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opengl - GL_TRIANGLE FAN Explanation

I am trying to "fill" a surface of a sphere that I drew using this algorithm here: http://paulbourke.net/miscellaneous/sphere_cylinder/, the first method. I know that GL_QUADS is no longer used in OpenGL 3+. SO I used GL_TRIANGLE_FAN. Is it the same thing? The problem here is that my sphere facets are squares. So if I use GL_TRIANGLE then I get a weird "filling", not all the surface is covered. Am I doing it the wrong way? How does GL_TRIANGLE_FAN work exactly?

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How does GL_TRIANGLE_FAN works exactly?

The first vertex of a triangle fan acts like a hub. The vertices following connect with the previous non-starting vertex and the hub.

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