# CASA Function: toPara

Converts an algebraic set given in implicit or projected representation to an algebraic set in parametric representation.

### Calling Sequence:

• B := toPara(A)
• B := toPara(A,var)
• B := toPara(A,var,options)

### Parameters:

A : {algset("impl"),algset("proj"),algset("para")}
• Algebraic set in impl para or projected representation.
var : name
• A variable name that is used for the parametrization.
options : list
• A list of options. Where an option is either one of the strings "optimal" (compute a parametrization over an optimal extension field), "realpar" (compute a real parametrization), "check" (plug in the resulting parametrization in the given equations), "points"= <list of simple points on the curve> (a list of points on the curve that could be helpful for the parametrization. The points which are not on the curve will be automatically removed.
• Note, the user should know that supplying simple points may not be a good idea for a non-expert since it may lead to a non-optimal parametrization.

### Result:

B : algset("para")
• A parametric representation of the algebraic set A.

### Description:

• This function takes an algebraic set and converts it into a parametric representation in affine space. Currently, it may be applied only to one-dimensional algebraic sets (i.e. both plane and space algebraic curves).
• The variables vars will be taken for the parametrization. If no variables are given, the function tries to take the variables from the given algebraic set.

### Examples:

> A:=mkImplAlgSet([y^2*z^3-x^5],[x,y,z],["basespace"="projective"]); > toPara(A); > A:=mkImplAlgSet([y^4-10*x*y^3+35*x^2*y^2-50*x^3*y+24*x^4+x^3],[x,y]); > toPara(A);        > toPara(A,t,["optimal"]);  > A:=mkImplAlgSet([x^2-y^3,z+y+x],[x,y,z]); > toPara(A); > A := mkProjAlgSet([[u^2-v^3+v^2],[u+v,u-v,1/v]],[u,v]); > toPara(A); 