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I have a need to increase the number of polygons in this mesh, maintaining mesh curvature: enter image description here

I tried increasing the number of polygons with the Subdivide tool in edit mode, with this result: enter image description here

The problem is that the points added with the subdivision, do not maintain the same curvature of the mesh, but were created in a straight line between one existing mesh point and another: enter image description here

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In your case a Subdivide operation with the Smoothness at 1 should give the desired result, but for more complicated shapes, don't use the Subdivide tool in Edit mode, it will tend to flatten the faces as you show, instead give a Subdivision Surface to your object and apply it, contrary to the simple Subdivide operation, the modifier will better interpolate the roundness on the whole shape.

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