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:
$$ x_d(t) = t + \frac{d \cos (t)}{\sqrt {1 + \cos (t)^2}} $$
$$ y_d(t) = \sin (t) - \frac{d}{\sqrt {1 + \cos (t)^2}} $$
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: Can Mathematica find 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/3-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.
http://postimg.org/image/un06qseiv/
http://s10.postimg.org/jn8jvl095/braid_colored_in.jpg
Thank you in advance for any help.
ParametricPlot
to graph the curve for $d = 1$, for example.) $\endgroup$