# Parametric equation for a space curve

With reference to the following image: the blue curve has trivially a parametrization:

$$(x, y, z) = (\cos\theta, \, \sin\theta, \, 0) \; \; \; \text{with} \; \theta \in [0,\,2\pi)$$

I would like to determine the parametric equations of the red curve, very badly drawn in Paint, where I mean a sinusoidal curve along the blue circumference.

Although I thought about it a lot, I still couldn't figure out how to derive these parametric equation. Any ideas?

• It looks like a curve of the form $$r(\theta) = r_0 + A \cos{(\omega \theta )}$$ But it's not very clear what you mean with the hand-drawn curve. Is it supposed to come out of the $z=0$-plane? Jan 2, 2019 at 12:13

You can try$$\theta\mapsto\left(\cos\theta,\sin\theta,\frac{\cos(8\theta)}8\right),$$for instance. • The $z$ coordinate had to be a waving line again, and so I thought about $\cos(8\theta)$, but then the waves would go too high and too low. That's why I divided by $8$. Jan 2, 2019 at 12:45
• No need for that. Just consider:$$\theta\mapsto\left(\cos\theta+\frac{\cos(8\theta)}8,\sin\theta,0\right).$$ Jan 2, 2019 at 13:02