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Hartmann. Der Aufbau der Realen Welt. de Gruyter, Berlin, 1964. 16. J. J. Koenderink and A. J. van Doorn. Relief: Pictorial and otherwise. Image and Vision Computing, 13:321{334, 1995. 17. K. Lorenz. Behind the Mirror: A Search for a Natural History of Human Knowledge. Harcourt Brace Jovanovich, New York, 1977. 18. H. Ouchi. Japanese and Geometrical Art. Dover, 1977. 19. M. Polanyi. Personal Knowledge: Towards a Post-Critical Philosophy. The University of Chicago Press, Chicago, 1958. 20. S. M.

Zeki. Phenomenal motion seen through arti cial intra-ocular lenses. Proc. Royal Society, London B, 260:165{166, 1995. nl Abstract. The sense of touch and the capability to analyze potential contacts is important to many interactions of robots, such as planning exploration, handling objects, or avoiding collisions based on sensing of the environment. It is a pleasant surprise that the mathematics of touching and contact can be developed along the same algebraic lines as that of linear systems theory.

So this important operation is part of both systems theories. In linear systems, translation over t is represented by convolution with the shifted delta-function δt (x) ≡ δ(x − t), of which the Fourier transform is e−iωt : F [δt ](ω) = du δ(u − t)e−iωu = dv δ(v)e−iωv × eiωt = e−iωt . This gives ft (x) ≡ f (x − t) F [·] F −1 F [ft ](ω) = F [f ](ω) × e−iωt [·] In contact systems, translation over t is represented as tangential dilation by the shifted delta-function δt (x) ≡ δ(x − t), of which the Legendre transform is −ωt: L[δt ](ω) = statu [δ(u − t) − ω u] = statv [δ(v) − ω v] − ωt = −ωt This gives ft (x) ≡ f (x − t) L[·] L−1 [·] L[ft ](ω) = L[f ](ω) − ωt Note that the computations in both frameworks run completely analogously.

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