Thaks for the reply, It was really interesting but what about the linear velocities on x, y and z and the angular velocities on x y and z.Charles Link said:The following Insights article that I authored might be of interest to you. https://www.physicsforums.com/insights/frenet-equations-2-d-result-cornu-spiral/
For the OP @Benyoucef Rayane A google shows the Archimedes spiral has ## r=\theta^a ## with ## a=1 ##. This means ## r=\theta ## for this spiral. We can write the velocity ## \vec{v}=(\frac{dr}{dt}) \hat{a}_r+(r \dot{\theta}) \hat{a}_{\theta} ##. We have for ## r=\theta ##, that ## \frac{dr}{dt}=\dot{\theta}=\omega ##. This gives ## \vec{v}=\omega \hat{a}_r+(r \omega) \hat{a}_{\theta} ##. As ## r ## gets large, ## \vec{v} \approx (r \omega ) \hat{a}_{\theta} ## as @rumborak pointed out.rumborak said:Since the usual definition of the Archimedes Spiral takes the angle as the input, it means you choose the angular velocity ω. Regarding linear velocity, I don't have an exact formula at hand, but since it gets closer to a circle with increasing angle, the linear velocity will asymptomatically approach ωr.
Thanks guys, I got it now.Charles Link said:For the OP @Benyoucef Rayane A google shows the Archimedes spiral has ## r=\theta^a ## with ## a=1 ##. This means ## r=\theta ## for this spiral. We can write the velocity ## \vec{v}=(\frac{dr}{dt}) \hat{a}_r+(r \dot{\theta}) \hat{a}_{\theta} ##. We have for ## r=\theta ##, that ## \frac{dr}{dt}=\dot{\theta}=\omega ##. This gives ## \vec{v}=\omega \hat{a}_r+(r \omega) \hat{a}_{\theta} ##. As ## r ## gets large, ## \vec{v} \approx (r \omega ) \hat{a}_{\theta} ## as @rumborak pointed out.