# How to convert a system of parametric equations?

I have a system of parametric equations (t is a parameter)

    z = ((1 - t)^2 (1 + 4 t - 2 t^2))/(2 (1 - 2 t + 2 t^2))
y = ((-1 + t) Sqrt[1 - t^2] (1 - 3 t + t^2))/(1 - 2 t + 2 t^2)
{t, 0.7071, 1}


I am going to find the normal equation of z=f(y) without parameter t. How can I do this in Mathematica? As I told before, I found only y=f(z) as analytical expression like:

    Y = ((Sqrt[3] a1 - 6 a2) Sqrt[-a1^2 + 4 Sqrt[3] a1 (2 + a2) -
12 a2 (4 + a2)] (-a1^2 + 4 Sqrt[3] a1 (-1 + a2) -
12 (-4 - 2 a2 + a2^2)))/(
96 Sqrt[3] (a1^2 - 4 Sqrt[3] a1 (1 + a2) + 12 (2 + 2 a2 + a2^2))),


where :

    a1 = Sqrt[(2^(2/3) a^2 - 4 a (-3 + 4 Z) + 2 2^(1/3) (3 + 4 Z)^2)/a],
a2 = Sqrt[
2 - a/(6 2^(1/3)) - (8 Z)/3 + (8 Sqrt[3] Z)/a1 - (3 + 4 Z)^2/(
3 2^(2/3) a)],
a = (-54 + Sqrt[k] + 1080 Z - 576 Z^2 + 128 Z^3)^(1/3),
k = -139968 Z + 1150848 Z^2 - 1396224 Z^3 + 470016 Z^4 -221184 Z^5


but for my task I need z=f(y).

• Have you looked at Eliminate[]? May 19, 2015 at 15:43
• yes, but there is no elimination. May 19, 2015 at 15:46
• What do you mean? Is there no result returned? May 19, 2015 at 15:49
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• Please show your code in the Question. May 19, 2015 at 15:50

## 1 Answer

Here is the implicit equation satisfied by (y,z).

ratpolys = {z - (1 - t)^2 (-2 t^2 + 4 t +
1)/(2 (2 t^2 - 2 t + 1)),
y - (t - 1) *Sqrt[1 - t^2]*(t^2 - 3 t + 1)/(2 t^2 - 2 t + 1)};
gb = First[
GroebnerBasis[ratpolys, {y, z}, t,
MonomialOrder -> EliminationOrder]]

(* Out[93]= -135 y^4 - 1728 y^6 + 1024 y^8 - 720 y^4 z + 1536 y^6 z -
360 y^2 z^2 - 4768 y^4 z^2 + 4096 y^6 z^2 + 160 z^3 - 112 y^2 z^3 +
3840 y^4 z^3 + 720 z^4 - 6176 y^2 z^4 + 6400 y^4 z^4 - 320 z^5 +
3584 y^2 z^5 - 2880 z^6 + 4608 y^2 z^6 + 1280 z^7 + 1280 z^8 *)


One can use Solve to get branches of algebraic roots for either variable in terms of the other.

• Ah, this was what I was going to suggest to the OP after verifying that Eliminate[] did not work for his example… May 20, 2015 at 0:42
• ...but Solve cant get algebraic roots z=f(y): Solve[-135 y^4 - 1728 y^6 + 1024 y^8 - 720 y^4 z + 1536 y^6 z - 360 y^2 z^2 - 4768 y^4 z^2 + 4096 y^6 z^2 + 160 z^3 - 112 y^2 z^3 + 3840 y^4 z^3 + 720 z^4 - 6176 y^2 z^4 + 6400 y^4 z^4 - 320 z^5 + 3584 y^2 z^5 - 2880 z^6 + 4608 y^2 z^6 + 1280 z^7 + 1280 z^8 == 0, z]. Jun 16, 2015 at 7:58
• The result, in terms of Root objects, is comprised of algebraic roots. That's by definition of what an algebraic root is. They are not in terms of explicit radicals but that's a different matter. Jun 17, 2015 at 18:04