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So I was computing the envelope of the family of lines $x \cos{C} + y \sin{C} = p$, where $p$ is a constant and $C$ is the parameter that defines the family. According to the normal procedure, we would differentiate the equation with respect to $C$ and get $-x \sin{C} + y \cos{C} =0$, and we would have a system of equations that when solved would give us the circumference of radius $p$ centered at the origin.

My problem is this: suppose we tried to differentiate with respect to $C$ again; we would have $-x \cos{C} - y\sin{C}=0$. But this is just the symmetric of the first equation, so combining them we would have $p=0$. Now this conclusion obviously makes no sense, so what is the problem with differentiating the equation twice?

ImHackingXD
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    What do you expect the second derivative with respect to the parameter to mean? The definition of an envelope involves tangent lines, hence is about first derivatives. – Ted Shifrin Oct 10 '23 at 17:09
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    @TedShifrin: I think OP is expecting it to give the curvature of the envelope? – Brian Tung Oct 10 '23 at 18:15
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    @BrianTung: Ah, that would never have occurred to me. That's going to be a far more complicated computation with the implicit function theorem. It will involve various third-order partial derivatives of the function defining the envelope. – Ted Shifrin Oct 10 '23 at 18:28

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