Principle Component Analysis on multiple functions I am working on a research project that uses data of geometrical structure complexities of different land coverage types (primarily cities, pastures and natural structures). My supervisor has instructed me to use PCA for the reduction of dimensions, but I have trouble understanding how it would work on my data.

That data is a collection of a hundred 2D plots of which the x-axis runs from 0 to 255 (in steps of 1) and the y-axis from 1 to 2 (in non-integer steps). The individual plots are not linear, but to some extent have the same shapes. 

My problem is that for as far as I know, PCA won't work here because for every x-value there are multiple y-values, if I plot all of the individual datasets into one big graph. Also, isn't it a problem that the individual plots are nonlinear?

So my question is: would PCA work on this 'multivalued and nonlinear' dataset? If so, where could I look for an explanation on how exactly it would work?
Variations of PCA can still be applicable.

If the data is not linear, use nonlinear PCA.

It is not an issue there a multiple "y"s for every "x". PCA is unsupervised, there is no notion of targets. In PCA, there are only dimensions. 

Typically, the dimensions are standardized so each dimension can be weighted independent of scale.

One caveat - PCA is domain agnostic. Given that you have physical land data, you can use domain-specific methods such as hierarchical spatial indexing.