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$$ x \cos \alpha + y \sin \alpha -p = 0$$ represents a straight line in polar form (or even taken in any other form),

$$ (x \cos \alpha + y \sin \alpha -p )^3 = 0$$ represents 3 straight lines repeated, but why does not, $ (n\in \mathbb Z) $

$$ ( x \cos \alpha + y \sin \alpha -p)^{2\,n} = 0 $$ at all plot for even powers ? I used Mathematica but other CAS could be written the same way in this respect. Why does it not represent evenly repeated straight lines?

Martin Sleziak
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Narasimham
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  • $f(x,y)=0 \iff (f(x,y))^k=0$ for any function $f$ and $k \in \mathbb{N} \setminus {0}$. It's not clear what you mean by does not at all plot for even numbers. – dxiv Oct 23 '16 at 19:02

1 Answers1

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There are two distinct issues here: what the equation represents and what plots (on particular software).

For any positive integer $n$, $f(x,y)^n = 0$ has exactly the same solutions as $f(x,y) = 0$.

On typical software programs that use numerical methods, implicit plots depend on detecting sign changes, so they will often miss local minima of $0$. See e.g. this recent question

Community
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Robert Israel
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  • Detection succeeds for odd powers and fails for even, right? Also does Maple give repeated $y$ roots for example to solve a simple $ x=1?$ with the evenly repeated straight line? – Narasimham Oct 23 '16 at 19:42