Algebraic curves defined by a single polynomial equation in two variables. e.g. electric motor y^2(y^2-9)-x^2(x^2-10);

Algebraic surfaces defined by a single polynomial equation in three variables. e.g. a Chubs surface x^4 + y^4 + z^4 - x^2 - y^2 - z^2 + 0.5;

Parameterised curves defined by a 3D vector expression in a single variable. e.g. a helix [cos(pi t), sin(pi t), t];

Parameterised surfaces defined by a 3D vector expression in two variables. e.g. a cross-cap [x,x y,y^2]

Intersection of surfaces with sets defined by another equation. For example the intersection of a conical surface with the set defined by a plane a x b y + cz =d.

This module can be used to calculate non-polynomial curves. For example a super ellipse pow(abs(x/a),p)+pow(abs(y/b),p)-1

Clipping, part of a surface inside a set define by an implicit equation, like the set inside a box min(min(min(xh-x,x-xl),min(yh-y,y-yl)),min(zh-z,z-zl)), or clipped by a sphere x^2+y^2+z^2-r^2

Mapping from R^3 to R^3 defined by 3D vector equation in three variables. e.g. a rotation [cos(pi th) x - sin(pi th) y,sin(pi th) x + cos(pi th) y,z];

Vector Fields, including unoriented vector field, and binary differential equations

Integral Curves. Uses the points in a geometry to define the starting points

Colourise: sets the colour of a surface depending on an expression. For example to colour by the z coordinate [(z+1), 0,(1-z)]; setting the red, green, and blue components for each point.

Extrude: produces surfaces of revolution and similar surfaces which depend on a curve and an equation. Can be used to produce families of curves.

Generalised Mappings where the equation depends on another surface. For example projection of a curve onto a surface. For example Gauss Map of a surface

    N / sqrt(N.N);   // Unit normal
    N = Sx ^^ Sy;    // calculate normal using cross product
    Sx = diff(S,x);  // derivatives of surface S
    Sy = diff(S,y);  // Definition of S read from the input surface

Generalised Intersections where the equation depends on the definition of another curve or surface. e.g. The profile of a surface, or parabolic lines

        // The profile of a surface
        N . [A,B,C];
        N = diff(S,x) ^^ diff(S,y);	

Generalised Clipping: e.g. the part of surface contained inside another already defined implicit surface

Generalised Colourise: colour by Gaussian or mean curvature

Generalised Extrude: e.g. tangent developable of a curve, or envelope of normals

        S + t T;            // Point on surface plus a multiple of unit tangent
        T = TT/sqrt(TT.TT); // unit length
        TT = diff(S,x);     // tangent to curve

Generalised Vector Fields: e.g. principle directions which are calculated using the definition of the input surface

Generalised Integrals Curves: e.g. principle curves of a surface calculated using the definition of the input surface

Specialised modules

BiIntersection Intersections where the equations depends on a pair of curves. For example the pre-symmetry set of a curve.

BiMap Mapping where the equation depends on a pair of curves. For example the Symmetry set.

Projective varieties: algebraic surfaces defined in real projective space, with options for stereographic projections and rotations in 4D