Envelope computation in the plane by approximate implicitization

Given a rational family of planar rational curves in a certain region of interest, we are interested in computing an implicit representation of the envelope. The points of the envelope correspond to the zero set of a function (which represents the envelope condition) in the parameter space combining the curve parameter and the motion parameter. We analyze the connection of this function to the implicit equation of the envelope. This connection enables us to use approximate implicitization for computing the (exact or approximate) implicit representation of the envelope. Based on these results, we formulate an algorithm for computing a piecewise algebraic approximation of low degree and illustrate its performance by several examples.