Dividing CDF rather than PDF equally in Latin Hypercube Sampling

Returning back to the basics of Latin Hypercube Sampling, I wanted to ask why LHS is a division of CDF into N intervals rather than PDF. I'm having some trouble understanding what the difference is between the division of CDF vs. PDF. Also, since CDF always ranges from 0 to 1, is that the reason LHS has the name hyper"cube" even when the different parameters have different ranges?

Latin hypercube samples are normally constructed uniformly on the (0,1) interval, and then transformed to the marginal desired distributions. So, you are correct, that is why they are referred to as "hyper" (multiple dimensions) "cubes" (sides of equal length). If $$f(x)$$ is the PDF, $$F(x)$$ is the CDF, and $$X$$ is a ($$n$$ x $$p$$) matrix that is a LHS, then $$F^{-1}(X[,i])$$ are the quantiles of the desired distribution, transforming the (0,1) interval to the range of the distribution of interest. The bin endpoints around $$F^{-1}(X[,i])$$ are also the points that divide the marginal PDF in to $$n$$ equal intervals.