Rational Bézier curves are a fundamental tool in computer-aided geometric design, yet their standard representation via control points and weights only allows indirect shape control, since the curve cannot easily be forced to pass through a prescribed point. This work presents an alternative, barycentric representation of a rational curve, expressed through nodes, interpolation points, and weights, which exactly interpolates the curve at every interpolation point. We derive explicit conversions between the Bézier and barycentric forms in both directions, and show how a curve can be brought into standard form through a suitable linear rational reparametrization. We further analyze three local design mechanisms available in the barycentric form: moving an interpolation point, changing a node while simultaneously preserving the shape and parametrization of the curve, and modifying a weight, for which we also provide a precise nonsingularity criterion for the modified curve. We additionally describe a degree elevation procedure that adds a new interpolation point. All derived results are illustrated by numerical examples and figures, confirming that the barycentric form enables precise, local shaping of complex curves in ways that are not directly possible with classical Bézier representation.
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