2021-02-27 09:08:38 +0100 received badge ● Popular Question (source) 2019-03-23 02:12:35 +0100 asked a question 3D line from equations to parametric In 3D space, given a line defined as the solution of two equations (two planes intersection) like in: sage: x,y,z=var('x y z') sage: eqns = [x + y + 2*z - 25 == 0, -x + y - 25 == 0]  how to obtain the direction vector and one (any) line point (parametric form) ? This solution from solve: sage: solve( eqns, [x,y,z] ) [[x == -r13, y == -r13 + 25, z == r13]]  has an answer in parametric form, but with parameter "r13" that has a name unpredictable and not usable in next steps. This solution from solve: sage: solve( eqns, [y,z] ) [[y == x + 25, z == -x]]  solves the issues of the previous, but it has been assumed that "x" is a valid parameter for the line ( something not true, by example, in case of vertical line: [ x==10 , y==2 ] ) The target is, by example, to obtain a parametric expression of any line that after can be used in a call to "parametric_plot3d".