My presentation of Dr. Frank's 1995 paper. Not suitable as a substitute for reading the original work, but the visualizations may be helpful.

1. Qualitative Spatial Reasoning:
Cardinal Directions as an Example
Andrew U. Frank
1995

2. Outline
2

3. Outline
• Introduction
2

4. Outline
• Introduction
• Motivation: Why qualitative? Why cardinal?
2

5. Outline
• Introduction
• Motivation: Why qualitative? Why cardinal?
• Method: An algebraic approach
2

6. Outline
• Introduction
• Motivation: Why qualitative? Why cardinal?
• Method: An algebraic approach
• Two cardinal direction systems
2

7. Outline
• Introduction
• Motivation: Why qualitative? Why cardinal?
• Method: An algebraic approach
• Two cardinal direction systems
- Cone-shaped directions
2

8. Outline
• Introduction
• Motivation: Why qualitative? Why cardinal?
• Method: An algebraic approach
• Two cardinal direction systems
- Cone-shaped directions
- Projection-based directions
2

9. Outline
• Introduction
• Motivation: Why qualitative? Why cardinal?
• Method: An algebraic approach
• Two cardinal direction systems
- Cone-shaped directions
- Projection-based directions
• Assessment
2

10. Outline
• Introduction
• Motivation: Why qualitative? Why cardinal?
• Method: An algebraic approach
• Two cardinal direction systems
- Cone-shaped directions
- Projection-based directions
• Assessment
• Research envisioned
2

11. Introduction
Geography utilizes large scale spatial reasoning
extensively.
•
Formalized qualitative reasoning processes are
essential to GIS.
•
An approach to spatial reasoning using qualitative
cardinal directions.
3

12. Motivation: Why qualitative?
Spatial relations are typically formalized in a
quantitative manner with Car tesian
coordinates and vector algebra.
4

13. Motivation: Why qualitative?
5

14. Motivation: Why qualitative?
5

15. Motivation: Why qualitative?
5

16. Motivation: Why qualitative?
“thirteen centimeters”
5

17. Motivation: Why qualitative?
Human spatial reasoning is based on qualitative
comparisons.
6

18. Motivation: Why qualitative?
Human spatial reasoning is based on qualitative
comparisons.
6

19. Motivation: Why qualitative?
Human spatial reasoning is based on qualitative
comparisons.
“longer”
6

20. Motivation: Why qualitative?
Human spatial reasoning is based on qualitative
comparisons.
“longer”
• precision is not always desirable
6

21. Motivation: Why qualitative?
Human spatial reasoning is based on qualitative
comparisons.
“longer”
• precision is not always desirable
• precise data is not always available
6

22. Motivation: Why qualitative?
Human spatial reasoning is based on qualitative
comparisons.
“longer”
• precision is not always desirable
• precise data is not always available
• numerical approximations do not account for
uncertainty 6

23. Motivation: Why qualitative?
7

24. Motivation: Why qualitative?
• For malization required for GIS
implementation.
7

25. Motivation: Why qualitative?
• For malization required for GIS
implementation.
• Interpretation of spatial relations
expressed in natural language.
7

26. Motivation: Why qualitative?
• For malization required for GIS
implementation.
• Interpretation of spatial relations
expressed in natural language.
• Comparison of semantics of spatial
terms in different languages.
7

27. Motivation: Why cardinal?
8

28. Motivation: Why cardinal?
Pullar and Egenhofer’s geographical scale
spatial relations (1988):
8

29. Motivation: Why cardinal?
Pullar and Egenhofer’s geographical scale
spatial relations (1988):
• direction north, northwest
8

30. Motivation: Why cardinal?
Pullar and Egenhofer’s geographical scale
spatial relations (1988):
• direction north, northwest
• topological disjoint, touches
8

31. Motivation: Why cardinal?
Pullar and Egenhofer’s geographical scale
spatial relations (1988):
• direction north, northwest
• topological disjoint, touches
• ordinal in, at
8

32. Motivation: Why cardinal?
Pullar and Egenhofer’s geographical scale
spatial relations (1988):
• direction north, northwest
• topological disjoint, touches
• ordinal in, at
• distance far, near
8

33. Motivation: Why cardinal?
Pullar and Egenhofer’s geographical scale
spatial relations (1988):
• direction north, northwest
• topological disjoint, touches
• ordinal in, at
• distance far, near
• fuzzy next to, close
8

34. Motivation: Why cardinal?
Pullar and Egenhofer’s geographical scale
spatial relations (1988):
• direction north, northwest
• topological disjoint, touches
• ordinal in, at
• distance far, near
• fuzzy next to, close
Cardinal direction chosen as a major example.
8

35. Method: An algebraic approach
9

36. Method: An algebraic approach
• Focus on not on directional relations
between points...
9

37. Method: An algebraic approach
• Focus on not on directional relations
between points...
• Find rules for manipulating directional
symbols & operators.
9

38. Method: An algebraic approach
• Focus on not on directional relations
between points...
• Find rules for manipulating directional
symbols & operators.
Directional symbols: N, S, E, W... NE, NW...
Operators: inv ∞ ()
9

39. Method: An algebraic approach
• Focus on not on directional relations
between points...
• Find rules for manipulating directional
symbols & operators.
Directional symbols: N, S, E, W... NE, NW...
Operators: inv ∞ ()
• Operational meaning in a set of formal
axioms.
9

40. Method: An algebraic approach
Inverse
Composition
Identity
10

41. Method: An algebraic approach
Inverse P2
P1
Composition
Identity
10

42. Method: An algebraic approach
Inverse P2
dir(P1,P2)
P1
Composition
Identity
10

43. Method: An algebraic approach
Inverse P2
dir(P1,P2)
inv(dir(P1,P2))
P1
Composition
Identity
10

44. Method: An algebraic approach
Inverse P2
dir(P1,P2)
inv(dir(P1,P2))
P1
Composition P2
P1 P3
Identity
10

45. Method: An algebraic approach
Inverse P2
dir(P1,P2)
inv(dir(P1,P2))
P1
Composition P2
dir(P1,P2)
P1 P3
Identity
10

46. Method: An algebraic approach
Inverse P2
dir(P1,P2)
inv(dir(P1,P2))
P1
Composition P2
dir(P1,P2) dir(P2,P3)
P1 P3
Identity
10

47. Method: An algebraic approach
Inverse P2
dir(P1,P2)
inv(dir(P1,P2))
P1
Composition P2
dir(P1,P2) dir(P2,P3)
P1 P3
dir(P1,P2) ∞ dir(P2,P3)
dir (P1,P3)
Identity
10

48. Method: An algebraic approach
Inverse P2
dir(P1,P2)
inv(dir(P1,P2))
P1
Composition P2
dir(P1,P2) dir(P2,P3)
P1 P3
dir(P1,P2) ∞ dir(P2,P3)
dir (P1,P3)
Identity P1
10

49. Method: An algebraic approach
Inverse P2
dir(P1,P2)
inv(dir(P1,P2))
P1
Composition P2
dir(P1,P2) dir(P2,P3)
P1 P3
dir(P1,P2) ∞ dir(P2,P3)
dir (P1,P3)
Identity P1
dir(P1,P1)=0
10

50. Method: Euclidean exact reasoning
11

51. Method: Euclidean exact reasoning
• Comparison between qualitative reasoning
and quantitative reasoning using analytical
geometry
11

52. Method: Euclidean exact reasoning
• Comparison between qualitative reasoning
and quantitative reasoning using analytical
geometry
• A qualitative rule is called Euclidean exact if
the result of applying the rule is the same as
that obtained by analytical geometry
11

53. Method: Euclidean exact reasoning
• Comparison between qualitative reasoning
and quantitative reasoning using analytical
geometry
• A qualitative rule is called Euclidean exact if
the result of applying the rule is the same as
that obtained by analytical geometry
• If the results differ, the rule is considered
Euclidean approximate
11

54. Two cardinal system examples
Cone-shaped Projection-based
N
NW NE NW N NE
W E W Oc E
SW SE SW S SE
S
“relative position of points
“going toward”
on the Earth”
12

55. Directions in cones
N
NW NE
W E
SW SE
S
13

56. Directions in cones
N • Angle assigned to nearest
NW NE named direction
• Area of acceptance increases
W E with distance
SW SE
S
13

57. Directions in cones
N
NW NE
W E
SW SE
S
13

58. Directions in cones
N
NW NE
W E
SW SE
S
14

59. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
W E
SW SE
S
14

60. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W E e(N)=NE
SW SE
S
14

61. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W E e(N)=NE
SW SE
S
14

62. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W E e(N)=NE
• 4/8 turn gives the inverse symbol:
SW SE e⁴(N)= inv(N) = S
S
14

63. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W E e(N)=NE
• 4/8 turn gives the inverse symbol:
SW SE e⁴(N)= inv(N) = S
S
14

64. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W E e(N)=NE
• 4/8 turn gives the inverse symbol:
SW SE e⁴(N)= inv(N) = S
S
14

65. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W E e(N)=NE
• 4/8 turn gives the inverse symbol:
SW SE e⁴(N)= inv(N) = S
S
• 8/8 turn gives the identity symbol:
e⁸(N)= N
14

66. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W E e(N)=NE
• 4/8 turn gives the inverse symbol:
SW SE e⁴(N)= inv(N) = S
S
• 8/8 turn gives the identity symbol, 0:
e⁸(N)= N = 0
15

67. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W E e(N)=NE
• 4/8 turn gives the inverse symbol:
SW SE e⁴(N)= inv(N) = S
S
• 8/8 turn gives the identity symbol, 0:
e⁸(N)= N = 0
15

68. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W E e(N)=NE
• 4/8 turn gives the inverse symbol:
SW SE e⁴(N)= inv(N) = S
S
• 8/8 turn gives the identity symbol, 0:
e⁸(N)= N = 0
15

69. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W 0 E e(N)=NE
• 4/8 turn gives the inverse symbol:
SW SE e⁴(N)= inv(N) = S
S
• 8/8 turn gives the identity symbol, 0:
e⁸(N)= N = 0
15

70. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W E e(N)=NE
• 4/8 turn gives the inverse symbol:
SW SE e⁴(N)= inv(N) = S
S
• 8/8 turn gives the identity symbol, 0:
e⁸(N)= N = 0
16

71. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W E e(N)=NE
• 4/8 turn gives the inverse symbol:
SW SE e⁴(N)= inv(N) = S
S
• 8/8 turn gives the identity symbol, 0:
e⁸(N)= N = 0
• Composition can be computed with averaging rules:
16

72. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W E e(N)=NE
• 4/8 turn gives the inverse symbol:
SW SE e⁴(N)= inv(N) = S
S
• 8/8 turn gives the identity symbol, 0:
e⁸(N)= N = 0
• Composition can be computed with averaging rules:
e(N) ∞ N = n
16

73. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W E e(N)=NE
• 4/8 turn gives the inverse symbol:
SW SE e⁴(N)= inv(N) = S
S
• 8/8 turn gives the identity symbol, 0:
e⁸(N)= N = 0
• Composition can be computed with averaging rules:
e(N) ∞ N = n
16

74. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W E e(N)=NE
• 4/8 turn gives the inverse symbol:
SW SE e⁴(N)= inv(N) = S
S
• 8/8 turn gives the identity symbol, 0:
e⁸(N)= N = 0
• Composition can be computed with averaging rules:
e(N) ∞ N = n
16

75. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W E e(N)=NE
• 4/8 turn gives the inverse symbol:
SW SE e⁴(N)= inv(N) = S
S
• 8/8 turn gives the identity symbol, 0:
e⁸(N)= N = 0
• Composition can be computed with averaging rules:
e(N) ∞ N = n
16

76. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W E e(N)=NE
• 4/8 turn gives the inverse symbol:
SW SE e⁴(N)= inv(N) = S
S
• 8/8 turn gives the identity symbol, 0:
e⁸(N)= N = 0
• Composition can be computed with averaging rules:
e(N) ∞ N = n
16

77. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W E e(N)=NE
• 4/8 turn gives the inverse symbol:
SW SE e⁴(N)= inv(N) = S
S
• 8/8 turn gives the identity symbol, 0:
e⁸(N)= N = 0
• Composition can be computed with averaging rules:
e(N) ∞ N = n e(N) ∞ inv (N)
16

78. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W E e(N)=NE
• 4/8 turn gives the inverse symbol:
SW SE e⁴(N)= inv(N) = S
S
• 8/8 turn gives the identity symbol, 0:
e⁸(N)= N = 0
• Composition can be computed with averaging rules:
e(N) ∞ N = n e(N) ∞ inv (N)
16

79. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W E e(N)=NE
• 4/8 turn gives the inverse symbol:
SW SE e⁴(N)= inv(N) = S
S
• 8/8 turn gives the identity symbol, 0:
e⁸(N)= N = 0
• Composition can be computed with averaging rules:
e(N) ∞ N = n e(N) ∞ inv (N)
16

80. Directions in cones
N Algebraic operations can be
NW NE performed with symbols:
• 1/8 turn changes the symbol:
W 0 E e(N)=NE
• 4/8 turn gives the inverse symbol:
SW SE e⁴(N)= inv(N) = S
S
• 8/8 turn gives the identity symbol, 0:
e⁸(N)= N = 0
• Composition can be computed with averaging rules:
e(N) ∞ N = n e(N) ∞ inv (N)
16

81. Cone direction composition table
17

82. Cone direction composition table
17

83. Cone direction composition table
Out of 64 combinations, only 10 are Euclidean exact.
17

84. Projection-based directions
18

85. Projection-based directions
W E
18

86. Projection-based directions
N
S
18

87. Projection-based directions
NW NE
SW SE
18

88. Projection-based directions
• With half-planes, only trivial
NW NE cases can be resolved:
NE ∞ NE = NE
SW SE
18

89. Projection-based directions
NW N NE
W Oc E
SW S SE
19

90. Projection-based directions
• Assign neutral zone in the
NW N NE
center of 9 regions
W Oc E
SW S SE
19

91. Projection-based directions
Algebraic operations can be
NW N NE
performed with symbols:
W Oc E
SW S SE
19

92. Projection-based directions
Algebraic operations can be
NW N NE
performed with symbols:
W Oc E • The identity symbol, 0, resides in
the neutral area.
SW S SE
19

93. Projection-based directions
Algebraic operations can be
NW N NE
performed with symbols:
W Oc E • The identity symbol, 0, resides in
the neutral area.
SW S SE
• Inverse gives the symbol opposite
the neutral area:
inv(N) = S
19

94. Projection-based directions
Algebraic operations can be
NW N NE
performed with symbols:
W Oc E • The identity symbol, 0, resides in
the neutral area.
SW S SE
• Inverse gives the symbol opposite
the neutral area:
inv(N) = S
19

95. Projection-based directions
Algebraic operations can be
NW N NE
performed with symbols:
W Oc E • The identity symbol, 0, resides in
the neutral area.
SW S SE
• Inverse gives the symbol opposite
the neutral area:
inv(N) = S
19

96. Projection-based directions
Algebraic operations can be
NW N NE
performed with symbols:
W Oc E • The identity symbol, 0, resides in
the neutral area.
SW S SE
• Inverse gives the symbol opposite
the neutral area:
inv(N) = S
• Composition combines each projection:
19

97. Projection-based directions
Algebraic operations can be
NW N NE
performed with symbols:
W Oc E • The identity symbol, 0, resides in
the neutral area.
SW S SE
• Inverse gives the symbol opposite
the neutral area:
inv(N) = S
• Composition combines each projection:
NE ∞ SW = 0
19

98. Projection-based directions
Algebraic operations can be
NW N NE
performed with symbols:
W Oc E • The identity symbol, 0, resides in
the neutral area.
SW S SE
• Inverse gives the symbol opposite
the neutral area:
inv(N) = S
• Composition combines each projection:
NE ∞ SW = 0
19

99. Projection-based directions
Algebraic operations can be
NW N NE
performed with symbols:
W Oc E • The identity symbol, 0, resides in
the neutral area.
SW S SE
• Inverse gives the symbol opposite
the neutral area:
inv(N) = S
• Composition combines each projection:
NE ∞ SW = 0 S ∞ E = SE
19

100. Projection-based directions
Algebraic operations can be
NW N NE
performed with symbols:
W Oc E • The identity symbol, 0, resides in
the neutral area.
SW S SE
• Inverse gives the symbol opposite
the neutral area:
inv(N) = S
• Composition combines each projection:
NE ∞ SW = 0 S ∞ E = SE
19

101. Projection-based directions
Algebraic operations can be
NW N NE
performed with symbols:
W Oc E • The identity symbol, 0, resides in
the neutral area.
SW S SE
• Inverse gives the symbol opposite
the neutral area:
inv(N) = S
• Composition combines each projection:
NE ∞ SW = 0 S ∞ E = SE
19

102. Projection composition table
20

103. Projection composition table
20

104. Projection composition table
Out of 64 combinations, 32 are Euclidean exact.
20

105. Assessment
21

106. Assessment
• Both systems use 9 directional symbols.
21

107. Assessment
• Both systems use 9 directional symbols.
• Cone-shaped system relies on averaging rules.
21

108. Assessment
• Both systems use 9 directional symbols.
• Cone-shaped system relies on averaging rules.
• Introducing the identity symbol 0 increases the
number of deductions in both cases.
21

109. Assessment
• Both systems use 9 directional symbols.
• Cone-shaped system relies on averaging rules.
• Introducing the identity symbol 0 increases the
number of deductions in both cases.
• There are fewer Euclidean approximations using
projection-based directions:
21

110. Assessment
• Both systems use 9 directional symbols.
• Cone-shaped system relies on averaging rules.
• Introducing the identity symbol 0 increases the
number of deductions in both cases.
• There are fewer Euclidean approximations using
projection-based directions:
‣ 56 approximations using cones
21

111. Assessment
• Both systems use 9 directional symbols.
• Cone-shaped system relies on averaging rules.
• Introducing the identity symbol 0 increases the
number of deductions in both cases.
• There are fewer Euclidean approximations using
projection-based directions:
‣ 56 approximations using cones
‣ 32 approximations using projections
21

112. Assessment
22

113. Assessment
• Both theoretical systems were implemented and
compared with actual results to assess accuracy:
22

114. Assessment
• Both theoretical systems were implemented and
compared with actual results to assess accuracy:
‣ Cone-shaped directions correct in 25% of cases.
22

115. Assessment
• Both theoretical systems were implemented and
compared with actual results to assess accuracy:
‣ Cone-shaped directions correct in 25% of cases.
‣ Projection-based directions correct in 50% of
cases.
22

116. Assessment
• Both theoretical systems were implemented and
compared with actual results to assess accuracy:
‣ Cone-shaped directions correct in 25% of cases.
‣ Projection-based directions correct in 50% of
cases.
- 1/4 turn off in only 2% of cases
22

117. Assessment
• Both theoretical systems were implemented and
compared with actual results to assess accuracy:
‣ Cone-shaped directions correct in 25% of cases.
‣ Projection-based directions correct in 50% of
cases.
- 1/4 turn off in only 2% of cases
- deviations in remaining 48% never greater
than 1/8 turn
22

118. Assessment
• Both theoretical systems were implemented and
compared with actual results to assess accuracy:
‣ Cone-shaped directions correct in 25% of cases.
‣ Projection-based directions correct in 50% of
cases.
- 1/4 turn off in only 2% of cases
-deviations in remaining 48% never greater
than 1/8 turn
• Projection-based directions produce a result that is
within 45˚ of actual values in 80% of cases.
22

119. Research envisioned
23

120. Research envisioned
Formalization of other large-scale spatial
relations using similar methods:
23

121. Research envisioned
Formalization of other large-scale spatial
relations using similar methods:
• Qualitative reasoning with distances
23

122. Research envisioned
Formalization of other large-scale spatial
relations using similar methods:
• Qualitative reasoning with distances
• Integrated reasoning about distances and
directions
23

123. Research envisioned
Formalization of other large-scale spatial
relations using similar methods:
• Qualitative reasoning with distances
• Integrated reasoning about distances and
directions
• Generalize distance and direction relations
to extended objects
23

124. Conclusion
24

125. Conclusion
• Qualitative spatial reasoning is crucial for
progress in GIS.
24

126. Conclusion
• Qualitative spatial reasoning is crucial for
progress in GIS.
• A system of qualitative spatial reasoning with
cardinal directions can be formalized using an
algebraic approach.
24

127. Conclusion
• Qualitative spatial reasoning is crucial for
progress in GIS.
• A system of qualitative spatial reasoning with
cardinal directions can be formalized using an
algebraic approach.
• Similar techniques should be applied to other
types of spatial reasoning.
24

128. Conclusion
• Qualitative spatial reasoning is crucial for
progress in GIS.
• A system of qualitative spatial reasoning with
cardinal directions can be formalized using an
algebraic approach.
• Similar techniques should be applied to other
types of spatial reasoning.
• Accuracy cannot be found in a single
method.
24

129. Subjective impact
A new sidewalk decal designed to help pedestrians find their way
in New York City.
25

130. Questions?
Qualitative Spatial Reasoning:
Cardinal Directions as an Example
Andrew U. Frank
1995
26