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I understand that every curve has two parallel curves for any given distance: one on both sides.

Finding parallel curves to some algebraic objects are simple, especially lines and circles ($P(y,d)$ means the parallel curve to curve $y$ at uniform distance $d$).

Lines:$$y=mx+b$$$$P(y,d)=(mx+b)±(d*\sqrt{1 + m^2})$$

Circles:$$y=±\sqrt{r^2-x^2}$$$$P(y,d)=±\sqrt{(r±d)^2-x^2}$$

Is there any theorem, law, etc. to determine any arbitrary algebraically-defined curve, such as parabolas, hyperbolas, ellipses, exponential functions, etc.?

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If we denote by $\mathbf{t} $ and $\mathbf{n} $ respectively the unit tangent and (outward) normal vectors $$ \mathbf{t} = \frac{1} {{\sqrt {dx^{\,2} + dy^{\,2} } }}\left( {\begin{array}{*{20}c} {dx} \\ {dy} \\ \end{array} } \right)\quad \mathbf{n} = \frac{1} {{\sqrt {dx^{\,2} + dy^{\,2} } }}\left( {\begin{array}{*{20}c} {dy} \\ { - dx} \\ \end{array} } \right) $$ the parametric equation of the parallell curves shall be (changing $d$ to $s$, to avoid confusion with the differential) $$ \left\{ \begin{gathered} x_p = x(\lambda ) \pm s\,n_{\,x} = x(\lambda ) \pm s\,\frac{{dy}} {{\sqrt {dx^{\,2} + dy^{\,2} } }} \hfill \\ y_p = y(\lambda ) \pm s\,n_{\,y} = y(\lambda ) \mp s\,\frac{{dx}} {{\sqrt {dx^{\,2} + dy^{\,2} } }} \hfill \\ \end{gathered} \right. $$ By putting $x(\lambda)=\lambda$, or by other algebraic means the above can be converted to the case of having $y=y(x)$ or $F(x,y)=0)$

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  • $\begingroup$ I am used to parametric equations in the {x(t)=~, y(t)=~} form. Is λ (lambda) merely an exact replacement of "t," or does it represent some special case? and where was the "t" used in the parametric equation down below? $\endgroup$ – ilcj Dec 5 '16 at 20:57
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    $\begingroup$ Yes, $\lambda$ is the parameter describing the original curve: you can call it with whatever other name, including $t$ if you are more familiar with it (apart that it is normally associated with the parameter representing time). For your 2nd question, I do not get what you mean. $\endgroup$ – G Cab Dec 5 '16 at 21:26

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