# How to parameterize an equation with 3 variables

Hello everyone, I'm currently working on a project explorting how to find the distance between two points along a curve. My first method is try create a plane cutting through the surface and both points thus allowing me to create a new equation with only two variables and solving for the arclength. However does anyone know if its possible to create parametric equations of a surface: x^2=1/(y^2+z^2)?
Thank you

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• Erdos
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• Flexipie
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thank you for such a quick answer, if I may ask, if I want to use the arclength formula wouldn't I need a vector function for example r(t)=(f(t),g(t),h(t)? Is that possible

• Erdos
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This is a surface. Arc length formula is for paths. So it does not make sense to use arc length formula for a surface.

• Flexipie
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So if I take two points on this surface, it isn't possible to trace the path and then arclength between those two points along the surface? I'm still satisfied with the answer thank you

• Erdos
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There would be infinite such paths. If you want the shortest path, yes, there will be shortest paths, but that is a very complicated question that need high level geometry.

• Flexipie
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Ok, thank you anyway, if its that hard I think its better I stay away from finding the shortest path, does it involve geodesics or is that only for spheres?

• Erdos
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They are called geodesics, but get very complicated on surfaces other than spheres.