I wanted to ask the question "Can clothoids be made parallel with other clothoids" I have come across statements that this is not possible, but no proof. A previous poster "Jean Marie" states that:
"You may know that in general, parallel curves of non-polynomial curves have no explicit equation"
seems to corroborate this (clothoid is transcendental). I am using clothoids in the alignment of bridge decks. The offsets from the setting out line or "Master String" should be parallel to the Master String. I have gotten around this problem by generating coordinates along the offsets perpendicular to the clothoid and fitting a third order polynomial to the data using "Linest". This gives satisfactory results, however I would like to see a proof of "clothoids can not be made parallel to other clothoids" Can anyone supply this?
Let $$ s \mapsto C(s) $$ be a clothoid with $C(0) = (0,0), C'(0) = (1, 0)$, and $k_C(s) = s$, and $s$ is an arclength parameter, just to make things very explicit. Let $N(s)$ be the (oriented) unit normal to $C$ at $C(s)$, so that $N(0) = (0, 1)$. Let's look at the offset curve of $C$, along $N$, but distance $1$. That's $$ t \mapsto D(t) = C(t) + N(t) $$ where I've used $t$ to remind us that $D$ is not in general parameterized by arclength. Let's compute a few things. Letting $T(s) = C'(s)$ denote the (unit) tangent vector, and $R$ denote counterclockwise rotation by $\pi/2$, we have
\begin{align} T'(s) &= k_C(s) N(s) \\ N(s) &= R T(s) \\ R N(s) &= R^2 T(s) = -T(s) \\ N'(s) &= R T'(s) \\ N'(s) &= R (k_C(s)N(s)) = -k_C(s) T(s)\\ N''(s) &= -k_C'(s) T(s) - k_C(s) T'(s)\\ &= -T(s) - s^2 N(s) \end{align}
That should be enough to get us started. Let's compute $D'(t)$ and $D''(t)$ and see how those look. \begin{align} D'(t) &= C'(t) + N'(t) \\ &= T(t) -k_C(t) T(t)\\ &= (1 -t) T(t) \end{align} Actually, right there we have enough. For at $t = 1$, $D(t)$ reverses direction, hence cannot possibly be a clothoid (for the curvature there would have to be infinite).
To summarize: I've shown a (very simple) clothoid, and an offset curve from that clothoid that is not a clothoid. Hence in general, it's not true that every offset of a clothoid is itself a clothoid. It's possible that some offset of a clothoid is a clothoid -- for instance the zero offset! -- but in general, one cannot be confident that this will happen. Indeed, my suspicion is that a nonzero offset from a clothoid is never a clothoid, but that might take more work to prove (I think it probably follows with a few more lines from what I've already written down), and it's the answer to a different (and un-asked) question.