaré Disc Model

    • hyperbolic points:inside of the unit circle

    • hyperbolic lines:lines and circlesperpendicular to the unit circle

    • hyperbolic angle:identical to euclidean angle

    • hyperbolic distance:changes withdistance from center

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    SymmetriesHyperbolic Geometry

    Poincaré Disc Model

    • hyperbolic points:inside of the unit circle

    • hyperbolic lines:lines and circlesperpendicular to the unit circle

    • hyperbolic angle:identical to euclidean angle

    • hyperbolic distance:changes withdistance from center

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    SymmetriesHyperbolic Geometry

    Hyperbolic Rigid Motions

    Reflection Rotation Translation Glide Reflection

    N.B.: translations now have only a single fixed line.

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Outline

    1 BasicsSymmetriesHyperbolic Geometry

    2 ProgramIntuitive InputGroup CalculationsFast Drawing

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Tilings by regular Polygons

    • Square

    • Triangular

    • Hexagonal

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Tilings by regular Polygons

    • Square

    • Triangular

    • Hexagonal

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    From regular Polygons to Triangles

    regular heptagons 4(2, 3, 7) regular triangles

    angles2π3

    anglesπ

    2,π

    3,π

    7angles

    2π7

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    From regular Polygons to Triangles

    regular heptagons 4(2, 3, 7) regular triangles

    angles2π3

    anglesπ

    2,π

    3,π

    7angles

    2π7

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    From regular Polygons to Triangles

    regular heptagons 4(2, 3, 7) regular triangles

    angles2π3

    anglesπ

    2,π

    3,π

    7angles

    2π7

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    General Tesselations

    4(4, 6, 7) 4(2, 5,∞)

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Why All Angles are Different

    • 4(n, n, n) ⊂ 4(2, 3, 2n)

    • 4(n, 2n, 2n) ⊂ 4(2, 4, 2n)

    • 4(n, m, m) ⊂ 4(2, m, 2n)

    4(k , m, n) : πk

    m+

    π

    n< π

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Why All Angles are Different

    • 4(n, n, n) ⊂ 4(2, 3, 2n)

    • 4(n, 2n, 2n) ⊂ 4(2, 4, 2n)

    • 4(n, m, m) ⊂ 4(2, m, 2n)

    4(k , m, n) : πk

    m+

    π

    n< π

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Why All Angles are Different

    • 4(n, n, n) ⊂ 4(2, 3, 2n)

    • 4(n, 2n, 2n) ⊂ 4(2, 4, 2n)

    • 4(n, m, m) ⊂ 4(2, m, 2n)

    4(k , m, n) : πk

    m+

    π

    n< π

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Algebraic Calculations

    General triangle reflection group 4(k , m, n)• Coxeter group (finitely represented group for GAP)〈

    a, b, c | a2 = 1, b2 = 1, c2 = 1, (ab)k = 1, (ac)m = 1, (bc)n = 1〉

    • Subgroups with finite index arenon-euclidean crystallographic (N.E.C.) groups

    • Orientation preserving subgroups are Fuchsian

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Algebraic Calculations

    General triangle reflection group 4(k , m, n)• Coxeter group (finitely represented group for GAP)〈

    a, b, c | a2 = 1, b2 = 1, c2 = 1, (ab)k = 1, (ac)m = 1, (bc)n = 1〉

    • Subgroups with finite index arenon-euclidean crystallographic (N.E.C.) groups

    • Orientation preserving subgroups are Fuchsian

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Algebraic Calculations

    General triangle reflection group 4(k , m, n)• Coxeter group (finitely represented group for GAP)〈

    a, b, c | a2 = 1, b2 = 1, c2 = 1, (ab)k = 1, (ac)m = 1, (bc)n = 1〉

    • Subgroups with finite index arenon-euclidean crystallographic (N.E.C.) groups

    • Orientation preserving subgroups are Fuchsian

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Algebraic Calculations

    General triangle reflection group 4(k , m, n)• Coxeter group (finitely represented group for GAP)〈

    a, b, c | a2 = 1, b2 = 1, c2 = 1, (ab)k = 1, (ac)m = 1, (bc)n = 1〉

    • Subgroups with finite index arenon-euclidean crystallographic (N.E.C.) groups

    • Orientation preserving subgroups are Fuchsian

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Algebraic Calculations

    General triangle reflection group 4(k , m, n)• Coxeter group (finitely represented group for GAP)〈

    a, b, c | a2 = 1, b2 = 1, c2 = 1, (ab)k = 1, (ac)m = 1, (bc)n = 1〉

    • Subgroups with finite index arenon-euclidean crystallographic (N.E.C.) groups

    • Orientation preserving subgroups are Fuchsian

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Group Generation

    1 Generator entered by user2 Add inverse operations3 Find “all” combinations

    • Group representation• Orbit of centerpiece• Each element starts

    a new domain

    4 For all triangles that arenot yet part of any orbit

    • add triangle tocentral domain

    • combine triangle with allgroup elements to calculateits orbit, adding to domains

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Group Generation

    1 Generator entered by user2 Add inverse operations3 Find “all” combinations

    • Group representation• Orbit of centerpiece• Each element starts

    a new domain

    4 For all triangles that arenot yet part of any orbit

    • add triangle tocentral domain

    • combine triangle with allgroup elements to calculateits orbit, adding to domains

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Group Generation

    1 Generator entered by user2 Add inverse operations3 Find “all” combinations

    • Group representation• Orbit of centerpiece• Each element starts

    a new domain

    4 For all triangles that arenot yet part of any orbit

    • add triangle tocentral domain

    • combine triangle with allgroup elements to calculateits orbit, adding to domains

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Group Generation

    1 Generator entered by user2 Add inverse operations3 Find “all” combinations

    • Group representation• Orbit of centerpiece• Each element starts

    a new domain

    4 For all triangles that arenot yet part of any orbit

    • add triangle tocentral domain

    • combine triangle with allgroup elements to calculateits orbit, adding to domains

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Group Generation

    1 Generator entered by user2 Add inverse operations3 Find “all” combinations

    • Group representation• Orbit of centerpiece• Each element starts

    a new domain

    4 For all triangles that arenot yet part of any orbit

    • add triangle tocentral domain

    • combine triangle with allgroup elements to calculateits orbit, adding to domains

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Group Generation

    1 Generator entered by user2 Add inverse operations3 Find “all” combinations

    • Group representation• Orbit of centerpiece• Each element starts

    a new domain

    4 For all triangles that arenot yet part of any orbit

    • add triangle tocentral domain

    • combine triangle with allgroup elements to calculateits orbit, adding to domains

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Group Visualization

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Fast and Perfect Drawing

    Fast draw smooth lines in real timePerfect image looks as correct as display hardware allows

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Fast and Perfect Drawing

    Fast draw smooth lines in real timePerfect image looks as correct as display hardware allows

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Fast and Perfect Drawing

    Fast draw smooth lines in real timePerfect image looks as correct as display hardware allows

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Reverse Pixel Lookup

    1 Scan convert trianglesTriangle preprocessing

    2 Map into central domainGroup preprocessing

    3 Update only changesRealtime drawing

    4 SupersamplingAntialiasing

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Reverse Pixel Lookup

    1 Scan convert trianglesTriangle preprocessing

    2 Map into central domainGroup preprocessing

    3 Update only changesRealtime drawing

    4 SupersamplingAntialiasing

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Reverse Pixel Lookup

    1 Scan convert trianglesTriangle preprocessing

    2 Map into central domainGroup preprocessing

    3 Update only changesRealtime drawing

    4 SupersamplingAntialiasing

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Reverse Pixel Lookup

    1 Scan convert trianglesTriangle preprocessing

    2 Map into central domainGroup preprocessing

    3 Update only changesRealtime drawing

    4 SupersamplingAntialiasing

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Reverse Pixel Lookup

    1 Scan convert trianglesTriangle preprocessing

    2 Map into central domainGroup preprocessing

    3 Update only changesRealtime drawing

    4 SupersamplingAntialiasing

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Reverse Pixel Lookup

    1 Scan convert trianglesTriangle preprocessing

    2 Map into central domainGroup preprocessing

    3 Update only changesRealtime drawing

    4 SupersamplingAntialiasing

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Reverse Pixel Lookup

    1 Scan convert trianglesTriangle preprocessing

    2 Map into central domainGroup preprocessing

    3 Update only changesRealtime drawing

    4 SupersamplingAntialiasing

    Martin von Gagern Hyperbolic Ornaments

  • BasicsProgram

    Intuitive InputGroup CalculationsFast Drawing

    Reverse Pixel Lookup

    1 Scan convert trianglesTriangle preprocessing

    2 Map into central domainGroup preprocessing

    3 Update only changesRealtime drawing

    4 SupersamplingAntialiasing

    Martin von Gagern Hyperbolic Ornaments

  • BasicsSymmetriesHyperbolic Geometry

    ProgramIntuitive InputGroup CalculationsFast Drawing