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Criteria are discussed for interpolation which preserves local convexity and local monotonicity. Particular schemes are then given for interpolation by parametric piecewise polynomials. The interpolating curves can have any desired order of geometric continuity and they are convex in a region where the data are convex. If the data arise from a function, then the curve is also single-valued, and in this case the curve can also preserve local monotonicity, if desired.
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Goodman et al. (1988) studied this question.
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