I want to partition some 2d points into 2 groups (clustering). The way that I need to do it is by using PCA to find the first principle component. Then I project the data to find 1d projections. Then I find the middle point on the principal component and partition the 2d points based on the indexes of point on each side of the middle point.
How can I find the line which separates the data into 2 partitions in 2D space? in 1d projected space the partition line is the one perpendicular to the first principal component that passes through the middle point.
Is the line in 2d space unique? Is it possible to compute the best line if it is not unique?
Note: I am actually doing hierarchical clustering with first component of PCA. I want to check the performance of this method against k-d tree.