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Unformatted text preview: ints can be written as
N h
i=1 1 + ((yi+1 − yi )/h)2 ≤ L, y1 = 0, yN +1 = 0. In addition to these constraints, we will also require that our curve passes through a set of preﬁxed
speciﬁed points. Let F ⊆ {1, . . . , N + 1} be an index set. For j ∈ F , we require yj = yj , where
ﬁxed ∈ RN +1 (the entries of y ﬁxed whose indices are not in F can be ignored). Finally, we add a
y
constraint on maximum curvature,
−C ≤ (yi+2 − 2yi+1 + yi )/h2 ≤ C, i = 1, . . . , N − 1. Explain how to ﬁnd the curve, i.e., y1 , . . . , yN +1 , that maximizes the area enclosed subject to these
constraints, using convex optimization. Carry out your method on the problem instance with data
given in iso_perim_data.m. Report the optimal area enclosed, and use the commented out code
in the data ﬁle to plot your curve.
Remark (for your amusement only). The isoperimetric problem is an ancient problem in mathematics with a history dating all the way back to the tragedy of queen Dido and the founding of
Carthage. The story (which is mainly the account of the poet Virgil in his epic volume Aeneid ),
goes that Dido was a princess forced to ﬂee her home after her b...
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 Fall '13
 F.Borrelli
 The Aeneid

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