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The longitude / latitude parameterization of a sphere is described by: $x = cos(φ) * cos(θ) \quad y = cos(φ) * sin(θ) \quad z = sin(φ)\quad$ where $\quadθ = 2 π u$ and $φ = π v - π / 2$

I understand that the parameter $u$ serves to essentially step through every angle of each circle that comprises a sphere, but I'm unclear as to the purpose of the parameter $v$ in the equation for $φ$. It seems that v is used to step through each angle (starting at the center of the sphere) that points to the latitudinal lines (each circle) on a sphere, in effect factoring in to the radius for each circle. I'm also not sure about why $π$ is multiplied by $v$ or why $π/2$ is subtracted from it. Can anyone clear up these points of confusion?

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The idea of this particular parameterization is that $u, v$ both range from $0$ to $1$. Thus $\varphi$ ranges from $-\pi/2$ to $\pi/2$, with $\varphi = 0$ in the equatorial plane $z = 0$.

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