curve is of degree $d$ and circularity $c$, the bounds for links and joints are $3d-4c+2$ and $\frac{9}{2}d-6c+1$, respectively.

Another advantage of our spatial approach concerns a certain defect in Kempe’s original construction that later was even considered a flaw [@demaine07 Section 3.2]. Kempe used parallelogram and anti-parallelogram linkages as basic building blocks of his linkages. It is well known that the configuration curves of these linkages consists of two irreducible algebraic components that intersect at flat folded positions. That is, the linkage may switch between parallelogram and anti-parallelogram mode, thus entering unwanted components of the configuration curve. The effect is that Kempe’s linkages draw more than the originally intended curve. At the cost of introducing additional links and joints, this defect can be overcome by the “bracing constructions” of [@kapovich02; @demaine07]. The approach of [@gallet15] uses anti-parallelograms and is therefore subject to the same defect and its resolution.

The basic building blocks of our approach are Bennett linkages whose configuration curve has only one irreducible component. Thus, no additional links and joints are needed to prevent the linkage from switching modes. We also capture this in the phrase “the configuration curve is free of spurious components”.

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