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First of all, let me define what I mean by "imposing," and let me clarify that I've only studied this operation in 2D Euclidean space. Now then, to impose one function onto another, you need two things:

  • A function upon which to impose, called the receiver.
  • A function to impose, called the imposer.

Now, let me first explain the concept generally. The idea is that, rather than graphing some function with respect to the x-axis, we treat the receiver as the x-axis, graphing the imposer with respect to receiver.

So, what do I mean by "graphing some function with respect to the x-axis?" Well, first we'll let $p_0$ be the point on the x-axis at some $x$, and we'll let $l$ be the line which is normal to the x-axis at $p_0$. It should be clear that $p_0=(x,0)$ and that $l$ is a vertical line which passes through $p_0$. Then, for some function $g$, let $p_1$ be the point on $l$ whose distance is equal to $g(x)$. It should be clear that $p_1=(x,g(x))$ since the distance from $(x,0)$ to $(x,g(x))$ is equal to $g(x)$. If you do the previous procedure for all $x$ and plot every $p_1$ on a graph, you will have successfully graphed $g$ with respect to the x-axis.

So, to reiterate my second paragraph, the idea is that we can graph any function with respect to some other function. The way we do this is by following the same procedure we used in the last paragraph. However, there are two main differences:

  • Instead of letting $p_0$ be the point on the x-axis at some $x$, we let $p_0$ be the point on some parametric function $f(t)$, the receiver, for some $t$.
  • Instead of letting $l$ be the line which is normal to the x-axis at $p_0$, we let $l$ be the line which is normal to $f$, the receiver, at $p_0$.

For example, this is $g(t)=cos(t)$, the imposer, imposed upon $f(t)=(t,a \cdot sin(t))$, the receiver, where $a$ is simply a real value which oscillates between $-1$ and $1$ with time. Basically, $a$ is the reason functions below are moving. a graph of cosine of t imposed upon an oscillating sine of t

The black function which resembles a standing wave, as I said before, is an oscillating sine function, and it is also the receiver. The blue function which, if you look closely, occasionally looks like a cosine function is the function resulting from imposing $g(t)$ onto $f(t)$. The green line segments are to illustrate the act of finding the point on the normal of $f(t)$ at $t$ with a distance of $g(t)$, like we covered in the above paragraphs.

If you're interested, here's the raw math to impose one function onto another: Given some parametric equation $f:f(t) = (x(t),y(t))$ upon which we wish to impose some function $g(t)$:

$$ Let\;h(t)=\frac{\frac{d}{dt}(y(t))}{\frac{d}{dt}(x(t))}=f'(t) $$ $$ Let\;j(t)=tan^{-1}(h(t))\pm\frac{\pi}{2} $$ Then $g$ imposed upon $f$ becomes the following parametric equation, in terms of $t$. $$ (g(t)cos(j(t))+x(t),g(t)sin(j(t))+y(t)) $$

I feel as though I should clarify now that, for most $f$ and $g$, this operation will produce two resultant functions. This fact is a result of the way I have defined this operation. That is, we are looking for any point $p_1$ on the normal line of $f$ at $p_0$ such that the distance between $p_0$ and $p_1$ is equal to $g(t)$. Put more simply, we're looking on a line for a point which is a specific distance away from another point on the line. We already know that, for any point $p$ on a line $l$, there will always be exactly two points on $l$ with a distance $\delta$ away from $p$, for all $\delta > 0$. This fact is the reason for which we are adding or subtracting $\frac{\pi}{2}$ in $j$. The only case I can think of wherein this operation does not produce two resultant functions is when $g(t) = 0$, as each resulting function will be exactly equal to the parametric $f(t)$; however, there very well may be more.

So, ignoring any incorrect notation or terminology I might have used, is this type of transformation used anywhere in mathematics? If so, could you show me where I could get some more information on it?

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    $\begingroup$ And for a neater-looking example, check out desmos.com/calculator/xnjlvso5hd It's a graph of the sine function imposed upon a logarithmic spiral. $\endgroup$ – Steven Apr 17 '15 at 4:17
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    $\begingroup$ If $g$ is constant this is known as a parallel curve. $\endgroup$ – user856 Jul 26 '15 at 1:07
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    $\begingroup$ You might be interested in phase portraits, of importance in qualitative differential equations. For instance one might make the vertical axis $\dot{x}(t)$ and the horizontal axis $x(t)$, then plot various curves in this plane according to parameters in the differential equation, or according to initial conditions. $\endgroup$ – anon Sep 11 '15 at 0:16
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    $\begingroup$ Woaaah cool idea! Its now on my list of favorite questions and answers in my profile. $\endgroup$ – goblin Sep 11 '15 at 0:39
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    $\begingroup$ @goblin Thanks. :) It started out as one of those normal random thoughts you have everyday. I was working on math homework, and I basically thought to myself, "what would happen if the x-axis were like a string and functions were just connected to it with some kind of straight, steel beam? what would it look like to move the string around?" (except my thought was more visual and less wordy..) anyway, I eventually wanted to see what it would look like, so I came up with this. :) $\endgroup$ – Steven Sep 11 '15 at 0:51
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If I understand your verbal description, the "receiver" is a "regular curve" $$ f(t) = \bigl(x(t), y(t)\bigr), $$ i.e., a (continuously-)differentiable curve with non-vanishing velocity. There is a unit tangent field $$ T(t) = \frac{f'(t)}{\|f'(t)\|} = \frac{\bigl(x'(t), y'(t)\bigr)}{\sqrt{x'(t)^{2} + y'(t)^{2}}} $$ and a unit normal field $$ N(t) = \frac{\bigl(-y'(t), x'(t)\bigr)}{\sqrt{x'(t)^{2} + y'(t)^{2}}}. $$ The result of "imposing" $g$ on $f$ is the pair of parametric curves $$ \gamma(t) = f(t) \pm g(t)N(t). $$

The closest common construction that comes to mind is the evolute, in which the imposer is the reciprocal of the curvature function of the receiver. (As Rahul notes in the comments, if $g$ is constant, then $\gamma$ is a "parallel curve" to $f$.)

You might search for references to "differential geometry of plane curves", "(unit) normal field", and the like.

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  • $\begingroup$ Do Carmo's book "Differential Geometry of Curves and Surfaces" talks about all of these things in its first chapter, and I found it to be pretty lucid. The majority of the book is dedicated to surfaces though. $\endgroup$ – Alfred Yerger Jul 17 '16 at 2:32
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This doesn't answer your question, but let me just make some observations. Let $E$ denote a Euclidean space.

Observation 0. Given a sufficiently-nice function $$f : E \leftarrow \mathbb{R}$$ (the receiver) we get another function

$$N(f) : E \leftarrow \mathbb{R} \times \mathbb{R}$$

defined as follows: $N(f)(y,t)$ is the point of height $y$ above the point $f(t)$ in the direction of the normal to $f$ at $t$.

Observation 1. Given an (arbitrary) function $$g : \mathbb{R} \leftarrow \mathbb{R},$$ we get a function $$(g, \mathrm{id}) : \mathbb{R} \times \mathbb{R} \leftarrow \mathbb{R}$$

given as follows

$$(g(x),x)=(g,\mathrm{id})(x)$$

Observation 2. The result of imposing $g$ upon $f$ is:

$$N(f) \circ (g,\mathrm{id})$$

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  • $\begingroup$ Wow, that's so much simpler than my question, haha. $\endgroup$ – Steven Sep 11 '15 at 1:09
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    $\begingroup$ @StevenFontaine, glad to be of service. Salutes. $\endgroup$ – goblin Sep 11 '15 at 1:11
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There is a pretty natural relationship with physics, specifically with the notion of changing reference frames.

If you add a symmetry, say the length of the lines, then Einstein Relativity is all about this. Step into a valid reference frame and take the world-line of a rigid body, call this the receiving function.

The world line of any other particle in the reference frame of the receiving function is the distance preserving map taking the receiving trajectory into the y axis. Basically, to swap reference frames with some body, you need to take its trajectory and make that your y axis, because in the rest frame of the object, it moves only in time with velocity 1.

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