Multiple layers and linear forms

It is not appropriate to use the surface-plot images, as shown above, to represent multiple layers, because the resulting images would be too complicated. Instead, we generate 2D ``interference diagrams'' similar to those of Ramsay (1967, page 531), Ramsay and Huber (1987, page 496), Thiessen and Means (1980), Thiessen (1986), and Perrin et al. (1988).

Mathematically, we use the concept of ``linear form'', as well as inverses to the functions defined previously. A single horizontal layer in 3-D space is given by the set of points with coordinates $(x,y,z)$ for which $z=0$. Parallel planes are those for which $z=z_0^{}$, for some particular constant value $z_0^{}$. Slanted planes are given by solving equations such as $z+.5x = constant$, for planes parallel to that in the lefthand image of Fig. 18, or $z+.4x -.4y = constant$ for planes parallel to that in Fig. 19. Different parallel layers correspond to different values of the constant. For a point with coordinates $(x,y,z)$, the layer on which it lies is given by the value of the linear form; e.g. $z$ or $z+.5x$ or $z+.4x -.4y$ in these examples.

Under a refolding deformation, layers are no longer flat. However, a sequence of folding (or shearing, or rotational) deformations can be undone in reverse sequence, to determine the flat layer on which a point originally lay. Equivalently, a function can be defined to give these values directly. For example, this ``layer function'' for the refoldings of Figs. 15, 18, and 19 are given respectively by the following Mathematica expressions, involving the inner product of vectors.

layer1[x_,y_,z_]:=({x,y,z}//axiZX[-.5,1.3]//axiZY[-.7,1.5]).{0,0,1};
layer2[x_,y_,z_]:=({x,y,z}//axiXZ[-.5,1.3]//axiZY[-1,1.6]).{.5,0,1};
layer3[x_,y_,z_]:=({x,y,z}//axiYZ[-.4]//axiZY[-1,1.5]).{.4,-.4,1};

Side remark:Inverses for the simple deformation functions $\mathbf{axiZY}_{[u,v]}$, etc. are obtained by simply negating the value of $u$, to get $\mathbf{axiZY}_{[{-}u,v]}$. Similarly for the shearing functions, the inverse of $\mathbf{shrZY}[u]$ is $\mathbf{shrZY}_{[{-}u]}$. For rotations, the inverse of $\mathbf{rot}_{[a,b,c]}$ is $\mathbf{rot}_{[{-}c,{-}b,{-}a]}$.