Like before, the equations of the planes are System of integral equations Unfortunately, the above is not very useful in the computational model described “naive” derivation performed earlier. The problem is that after of the growth parameters all planes need to be moved so that each change they are tangent to the polyhedron (t ) (a plane is left invariant by this transformation if it already produces a P face of (t )). And if the parameters are changing continuously, this turns the system of (at most) 50 P independent algebraic equations into a system of 50 intertwined where the “integro-differential procedures”, calculation of (t ) requires the knowledge of its history over some past time interval (t ,t ). P − δt More precisely, for the polyhedron (t), define (P (t )) as a vertex of (t ) that is closest to the plane (t ) P wn P πn (and thus lies in that plane). Then the instantaneous velocity of this vertex is completely determined by the three faces it belongs to. Let us call the normal vectors of these faces = (nix , , = 1, 2, 3 (keep in ni niy niz), i mind the dependence of these on Note that, in general, the set of vectors for a given plane changes n). ni πn with time. Then the velocity vector of the point , which depends on time through the speeds (t wn explicitly Vni ) as well as through the shape of (t ), is the unique solution of the simple linear system implicitly P
Like before, the equations of the planes are System of integral equations Unfortunately, the “naive” derivation performed above is not very useful in the computational model described earlier. The problem is that after each change of the growth parameters all planes need to be moved so that they are tangent to the polyhedron P (t ) (a plane is left invariant by this transformation if it already produces a face of P (t )). And if the parameters are changing continuously, this turns the system of (at most) 50 independent algebraic equations into a system of 50 intertwined “integro-differential procedures”, where the calculation of P (t ) requires the knowledge of its history over some past time interval (t − δt ,t ). More precisely, for the polyhedron P (t), define wn (P (t )) as a vertex of P (t ) that is closest to the plane πn (t ) (and thus lies in that plane). Then the instantaneous velocity of this vertex is completely determined by the three faces it belongs to. Let us call the normal vectors of these faces ni = (nix , niy , niz), i = 1, 2, 3 (keep in mind the dependence of these on n). Note that, in general, the set of vectors ni for a given plane πn changes with time. Then the velocity vector of the point wn , which depends on time explicitly through the speeds Vni (t ) as well as implicitly through the shape of P (t ), is the unique solution of the simple linear system