We also investigate the question if algebraic closure decides consistency for

![0_image_0.png](0_image_0.png)

DRAop and DRAopp.

![0_image_1.png](0_image_1.png)

Proposition 56. For DRAopp *algebraic closure does not decide consistency.*
Proof. This proof is inspired by the one that shows that algebraic closure does not decide consistency for *OPRA* (ref. to [49]). Consider a DRAopp constraint network in three points A, B and C as shown in Fig. 29. Both A and B point at C. These three points are in the relations:
A LEFTright- B A FRONTleft C B FRONTleft C.

We add a point D to our constraint satisfaction problem with C RIGHTleftP D.

We claim that D also lies in front of A and B by introducing the constraints A FRONTleft D and B FRONTleft D. By inspecting the composition table of DRAopp, we can see that it is consistent. Since by the constraint A LEFTright- B the points A and B are not collinear, D has to lie on the intersection point of the rays lA and lB, but by A FRONTleft C and B FRONTleft C, C also has to lie on that intersection point. Hence, C and D have to have the same position, what is a contradiction to the constraint C RIGHTleftP D. Hence this scenario is algebraically closed, but inconsistent.

Proposition 57. For DRAop *algebraic closure does not decide consistency.*
Proof. This proof is analogous to the one of Prop. 56, with substituting LEFTright- by LEFTright and RIGHTleftP by RIGHTleft.

## 5 A Sample Application Of The Dipole Calculus

In this section, we want to demonstrate with an example how spatial knowledge expressed in DRAfp can be used for deductive reasoning based on constraint