Projective relations in a 3 D environment Roland

Projective relations in a 3 D environment Roland Billen 1 & Eliseo Clementini 2 1 University of Liège (Belgium) 2 University of L’Aquila (Italy)

TOC ¢ Background and motivations ¢ Ternary proj. relationships among points in R² ¢ Ternary proj. relationships among regions in R² ¢ Ternary proj. relationships among points in R³ ¢ Ternary proj. relationships among bodies in R³ ¢ Quaternary proj. relationships among points in R³ ¢ Quaternary proj. relationships among bodies in R³ ¢ Further research Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Background and Motivations Qualitative Spatial Reasoning ¢ What is projective geometry? ¢ A geometry more specific than topology and less specific than metric l E. g. , topological property: l E. g. , projective property: l E. g. , metric property: disconnected concave square Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Background and Motivations ¢ Why projective geometry? Definition of many qualitative relations l Topological: • Lakes inside Scotland l Projective: • • l Cities between Glasgow and Edinburgh Lakes surrounded by mountains Shops on the right of the road Flags above the tree Metric: • Edinburgh is east of Glasgow • Edinburgh is not far from Glasgow Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Background and Motivations ¢ Projective invariants l Collinearity properties • e. g. , three points belong to the same line RO 2 RO 1 PO Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Background and Motivations We wished to extend our model in 3 D ¢ Could be used in ¢ 3 D GIS l Virtual Reality l Augmented Reality l Robot Navigation l Navigation in Geographic environment l… l Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Ternary projective relationships among points in R² ¢ Deriving other projective properties from collinearity U collinear aside rightside inside outside between leftside nonbetween before after Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Ternary projective relationships among points in R² Partition of R² based on the two reference points ¢ Set of JEPD relationships (7) ¢ Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Ternary projective relationships among regions in R² Still based on collinearity and reference objects shapes ¢ Set of JEPD relationships (34) ¢ Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Ternary projective relationships among regions in R² ls(A, B, C) = (1 0 0 | 0 0), bf(A, B, C) = (0 1 0 0 0 | 0 0) Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Ternary projective relationships among regions in R² Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Ternary projective relationships among points in R³ ¢ Almost the same that in R² Except that … U collinear aside rightside inside outside between leftside nonbetween before after Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Ternary projective relationships among points in R³ ¢ ¢ The specialisation of the aside relation is not possible in R³ Set of JEPD relations (6) U collinear inside outside between aside nonbetween before after Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Ternary projective relationships among bodies in R³ ¢ ¢ The relation collinear among bodies is the generalisation of the same relation among points The partition of the space is based on tangent planes (similarity with regions in R²) Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Ternary projective relationships among bodies in R³ ¢ A collinearity subspace can be defined ¢ The space is divided into a between subspace, a non-between subspace and an aside subspace Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Ternary projective relationships among bodies in R³ ¢ ¢ Same basic relationships than for points set of JEPD relationships (18) U collinear inside outside between aside nonbetween before after Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Ternary projective relationships among bodies in R³ bf(A, B, C) bt(A, B, C) bf: as(A, B, C) Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Quaternary projective relationships among points in R³ ¢ ¢ ¢ Three non collinear points define one an only one plane in the space concept of coplanarity Such a plane (called hyperplane) divides the whole space in two regions , called halfspaces Depending on the order of the three reference points, the plane can be oriented in R³ Positive and negative halfspaces Based on this partition, one can define projective relations between a point and three reference points These relations are therefore quaternary above, below, internal, external, inside and outside is a JEPD set of relations in R³ (6) Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Quaternary projective relationships among points in R³ Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Quaternary projective relationships among points in R³ U coplanar non coplanar below inside outside internal above external Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Quaternary projective relationships among bodies in R³ ¢ The concept of coplanarity between four bodies can be introduced as a generalisation of the same relation among points ¢ We end up the same basic relationships than for points, and a set of JEPD relationships (18) Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Quaternary projective relationships among bodies in R³ ¢ To Build the coplanarity subspace … …We consider 8 internal and external tangent planes to the three reference bodies Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Quaternary projective relationships among bodies in R³ Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Quaternary projective relationships among bodies in R³ Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Quaternary projective relationships among bodies in R³ ¢ The all set of quaternary relations can be obtained based on the empty / non-empty intersections of the primary body A with the subspaces which satisfy the basic quaternary relations int(A, B, C, D) = (1 0 0 0 | 0 0), ab(A, B, C, D) = (0 0 1 0 | 0 0), in(A, B, C, D) = (0 0 | 1 0), ext(A, B, C, D) = (0 1 0 0 | 0 0), be(A, B, C, D) = (0 0 0 1 | 0 0), ou(A, B, C, D) = (0 0 | 0 1) Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Quaternary projective relationships among bodies in R³ ext(A, B, C, D) int(A, B, C, D) Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Quaternary projective relationships among bodies in R³ ab(A, B, C, D) ext: ab(A, B, C, D) Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Further research ¢ (at SDH 04) l Algorithms for the computation of projective relations. (done) l Reasoning system for all ternary relations, composition tables and proofs. (on going) l Extensions to n-ary relations: surrounded by, in the middle of, etc. l Extensions to other geometric types: region/line, line/line, etc. l Extensions to 3 D relations. Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Further research ¢ (currently) l Algorithms for the computation of projective relations. (done) l Reasoning system for all ternary relations, composition tables and proofs. (almost done) l Extensions to n-ary relations: surrounded by, in the middle of, etc. (partially done) l Extensions to other geometric types: region/line, line/line, etc. l Extensions to 3 D relations. (done) l Reasoning system for all quaternary relations, composition tables and proofs. l Mapping these concepts to specific environment Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Mapping in 2 D Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Mapping in 3 D Projective relations in a 3 D environment, Billen R. & Clementini E. , GIScience 06, Muenster

Thanks for attention Questions ? ?