The difference between an affine k-simplex and a rectilinear k-simplex

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The notion of rectilinear k-simplex appears in Theorem 10.27 of Rudin's book "Principles of Mathematical analysis", then what is the definition of a rectilinear k-simplex? I read the proof of Theorem 10.27 and think that the proof treats oriented affine k-simplex instead of rectilinear k-simplex, is it right?