Implicitization using moving curves and surfaces
SIGGRAPH '95 Proceedings of the 22nd annual conference on Computer graphics and interactive techniques
The moving line ideal basis of planar rational curves
Computer Aided Geometric Design
The mu-basis of a rational ruled surface
Computer Aided Geometric Design
The µ-basis of a planar rational curve: properties and computation
Graphical Models
Revisiting the µ-basis of a rational ruled surface
Journal of Symbolic Computation
Implicitization and parametrization of quadratic surfaces with one simple base point
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
The μ-basis and implicitization of a rational parametric surface
Journal of Symbolic Computation
Implicitizing rational surfaces of revolution using µ-bases
Computer Aided Geometric Design
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A rational surfaceS(s,t)=(a(t)a^@?(s),a(t)b^@?(s),b(t)c^@?(s),c(t)c^@?(s)) can be generated from two orthogonal rational planar directrices: P(t)=(a(t),b(t),c(t)) in the xz-plane and P^@?(s)=(a^@?(s),b^@?(s),c^@?(s)) in the xy-plane. Moving a scaled copy of the curve P^@?(s) up and down along the z-axis with the size controlled by the curve P(t), we get the surface S(s,t). For example, when P^@?(s) is a circle with center at the origin, the surface S(s,t) is a surface of revolution. Many other useful and interesting surfaces whose cross sections are not circles can also be generated in this manner. We provide a new technique to implicitize this kind of rational surface using @m-bases. Let P(t) be a rational planar curve of degree n with a @m-basis consisting of two moving lines of degree @m and n-@m, and let P^@?(s) be a rational planar curve of degree m with a @m-basis consisting of two moving lines of degree @m^@? and m-@m^@?. From the @m-bases for these two directrix curves P(t), P^@?(s), we can easily generate a @m-basis for the surface S(s,t) consisting of three moving planes that follow the surface with generic bidegrees (m-1,@m), (m-1,n-@m), (m,0). To implicitize the surface S(s,t), we construct a (2m-1)nx(2m-1)n sparse resultant matrix R"s","t for these three polynomials. We show that det(R"s","t)=0 is the implicit equation of the surface S(s,t) with a known extraneous factor of degree (m-1)n. To decrease the size of this matrix and to eliminate entirely the extraneous factor, we construct a new mnxmn Sylvester style sparse matrix S"s","t from four moving planes that follow the surface S(s,t). We prove that det(S"s","t)=0 is the exact implicit equation of the surface S(s,t) without any extraneous factors. Examples are presented to illustrate our methods.