Automatic parameterization of rational curves and surfaces IV: algebraic space curves

For an irreducible algebraic space curve <italic>C</italic> that is implicitly defined as the intersection of two algebraic surfaces, <italic>f</italic> (<italic>x</italic>, <italic>y</italic>, <italic>z</italic>) = 0 and <italic>g</italic> (<italic>x</italic>, <italic>y</italic>, <italic>z</italic>) = 0, there always exists a birational correspondence between the points of <italic>C</italic> and the points of an irreducible plane curve <italic>P</italic>, whose genus is the same as that of <italic>C</italic>. Thus <italic>C</italic> is rational if the genus of <italic>P</italic> is zero. Given an irreducible space curve <italic>C</italic> = (<italic>f</italic> ∩ <italic>g</italic>), with <italic>f</italic> and <italic>g</italic> not tangent along <italic>C</italic>, we present a method of obtaining a projected irreducible plane curve <italic>P</italic> together with birational maps between the points of <italic>P</italic> and <italic>C</italic>. Together with [4], this method yields an algorithm to compute the genus of <italic>C</italic>, and if the genus is zero, the rational parametric equations for <italic>C</italic>. As a biproduct, this method also yields the implicit and parametric equations of a rational surface <italic>S</italic> containing the space curve <italic>C</italic>. The birational mappings of implicitly defined space curves find numerous applications in geometric modeling and computer graphics since they provide an efficient way of manipulating curves in space by processing curves in the plane. Additionally, having rational surfaces containing <italic>C</italic> yields a simple way of generating related families of rational space curves.

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