# Variable Transformations of Functions in the Euclidean Plane

Let $A$ be an everywhere differentiable function in $\mathbf{R}$ whose graph is a subset of $\mathbf{R^2}$. Let $M$ also be an everywhere differentiable function in $\mathbf{R}$ whose graph is a subset of $\mathbf{R^2}$. If $A(x):=sin(x)$ and $M(x):=x^2$, and there also exists a relation $R$ such that from every point $(a,b)$ on $R$ there exists a line segment, connecting $(a,b)$ to at least one point $(c,d)$ on $A$, which is normal to $A$ at $(c,d)$ and has length $M(c)=c^2$. Is $R$ then everywhere differentiable, and furthermore, what is the derivative of $R$ at $(a,b)$?

My thinking is this: If one were to represent $A(x)$ and $M(x)$ parametrically, then it would be possible to manipulate $x$ and $y$ independently. Let

$$x=t,\qquad y=\sin(t)$$

Then it follows that $R$, represented parametrically, is

\begin{align} x'&=t - t^2/\sqrt{1 + 1/(\cos(t))^2} \\ y'&=\sin(t)+t^2/\sqrt{1 + (\cos(t))^2}\end{align}

when the appropriate transformations are applied. These transformations are achieved by resolving the perpendicular line from $(a,b)$ to $(c,d)$ to the $x$ and $y$ axes so that then the necessary transformations become readily apparent. Therefore, $dR/dx=(dy'/dt)/(dx'/dt)$. It would also follow that $R$ is not everywhere differentiable. Is there another protocol which would achieve the same results or even different results?

EDIT: Thanks to Willie Wong I have been made aware that "when M(x) is less than the radius of curvature for the graph of A at (x,A(x)) , you can guarantee that the corresponding R is differentiable." I am curious now as to if a general theory of simplifying analysis of R by breaking R down into its respective A and M functions has been explored? Or more generally in n dimensions?

EDIT: For instance, in clarification of my original edit, in n dimensions the normal lines of magnitude M would be normal to the tangential hypersurface at (a,a,) in A. And B would be the resulting relation after this transformation. Is there a general theory in this regard, or has extensive research been done in this field? The purpose of this question is to know if this is a viable field of research and what research papers in this field, if they exist, could you point me to?

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Yes, thank you very much. That was a mistake on my part. What do you think of my general process? –  analysisj Aug 24 '11 at 0:24
Ultimately, I am interested in generalizing this process to n-space as to allow for simplification of analysis in dealing with R relations. –  analysisj Aug 24 '11 at 0:26
Generally, when $M(x)$ is less than the radius of curvature for the graph of $A$ at $(x,A(x))$, you can guarantee that the corresponding $R$ is differentiable. Otherwise you can pick up self-intersections and cusps, if $R$ lies on the "concave side" of $A$. –  Willie Wong Aug 24 '11 at 0:29
Also posted to MathOverflow, mathoverflow.net/questions/73530/… –  Gerry Myerson Aug 24 '11 at 0:32
Thank you very much, has this been explored in simplifying analysis of R by breaking R down into its respective A and M functions? Or more generally in n dimensions? Two additional questions: 1) Is this worth exploring or is it treaded territory and 2) Is there a theorem that you can point me to about your previous comment? Thank you very much –  analysisj Aug 24 '11 at 0:34