Research in Computational Differential Geomet

Transcription

1 Research in Computational Differential Geometry November 5, 2014

2 Approximations Often we have a series of approximations which we think are getting close to looking like some shape.

3 Approximations Often we have a series of approximations which we think are getting close to looking like some shape. For example, pictures at higher and higher resolutions will come close to looking like the real picture.

4 Approximations Often we have a series of approximations which we think are getting close to looking like some shape. For example, pictures at higher and higher resolutions will come close to looking like the real picture. But what do we mean by close to looking like?

5 Approximation of Shapes Consider the following situation:

6 Approximation of Shapes Consider the following situation: Let S 0 be the polygonal path from (0, 0) to (1, 0) to (1, 1).

7 Approximation of Shapes Consider the following situation: Let S 0 be the polygonal path from (0, 0) to (1, 0) to (1, 1). Let S 1 be the polygonal path from (0, 0) to (.5, 0) to (.5,.5) to (1,.5) to (1, 1).

8 Approximation of Shapes Consider the following situation: Let S 0 be the polygonal path from (0, 0) to (1, 0) to (1, 1). Let S 1 be the polygonal path from (0, 0) to (.5, 0) to (.5,.5) to (1,.5) to (1, 1). Generally, let S n be the polygonal path which starts at (0, 0) and moves along segments of length 2 n, alternatively horizontal and vertical, until it reaches (1, 1).

9 Approximation of Shapes Consider the following situation: Let S 0 be the polygonal path from (0, 0) to (1, 0) to (1, 1). Let S 1 be the polygonal path from (0, 0) to (.5, 0) to (.5,.5) to (1,.5) to (1, 1). Generally, let S n be the polygonal path which starts at (0, 0) and moves along segments of length 2 n, alternatively horizontal and vertical, until it reaches (1, 1). As n does S n look like any shape?

10 S 2 and S 4

11 Does S n Look Like anything as n? In some ways the set starts to look like a straight line.

12 Does S n Look Like anything as n? In some ways the set starts to look like a straight line. But in many ways the set is never anything like a straight line.

13 Does S n Look Like anything as n? In some ways the set starts to look like a straight line. But in many ways the set is never anything like a straight line. One way to see this is to compare the lengths:

14 Does S n Look Like anything as n? In some ways the set starts to look like a straight line. But in many ways the set is never anything like a straight line. One way to see this is to compare the lengths: The line would have length 2...

15 Does S n Look Like anything as n? In some ways the set starts to look like a straight line. But in many ways the set is never anything like a straight line. One way to see this is to compare the lengths: The line would have length 2... But the sets S n always have length exactly 2.

16 Not All Approximations the Same So some approximations may maintain certain features, but not others.

17 Not All Approximations the Same So some approximations may maintain certain features, but not others. Let s look at the types of features we may want to maintain.

18 Useful Measurements Lengths

19 Useful Measurements Lengths Areas

20 Useful Measurements Lengths Areas Volumes (etc.)

21 Useful Measurements Lengths Areas Volumes (etc.) Curvatures

22 Useful Measurements Lengths Areas Volumes (etc.) Curvatures Tangent Directions

23 Useful Measurements Lengths Areas Volumes (etc.) Curvatures Tangent Directions Angles

24 Not Useful Measurements The measurements on the previous slides can be found in many cases with derivatives and integrals.

25 Not Useful Measurements The measurements on the previous slides can be found in many cases with derivatives and integrals. But not every derivative or integral leads to a useful calculation about shape.

26 Not Useful Measurements The measurements on the previous slides can be found in many cases with derivatives and integrals. But not every derivative or integral leads to a useful calculation about shape. Ex: The Angular Velocity of a point moving on a Circle may be useful in Physics, but it doesn t tell us anything about the circle.

27 Not Useful Measurements The measurements on the previous slides can be found in many cases with derivatives and integrals. But not every derivative or integral leads to a useful calculation about shape. Ex: The Angular Velocity of a point moving on a Circle may be useful in Physics, but it doesn t tell us anything about the circle. Generally we only care about measurements which are independent of coordinates.

28 Coordinates Coordinates are numbers assigned to points on a shape.

29 Coordinates Coordinates are numbers assigned to points on a shape. For example, the (x, y) coordinates of the plane or longitude and latitude on a sphere.

30 Coordinates Coordinates are numbers assigned to points on a shape. For example, the (x, y) coordinates of the plane or longitude and latitude on a sphere. Coordinates are not necessary for geometry.

31 Coordinates Coordinates are numbers assigned to points on a shape. For example, the (x, y) coordinates of the plane or longitude and latitude on a sphere. Coordinates are not necessary for geometry. But coordinates are practically necessary to use Calculus techniques.

32 Coordinates Coordinates are numbers assigned to points on a shape. For example, the (x, y) coordinates of the plane or longitude and latitude on a sphere. Coordinates are not necessary for geometry. But coordinates are practically necessary to use Calculus techniques. In geometry we prefer coordinates which are intrinsic to the shape.

33 Intrinsic versus Extrinsic Coordinates

34 Manifolds Manifolds are shapes viewed as independent of their surrounding environment.

35 Manifolds Manifolds are shapes viewed as independent of their surrounding environment. For example, a sphere is considered to be a two-dimensional object, since it can be described using only two variables.

36 Manifolds Manifolds are shapes viewed as independent of their surrounding environment. For example, a sphere is considered to be a two-dimensional object, since it can be described using only two variables. All calculations are done on the ground.

37 Tangents in Manifolds Most calculations in differential geometry deal with tangent vectors in some way.

38 Tangents in Manifolds Most calculations in differential geometry deal with tangent vectors in some way. A tangent vector can be thought of as traveling from a certain point on the manifold with a certain velocity.

39 Tangents in Manifolds Most calculations in differential geometry deal with tangent vectors in some way. A tangent vector can be thought of as traveling from a certain point on the manifold with a certain velocity. Thus it is key to remember that in differential geometry, tangents have an initial position

40

41 Local Metrics We would like to calculate lengths, angles, etc. from our coordinates.

42 Local Metrics We would like to calculate lengths, angles, etc. from our coordinates. But in most cases this cannot be done directly from coordinates.

43 Local Metrics We would like to calculate lengths, angles, etc. from our coordinates. But in most cases this cannot be done directly from coordinates. Ex. if think of a sphere as a rectangle in θ and φ, we will guess its area as 2π π = 2π 2, while the actual area is 4π.

44 Local Metrics We would like to calculate lengths, angles, etc. from our coordinates. But in most cases this cannot be done directly from coordinates. Ex. if think of a sphere as a rectangle in θ and φ, we will guess its area as 2π π = 2π 2, while the actual area is 4π. To calculate the correct area, we keep track of local distance.

45 Local Metrics We would like to calculate lengths, angles, etc. from our coordinates. But in most cases this cannot be done directly from coordinates. Ex. if think of a sphere as a rectangle in θ and φ, we will guess its area as 2π π = 2π 2, while the actual area is 4π. To calculate the correct area, we keep track of local distance. This is a change of inner product on tangent vectors.

46 Local Metrics We would like to calculate lengths, angles, etc. from our coordinates. But in most cases this cannot be done directly from coordinates. Ex. if think of a sphere as a rectangle in θ and φ, we will guess its area as 2π π = 2π 2, while the actual area is 4π. To calculate the correct area, we keep track of local distance. This is a change of inner product on tangent vectors. This is used to derive the ds, da, dv terms etc. that you see in Calculus.

47 Metrics Changing the Shape Ex, if we consider the set {(x, y) : x 2 + y 2 < 1}, we can simply apply the normal inner product to the tangent vectors.

48 Metrics Changing the Shape Ex, if we consider the set {(x, y) : x 2 + y 2 < 1}, we can simply apply the normal inner product to the tangent vectors. However, we can also decide to treat the tangent vectors to different points differently, say by scaling the inner product at (a, b) by the term 1 1 a 2 b 2.

49 Metrics Changing the Shape Ex, if we consider the set {(x, y) : x 2 + y 2 < 1}, we can simply apply the normal inner product to the tangent vectors. However, we can also decide to treat the tangent vectors to different points differently, say by scaling the inner product at (a, b) by the term 1 1 a 2 b 2. This changes our notions of lengths, areas, and lines (though angles are similar to before). A change in metric changes the shape.

50 Metrics Changing the Shape Ex, if we consider the set {(x, y) : x 2 + y 2 < 1}, we can simply apply the normal inner product to the tangent vectors. However, we can also decide to treat the tangent vectors to different points differently, say by scaling the inner product at (a, b) by the term 1 1 a 2 b 2. This changes our notions of lengths, areas, and lines (though angles are similar to before). A change in metric changes the shape. (A more complicated calculation could show that this is in some complicated sense equivalent to taking distances on a hyperboloid).

51 Comparing Distances

52 Curvature of a Curve At each point we can define a tangent direction (this depends on the way the curve is fit into its surrounding space).

53 Curvature of a Curve At each point we can define a tangent direction (this depends on the way the curve is fit into its surrounding space). If we direct the curve, this defines a unit tangent.

54 Curvature of a Curve At each point we can define a tangent direction (this depends on the way the curve is fit into its surrounding space). If we direct the curve, this defines a unit tangent. We can then differentiate this with respect to arc length.

55 Curvature of a Curve At each point we can define a tangent direction (this depends on the way the curve is fit into its surrounding space). If we direct the curve, this defines a unit tangent. We can then differentiate this with respect to arc length. The length of the resulting vector is then the unsigned curvature at that point.

56 Curvature of a Curve At each point we can define a tangent direction (this depends on the way the curve is fit into its surrounding space). If we direct the curve, this defines a unit tangent. We can then differentiate this with respect to arc length. The length of the resulting vector is then the unsigned curvature at that point. The signed curvature is positive or negative depending on the direction of the curve.

57 Curvature of a Surface At a point, define the lines in various directions from that point.

58 Curvature of a Surface At a point, define the lines in various directions from that point. Find the directions with the greatest and least curvature.

59 Curvature of a Surface At a point, define the lines in various directions from that point. Find the directions with the greatest and least curvature. Multiply these curvatures.

60 Curvature of a Surface At a point, define the lines in various directions from that point. Find the directions with the greatest and least curvature. Multiply these curvatures. This is the Gaussian curvature at this point.

61

62 Topological Invariants Sometimes geometric measures do not captures the shape of the object in the simplest way.

63 Topological Invariants Sometimes geometric measures do not captures the shape of the object in the simplest way. For example, consider the following:

64 Euler Characteristic A simple topological invariant is the Euler characteristic.

65 Euler Characteristic A simple topological invariant is the Euler characteristic. For a polyhedron, this is V E + F

66 Euler Characteristic A simple topological invariant is the Euler characteristic. For a polyhedron, this is V E + F Any polyhedron without holes has an Euler characteristic of exactly two.

67 Euler Characteristic

68 Betti Numbers Generally Euler Characteristic can be generalized by considering the Betti numbers of a shape.

69 Betti Numbers Generally Euler Characteristic can be generalized by considering the Betti numbers of a shape. The Betti numbers in some way count the k-dimensional holes of a shape.

70 Betti Numbers Generally Euler Characteristic can be generalized by considering the Betti numbers of a shape. The Betti numbers in some way count the k-dimensional holes of a shape. In the previous slide we have b 0 = 1, b 1 = 1.

71 Betti Numbers Generally Euler Characteristic can be generalized by considering the Betti numbers of a shape. The Betti numbers in some way count the k-dimensional holes of a shape. In the previous slide we have b 0 = 1, b 1 = 1. For a torus, b 0 = 1, b 1 = 2, b 2 = 1.

72 Betti Numbers Generally Euler Characteristic can be generalized by considering the Betti numbers of a shape. The Betti numbers in some way count the k-dimensional holes of a shape. In the previous slide we have b 0 = 1, b 1 = 1. For a torus, b 0 = 1, b 1 = 2, b 2 = 1. Generally these are calculated by Homology.

73 Topological Measures The most important topological invariant for geometry is homology. Homology counts the number of holes in various dimensions. Ex. A torus has 4 holes : 1 0-dimensional, 2 1 dimensional, and 1 2-dimensional.

74 The Normal Cycle The Normal Cycle (or Current) is an object which in some way encodes all of the previous measures.

75 The Normal Cycle The Normal Cycle (or Current) is an object which in some way encodes all of the previous measures. An n-current is a generalization of an n-dimensional shape (in the sense that we can integrate n dimensional differential forms over them).

76 The Normal Cycle The Normal Cycle (or Current) is an object which in some way encodes all of the previous measures. An n-current is a generalization of an n-dimensional shape (in the sense that we can integrate n dimensional differential forms over them). By integrating certain forms over the normal cycle you can retrieve the previous information.

77 The Normal Cycle The Normal Cycle (or Current) is an object which in some way encodes all of the previous measures. An n-current is a generalization of an n-dimensional shape (in the sense that we can integrate n dimensional differential forms over them). By integrating certain forms over the normal cycle you can retrieve the previous information. Therefore convergence of normal cycles implies convergence of the previous measures.

78 The Normal Cycle The Normal Cycle (or Current) is an object which in some way encodes all of the previous measures. An n-current is a generalization of an n-dimensional shape (in the sense that we can integrate n dimensional differential forms over them). By integrating certain forms over the normal cycle you can retrieve the previous information. Therefore convergence of normal cycles implies convergence of the previous measures. This is a convenient (but tricky) way to show that an approximation is good.

79 Normal Cycles

80 Ways Approximations can Go Wrong It is not in the scope of this talk to discuss how to approximate correctly

81 Ways Approximations can Go Wrong It is not in the scope of this talk to discuss how to approximate correctly So let s look at ways to approximate incorrectly (or at least imperfectly).

82 Ways Approximations can Go Wrong It is not in the scope of this talk to discuss how to approximate correctly So let s look at ways to approximate incorrectly (or at least imperfectly). Along the way we can talk about possible fixes.

83 Pointwise Convergence, Length Does Not Converge

84 Approximate with secant lines

85 The Slope is Incorrect

86 Be More Careful About How You Pick Slopes

87 Topological Information is Incorrect

88 Another Example

89 Morse Scanning

90 Papers to Read If you like the following techniques, they are gathered together in the paper Pixelations of planar semialgebraic sets and shape recognition by Nicolaescu and Rowekamp

91 Papers to Read If you like the following techniques, they are gathered together in the paper Pixelations of planar semialgebraic sets and shape recognition by Nicolaescu and Rowekamp (Available online and appearing in a forthcoming issue of Algebraic and Geometric Topology)

92 In Three Dimensions Three dimensions can be related to two dimensions by using scanning.

93 In Three Dimensions Three dimensions can be related to two dimensions by using scanning. (Think of CAT scans)

94 In Three Dimensions Three dimensions can be related to two dimensions by using scanning. (Think of CAT scans) Mathematically this relates to Morse theory.

95 In Three Dimensions Three dimensions can be related to two dimensions by using scanning. (Think of CAT scans) Mathematically this relates to Morse theory. But problems can arise much more easily.

96 General Approximation Generally problems are made harder by increasing the dimension or requiring finer approximation.

97 General Approximation Generally problems are made harder by increasing the dimension or requiring finer approximation. I discuss very good approximations, but only in low dimensions.

98 General Approximation Generally problems are made harder by increasing the dimension or requiring finer approximation. I discuss very good approximations, but only in low dimensions. The opposite problem (loose approximations in high dimensions) is very active currently.

99 Suggested Works Edelsbruner, Harer, Computational Topology, An Introduction Fu, Convergence of curvatures in secant approximations Hatcher, Algebraic Topology Morvan, Generalized Curvatures Nicolaescu, An Invitation to Morse Theory Nicolaescu, Lectures on the Geometry of Manifolds