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Say I have a hemisphere with some radius R. Is there an equation that could represent any latitudinal line that lies along the surface of the hemisphere?

I want to be able to have an equation that could describe the curve that runs along my hemisphere in cartesian coordinates. From that equation, I could choose any x,y,z values and be able to plot that curve overlayed on my hemisphere.

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The $z$ component of the latitudinal line remains constant at $R\cos\phi$ where $\phi$ is the angle of the line from the $z$-axis. At this point, the $(x,y)$ coordinates form a circle with radius $z$ in the $x,y$-plane, so we have the curve $y = \pm\sqrt{R^2\cos^2\phi - x^2}$.

In sum, each latitudinal line is described as $$ (x,y,z) = \left(x,\pm\sqrt{R^2\cos^2\phi-x^2}, R\cos\phi\right) $$ where $x\in[-R\cos\phi,R\cos\phi]$ and $\phi$ tells you which latitudinal line you care about.

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