# Thread: Parametrization of the intersection of cylinders

1. ## Parametrization of the intersection of cylinders

Use sine and cosine to parametrize the intersection of the cylinders x^2+y^2=1 and x^2+z^2=1 (use two vector-valued functions). Then describe the projections of this curve on the three coordinate planes.

I am not sure how to do this problem at all any help would be great. Thanks

2. ## Re: Parametrization of the intersection of cylinders

Originally Posted by acasas4
Use sine and cosine to parametrize the intersection of the cylinders x^2+y^2=1 and x^2+z^2=1 (use two vector-valued functions). Then describe the projections of this curve on the three coordinate planes.

I am not sure how to do this problem at all any help would be great. Thanks
The projection of the cylinder $x^2+y^2=1$ on the Oxy-plane is a circle $x^2+y^2=1$, which has this parametric equation $\begin{cases}x=\cos{t},\\y=\sin{t},\end{cases}\!\! t\in[0;2\pi].$

Also $x^2+z^2=1~\Rightarrow~z=\pm\sqrt{1-x^2}=\pm\sqrt{1-\cos^2t}$

So, $\begin{cases}x=\cos{t},\\y=\sin{t},\\z=\pm\sqrt{1-\cos^2t},\end{cases}\!\!t\in[0;2\pi].$

See the plot, where
- the blue line is $\left(\cos{t},\sin{t},-\sqrt{1-\cos^2t}\right),$
- the red line is $\left(\cos{t},\sin{t},\sqrt{1-\cos^2t}\right).$

For Maple

with(plots):
A := spacecurve([cos(t), sin(t), sqrt(1-cos(t)^2)], t = 0..2*Pi, color=red, thickness=3):
B := spacecurve([cos(t), sin(t), -sqrt(1-cos(t)^2)], t = 0..2*Pi, color=blue, thickness=3):
display(A, B, axes=normal, scaling=constrained, view=[-2..2, -2..2, -2..2], numpoints=1000);

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# curve of intersection of two cylinders

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