convert parametric equations to a normal one

asked 2017-10-04 00:53:59 -0500

lancaster gravatar image

updated 2017-10-05 10:08:42 -0500

I solved my problem into parametric equations as shown below:

x == 3/4*sqrt(2*m^2 + 2)*m/(m^2 + 1), y == 1/4*sqrt(2*m^2 + 2)/(m^2 + 1)

But I meet difficult on how to convert into a normal one, I want to get an equation with x, y only. For another general description: is there any function will help solve parametric equations into a normal one like solve: x == cos(theta), y == sin(theta) into x ^ 2 + y ^ 2 == 1. I learned how to solve it with my pen, but I want a solution with Sage code.


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Please give a clear sense to

But I meet difficult on how to convert into a normal one.

What should be converted into a "normal one"? Note that the (non-algebraic!) parametrization works for $$x^2+9y^2 = 9/8$$ and not for $x^2+y^2=1$:

sage: var('m');
sage: x = 3/4*sqrt(2*m^2 + 2)*m/(m^2 + 1)
sage: y = 1/4*sqrt(2*m^2 + 2)/(m^2 + 1)
sage: (x^2+9*y^2).canonicalize_radical()

Do we really need the $m$-parametrization for the further question with parameter $\theta$?

Note also that $\theta$ appears in the transcendental functions $x,y$, which live in a domain outside algebra. So the suggested question asks for finding an algebraic relation between non-algebraic objects?

dan_fulea gravatar imagedan_fulea ( 2017-10-04 13:28:02 -0500 )edit

I have edited my question, hope it is clearer now. I mean "normal" to be something like a Conic Curve equation like ellipse is x^2/a^2 + y^2/b^2 = 1

lancaster gravatar imagelancaster ( 2017-10-05 10:20:34 -0500 )edit