Parallel curve to a sine wave I've been trying to find the formula for the offset/parallel to a sine wave.  Not just the parametric equation, but the y = f(x) form.
Here's what I've done so far:
Read up on the parametric form and plugged in the x(t) and y(t) formulas. What I get is of course a parametric equation in terms of t.
If $$ y =  \sin x $$
then the parameterization would be
$$ x = t $$
$$ y = \sin  t $$
Plugging in the offset formula:
$$ x_d(t)  = t + \frac{d\cdot\cos t}{\sqrt{1 + \cos^2 t}} $$
$$ y_d(t)  = \sin t  - \frac{d}{\sqrt {1 + \cos^2 t}} $$
Now, that's all accurate, but it doesn't put it into a function form.  According to my calculus book, the next step is to solve each of these for t and then set them equal to one another.  The problem is that they are kind of a mess, with those sinusoidal functions involved.
My question is:  What's the y =  f(x) form for an offset curve of a sine wave?
A little background:  I need this because I'm trying to find the intersection point when 3 offsets of three $\pi\over3$-out-of-phase to each other sine waves intersect.  Basically where the green, blue and red intersect at the same time in the link below.  I can find it numerically, but I'd like it exactly because it's something of discovery to find out how the ancient people drew braids using just compass and straight edges.
I can draw it no problem in C#:
The intersection point was found using trial and error and is approximately 0.63. There are two blue lines, two red lines and two green lines, because I used +0.63 offset and -0.63 offset from the sine wave.

Thank you in advance for any help.
 A: visualizing things in WebGl can help: 
https://www.shadertoy.com/view/XsByzd
shows the assumption that the 2 points, p.xy, and the point on sin(x) that is closest to p.xy, are both on a line that is is 90° to y=cos(x);
you are not alone, finding the euclidean distance (or a good upper bound) of a point to any integral or surface is a core problem of procedural textures and raymarching (=sphere tracking), the iterative approach to raytracing, finding the intersection of a ray with anything (analytically).
there are some helpful identities for inequalities, useful for good upper bounds. like, knowing that: something => distance => somethingElse.
A: You cannot solve for $t$ from second equation and plug it into the first ( or the other way, first into the second).  So $y=f(x)$ form is not possible.
As it is parametric form is fine. If the base curve is not closed form like the sine curve but available in a parametrized form then a differential equation of a parallel curve can be defined using the normal offset direction.Entire thing can be numerically integrated. And, as it is here only in a plot we can see it composed together.
Maybe the ancients appreciated art more than accuracy not available then.
The lines are not strictly parallel (In 3-space called Bertrand surfaces) . Braids can be filled with 3 colors, two colors overlap, first orange one (needles) remain so.

A: If $\gamma(t) = \bigl(u(t), v(t)\bigr)$ is a parametrization of a plane curve ($t$ in some interval of real numbers), and if the component function $u$ is one-to-one (i.e., does not achieve the same value at two distinct $t$, so that $\gamma$ traces the graph of some function), then $(x, y) = \gamma(t)$ satisfies $y = f(x)$ is and only if $v(t) = f\bigl(u(t)\bigr)$ for all $t$ in $I$, in and only if
$$
f = v \circ u^{-1}.
$$
In words, an explicit formula $y = f(x)$ for a graph traced by a parametric curve relies implicitly on one's ability to invert the component function $u$ (i.e., to solve for the parameter), and in practice "usually" depends explicitly on being able to invert $u$.
