The parametric curve $(x,y) = (\sin t, \sin 2t)$ is plotted below.

[asy]
unitsize(2 cm);

pair moo (real t) {
  return (sin(t),sin(2*t));
}

real t;
path foo = moo(0);

for (t = 0; t <= 2*pi + 0.1; t = t + 0.1) {
  foo = foo--moo(t);
}

draw((-1.2,0)--(1.2,0));
draw((0,-1.2)--(0,1.2));
draw(foo,red);
[/asy]

This curve can also be expressed in the form
\[ax^4 - bx^2 + cy^2 = 0,\]where $a,$ $b,$ and $c$ are positive integers.  Find the smallest possible value of $a + b + c.$
If $x = \sin t$ and $y = \sin 2t = 2 \sin t \cos t$ then
\begin{align*}
y^2 &= (2 \sin t \cos t)^2 \\
&= 4 \sin^2 t \cos^2 t \\
&= 4x^2 (1 - x^2) \\
&= 4x^2 - 4x^4.
\end{align*}Thus,
\[4x^4 - 4x^2 + y^2 = 0,\]so the smallest possible value of $a + b + c$ is $4 + 4 + 1 = \boxed{9}.$