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

Outline
2

Outline
• Introduction
2

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

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

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

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

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

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

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

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

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

Motivation: Why qualitative?
5

Motivation: Why qualitative?
5

Motivation: Why qualitative?
5

Motivation: Why qualitative?
“thirteen centimeters”
5

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

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

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

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

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

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

Motivation: Why qualitative?
7

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

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

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

Motivation: Why cardinal?
8

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

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

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

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

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

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

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

Method: An algebraic approach
9

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

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

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

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

Method: An algebraic approach
Inverse
Composition
Identity
10

Method: An algebraic approach
Inverse P2
P1
Composition
Identity
10

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

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

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

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

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

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

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

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

Method: Euclidean exact reasoning
11

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cone direction composition table
17

Cone direction composition table
17

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

Projection-based directions
18

Projection-based directions
W E
18

Projection-based directions
N
S
18

Projection-based directions
NW NE
SW SE
18

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Projection composition table
20

Projection composition table
20

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

Assessment
21

Assessment
• Both systems use 9 directional symbols.
21

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

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

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

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

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

Assessment
22

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

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

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

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

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

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

Research envisioned
23

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

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

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

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

Conclusion
24

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

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

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

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

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

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