Tessellation Data StructuresTessellation Data Structures

Squares

Triangles

Hexagons

Properties:

• Orientation consistency

• Recursively subdividable?

• Radial symmetry

• Compatible with array data structures and a cartesian coorninate system.

Rasterization of a MapRasterization of a Map

Scan Line StructureScan Line Structure

AB

B

B

AB B

AB B

B P B A B

B A B

B A B

B A B

B A B

B A L B

B A B L

B A B

B A B

B A B A B

Quad-Tree StructureQuad-Tree Structure

1 2

3 4

5

6

7 8

9 10

A

B C

5 6

1 2 3 4 7 8 9 10

Linear Indexing TransformationsLinear Indexing Transformations

• Transformation of a graph topology into a path topology with the aim to order the objects in a linear sequence while preserving some of the spatial connections between the objects

• The requirements to space-filling curves (Peano scans): 1. passes once through every object2. objects close in the space should be near one another in the path transform3. An inverse operation exists to recover the original space

A Y-Coordinate Projection A Y-Coordinate Projection Path TransformationPath Transformation

P1

P2

P3

P4

P5

P6

P7

P8

P9

P10

P11

P12

P13

P14

P15

A Minimal Distance A Minimal Distance Path TransformationPath Transformation

P1

P3

P4

P5

P6

P7

P8

P9

P10

P11

P12

P13

P14

P15

P2

Row OrderRow Order

Row-PrimeRow-Prime

Morton OrderMorton Order

Pi OrderPi Order

Diagonal OrderDiagonal Order

Spiral OrderSpiral Order

Address Address Computation of Morton OrderComputation of Morton Order

0 000

1 001

2 002

3 003

4 010

5 011

63 333

6 012

7 013

8 020

9 021

10 022

11 023

12 030

13 031

14 032

15 033

16 100

17 101

18 102

19 103

20 110

21 111

22 112

23 113

24 120

25 121

26 122

27 123

28 130

29 131

30 132

31 133

32 200

33 201

34 202

35 203

36 210

37 211

38 212

39 213

40 220

41 221

42 222

43 223

44 230

45 231

46 232

47 233

48 300

49 301

50 302

51 303

52 310

53 311

54 312

55 313

56 320

57 321

58 322

59 323

60 330

61 331

62 332

000

000

001

001

002

002

003

003

004

004

005

005

006

006

007

007

0

0

1

1

2

2

3

3

4

4

5

5

6

6

7

7X |Y

X

Y

0 1 1

1 0 0

1 0 0 1 0 1

SummarySummary

• Data organization remains an important topic in digital cartography. The structure that is chosen depends on the type of base map being stored and the purpose of the digital cartographic information processing system.

Questions for ReviewQuestions for Review• Which are the mainly used three tessellation

data structures? Which properties do they have? What is the mostly used data structure for raster data?

• What are the data representations in raster data for points, lines, and regions?

• What is the scan line structure (or run- length coding)?

• How does a quad-tree record and restore a map image?

• What are the requirements for the space-filling curves (Peano scans)?

• According to your knowledge, what are the commonly used Peano scans? What are their advantages and disadvantages?

• Can you calculate the linear path topology address from the original cartesian coordinate system using Morton order? What is the reversal formula for the address transformation using the Morton order?

Questions for ReviewQuestions for Review