> <\body> > We describe the type of curves and surfaces provided in the plugin > of the package for . > A is defined as the set of points <\equation*> \=|f>, \, |f>|)>\\, t\ U|}> where is the dimension of the ambient space and , \, f> are univariate polynomials (with coefficients in >). If =1>, the rational curve is also called a . \; Bézier curves are special types of rational curves, which use Bernstein bases to represent the polynomial functions. \ A Bézier parametric curve is the image of a map : *\ \> of the form <\equation*> s\*\>\ B> where \\> are the control points, and = >, d|)>> are the Bernstein basis elements of degree for the interval >. A rational \ Bézier curve is constructed as follows: <\equation*> s\*\>w \ B> \ |>w B> > where \\> are the of the control points (usually \0>). <\cpp-code> \curve type="rational" "0 255 0"\ \ \ \domain\0 1\/domain\ \ \ \polynomial\t\/polynomial\ \ \ \polynomial\t^2\/polynomial\ \ \ \polynomial\t^3\/polynomial\ \ \ polynomial\1\/polynomial\> \/curve\ \; > A rational curve is defined as the image of a map : U\\ \ \> of the form <\equation*> \=|f>, \, |f>|)>, \U|}> where is the dimension of the ambient space and , \, f> are bivariate polynomials (with coefficient in >). <\cpp-code> \surface type="rational" "0 255 0"\ \ \domain\0 0 1 0 0 1\/domain\ \ \ \polynomial\s\/polynomial\ \ \ \polynomial\t\/polynomial\ \ \ \polynomial\s^2+t^2\/polynomial\ \ \ polynomial\1\/polynomial\> \/surface\ An algebraic curve in > is defined \ by one equation of the form: <\equation*> f=0 where > is a polynomial with coefficieints in >.\ Here is an image of the curve defined by the equation <\eqnarray*> +x-7 xy+21 xy-35 xy+35 xy-21 xy+7 x y-y>||>|- 7 x+35 \ xy -70 xy+ 70 x y-35 x y + 7y>||>|+14 x-42 xy +42 x y-14 y+16 y-7 x - 7y-2>||>>> <\with|par-mode|center> <\cpp-code> \curve "algebraic" "nice_curve_10"\ \ \ \domain\-4.1 4.2 -3.1 3\/domain\ \ \ \polynomial\-y^8+x^7-7*x^6*y+21*x^5*y^2-35*x^4*y^3+35*x^3*y^4 -21*x^2*y^5+7*x*y^6-y^7+8*y^6-7*x^5+35*x^4*y-70*x^3*y^2+70*x^2*y^3 -35*x*y^4+7*y^5-20*y^4+14*x^3-42*x^2*y+42*x*y^2-14*y^3+16*y^2 -7*x+7*y-2\/polynomial\ \/curve\ An algebraic curve in > is defined \ by two of more equations of the form: <\equation*> \=\\;f=0, f=0, \|}> The solution set of these equations is of dimension . In other words, the tangent linear space at almost all solution points \> is a line. Here we see two surfaces intersecting in a blue curve: \; <\with|par-mode|center> |algebraic_curve3d.axl> <\cpp-code> \curve >"algebraic" "0 0 255"\ \ \ \domain\-3 3.1 -3 3.1 -3 3.1\/domain\ \ \ \polynomial\10000*x^4+10000*y^4+10000*z^4-40000*x^2-40000*y^2*z^2-40000*y^2 -40000*z^2*x^2-40000*z^2-40000*x^2*y^2+207846*x*y*z+10000\/polynomial\ \ \ \polynomial\40000*x^3-80000*x-80000*x*z^2-80000*x*y^2+207846*y*z\/polynomial\ \/curve\ A general is defined as the solution set of one equation: <\equation*> \ = \U\\; f=0|}> where is a subdomain of >. Usually is continous and enough differentiable so\ An algebraic surface is a special case of an implicit surface where > is a polynomial with coefficients in >. \; Here is an example of a surface defined by the equation: <\equation*> x y- z =0 which is known as the . <\with|par-mode|center> |algebraic_surface_whitney.axl> <\cpp-code> \surface "algebraic" "whitney umbrella" "0 170 255" \ \ \ \domain\-3 3 -3 3 -3 3\/domain\ \ \ \polynomial \x^2*y-z^2\/polynomial\ \/surface\ <\itemize-dot> Topology and arrangement of algebraic curves Topology of surfaces Intersection of parametric surfaces Auto-intersection of parametric surfaces <\initial> <\collection>