A Polynomial Solution for Potato Peeling Problem
A Polynomial Solution for Potato Peeling Problem
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The same argument in V-Lcmma applies here: perturbing Cj toward the balance position (this is possible since C, - is not extremal), while keeping all other chords unchanged, increases the area of the series and therefore violates the local optimality. A Lemma 4. Let Q be a maximal convex polygon in P bounded by the sequence (Cq, C I, . . . . C„) of chords. Then Q contains at least two extremal chords of P. Furthermore, there exists a maximal convex polygon with three or more extremal chords.
Pr...oof. To show the first part, assume to the contrary that Q has or 1 extremal chord. Note that (C2, . . . , C„) forms a (Cg, Ci)-chain. Without loss of generality, assume that this chain contains no extremal chords. Clearly (C2, . . . , C„) is a V-chain, and by the V-Lemma, it is not optimal. For the second part, note that if there are only' two extremal chords, say Cq and Cj^, then they must be parallel (otherwise the V-Lemma shows that either the (Cq, Cjt)-chain or the (C^, Co)-chain is non-optimal).
What to read after A Polynomial Solution for Potato Peeling Problem?
You can find similar books in the "Read Also" column, or choose other free books by J S Chang to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
The same argument in V-Lcmma applies here: perturbing Cj toward the balance position (this is possible since C, - is not extremal), while keeping all other chords unchanged, increases the area of the series and therefore violates the local optimality. A Lemma 4. Let Q be a maximal convex polygon in P bounded by the sequence (Cq, C I, . . . . C„) of chords. Then Q contains at least two extremal chords of P. Furthermore, there exists a maximal convex polygon with three or more extremal chords.
Pr...oof. To show the first part, assume to the contrary that Q has or 1 extremal chord. Note that (C2, . . . , C„) forms a (Cg, Ci)-chain. Without loss of generality, assume that this chain contains no extremal chords. Clearly (C2, . . . , C„) is a V-chain, and by the V-Lemma, it is not optimal. For the second part, note that if there are only' two extremal chords, say Cq and Cj^, then they must be parallel (otherwise the V-Lemma shows that either the (Cq, Cjt)-chain or the (C^, Co)-chain is non-optimal).
What to read after A Polynomial Solution for Potato Peeling Problem?
You can find similar books in the "Read Also" column, or choose other free books by J S Chang to read onlineMoreLess
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