I'm trying to brush up on some math for my latest Computer Science class. We are given a shape (curve) in parametric form. How can we show that this is symmetric (or not) about the x-axis or y-axis?
If we have an equation in explicit form like: $y = mx + b$
Then to show that this line is symmetric about some axis, we can set either $x=-x$ or $y = -y$ and show if the equation remains the same. However, if we have an equation in parametric form, such as:
$x(t) = sin(t), y(t) = cos(t)$
Then how do we go about showing if this resulting shape is symmetric or not? Is there a way to do so without converting back to the explicit form?
Please excuse my terminology, as I'm sure I'm mixing up some terms.
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$\begingroup$Maybe a bit of a stupid but immediate method. Assume your function is given in parametric form $$x = \varphi(t)$$ $$y=\psi(t)$$ where $t \in (a,b)$. Assume you want to check whether your curve is symmetric with respect to $x \mapsto -x$ and $y \mapsto y$, i.e. reflection in the $y$ axis. For each value $t_0 \in (a,b)$ form the function $$f(t, t_0) = \big(\varphi(t) + \varphi(t_0)\big)^2 + \big(\psi(t) - \psi(t_0)\big)^2$$ Observe $f(t,t_0) \geq 0$ and $f(t,t_0) = 0$ if and only if for some $t \in (a,b),$ $\,\, \varphi(t) = -\varphi(t_0)$ and $\psi(t) = \psi(t_0)$. So if for each $t_0 \in (a,b)$ you can find a solution to the equation $f(t,t_0) = 0$ in terms of $t \in (a,b)$ then your curve is symmetric with respect to the $y$ axis.
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