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Why only in parametric equation of parabola, angle is not involved unlike parametric equation in ellipse and hyperbola, angle is involved. Thank you.

[edit]:

In parametric equation of parabola $y^2=4ax$, any coordinate on the parabola is taken as ($at^2,2at$).

$t$ is not an angle here.

Where as in ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2}$$ it is ($a\cos(t), b\sin(t)$), $t$ is not the angle made on the ellipse but on the circle of radius $a$.

Similarly on a hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2}$$ it is ($a\sec t,\pm b\tan t$)

UNAN
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Rajesh Marndi
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  • Do you know about angular symmetry? – ਮੈਥ Jul 09 '20 at 10:04
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    I would be helpful if you write down the equations, you are asking about. – user Jul 09 '20 at 10:19
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    There are many parametric equations for the parabola (and, indeed, any curve), so you'll need to be specific. ... In any case, polar coordinates provide a nice, unified form for all conic sections; namely $$r =\frac{\ell}{1+e\cos\theta}$$ (for a conic with eccentricity $e$ and semi-latusrectum $\ell$). That form (obviously) involves an angle. Note that one can use the polar equation to write a cartesian parameterization via $(x,y)=(r\cos\theta,r\sin\theta)$. – Blue Jul 09 '20 at 11:27
  • Ah, you updated while I was typing my comment. :) ... I'll note here that $(at^2,2at)$ is one parameterization of the parabola $y^2=4ax$. If you wanted an angle-based parameterization, you could substitute $x\to r\cos\theta$ and $y\to r\sin\theta$ to get $$r^2\sin^2\theta=4ar\cos\theta \quad\to\quad r = 4 a \cot\theta \csc\theta$$ and then we have $$(x,y)=(r\cos\theta,r\sin\theta)=(4a\cot^2\theta, 4a\cot\theta)$$ – Blue Jul 09 '20 at 11:35
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    By the way ... It's worth noting this parameterization of the ellipse: $$(x,y)=\left(a\frac{1-t^2}{1+t^2}, b\frac{2t}{1+t^2} \right)$$ This form is useful when the trig functions in the angle-based parameterization just get in the way. (Can you derive a similar angle-free parameterization for the hyperbola?) Anyway, to restate a point from my first comment: there are many parameterizations for any curve, each typically suited for a particular use case. It's good to have options. – Blue Jul 09 '20 at 11:43

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