By R Cooke, K L. Lockett, J A Bellman
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The key idea is that, at any time, the turtle is in a face of the cube which one can identify with a ﬁxed square. The parallel projection then maps this square onto a parallelogram. 9(b) where we map the square A with vertices a, b, c, and d onto the parallelogram A¢ with vertices a¢, b¢, c¢, and d¢, respectively. If p is an arbitrary point of A, write p in the form a + s ab + t ad. The parallel projection will then map p to a¢ + s a¢b¢ + t a¢d¢. Therefore, the basic steps are: (1) Pick points ai in the plane onto which the vertices of the cube (one may as well use the standard unit cube [0,1] ¥ [0,1] ¥ [0,1]) get mapped.
1 implements the algorithm we have been describing. In our discussion above we have restricted ourselves to lines that start at the origin and end in the ﬁrst octant. Starting at another point simply amounts to adding a constant offset to all the points. Lines that end in a different octant can be handled in a similar way to the ﬁrst octant case – basically by interchanging the x and y. What this boils down to is that an algorithm which handles all lines is not much harder, involving only a case statement to separate between the case where the absolute value of the slope is either larger or less than or equal to 1.
Pn), q = (q1,q2, . . ,qn) Œ Zn. The points p and q are 2n-adjacent in Zn if and only if n Â q i - pi =1 i =1 They are (3n - 1)-adjacent in Zn if and only if p π q and |qi - pi| £ 1 for 1 £ i £ n. Properties of 2n- and (3n - 1)-adjacency are studied extensively in [Herm98]. Deﬁnition. A (discrete or digital) curve from point p to point q in Zn is a sequence rs, 1 £ s £ k, of points such that p = r1, q = rk, and rs is adjacent to rs+1, 1 £ s £ k - 1. Furthermore, with this notation, we deﬁne the length of the curve to be k - 1.