parametrise equation of a hyperbola

Question

Any point on an ellipse can be wrttien as $(a\cos\theta,b\sin\theta)$, How could we genarilse this to a hyperbola?

Answer 1

You are probably looking for $(a \sec \theta,b\tan \theta)$ if centered at the origin like your ellipse form. Note that I have applied no effort to answer this, here is a pretty decent expnanation.

If it is not centered at the origin, then the parametric form for the hyperbola, $$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$$ is $$(h+a\sec \theta, k+b\tan \theta).$$

Again, look at that link, and do edit your post if you want some sort of detailed derivation.

Answer 2

Alternately, since this is a hyperbola after all, use hyperbolic trig functions: $(a\cosh(t),b\sinh(t))$. The hyperbolic analogue of the Pythagorean identity is $\cosh^2 - \sinh^2 = 1$, so the curve satisfies $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. The parameter $t$ can be interpreted as hyperbolic angle, the area swept out by a ray from the origin to the curve as it traverses from the vertex of the hyperbola to $(a\cosh(t),b\sinh(t))$.

Answer 3

I know you asked for a transcendental parametrization, but don’t forget the rational parametrizations of the circle $X^2+Y^2=1$ such as $X=2t/(t^2+1)$, $Y=(t^2-1)/(t^2+1)$.

Similarly, if you want to parametrize the hyperbola $Y^2-X^2=1$, you can use $X=2t/(t^2-1)$, $Y=(t^2+1)/(t^2-1)$. In both cases, there are many other similar parametrizations that you can use.