Mathematica Asked on March 6, 2021
Given (assuming $0 leq t leq pi/2$)
x == (1 + a Sin[2 t]) Sin[t],
and
y == 1 - (1 + a Sin[2 t]) Cos[t]
How can I get $y(x)$ without the $t$ variable appearing?
Solving for, and eliminating, $t$ directly proved very awkward and this didn’t work:
Assuming[0 < t < [Pi]/2,
Eliminate[{x == (1 + a Sin[2 t]) Sin[t],
y == 1 - (1 + a Sin[2 t]) Cos[t] }, t]]
The usual procedure for obtaining a Cartesian equation from parametric equations like yours is to use GroebnerBasis[]
for eliminating variables:
First[GroebnerBasis[Append[TrigExpand[{x == (1 + a Sin[2 t]) Sin[t],
y == 1 - (1 + a Sin[2 t]) Cos[t]}],
Cos[t]^2 + Sin[t]^2 == 1],
{x, y, a}, {Cos[t], Sin[t]}]]
-4 a x + x^2 - 4 a^2 x^2 - 4 a x^3 + 2 x^4 + x^6 - 2 y + 12 a x y - 8 x^2 y +
8 a^2 x^2 y + 4 a x^3 y - 6 x^4 y + 9 y^2 - 12 a x y^2 + 16 x^2 y^2 -
4 a^2 x^2 y^2 + 3 x^4 y^2 - 16 y^3 + 4 a x y^3 - 12 x^2 y^3 + 14 y^4 +
3 x^2 y^4 - 6 y^5 + y^6
The result obtained is a degree-$6$ polynomial, so I am not very sanguine about solving for y
in this case.
Answered by J. M.'s ennui on March 6, 2021
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