Using scatteredInterpolant for 1-D data

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I have a simple 2xN vector of x and y values that are created by the countourc function; the values are samples from a single contour line on a plot and form a pseudo-circle. I want to be able to interpolate to find any value on the "circle" using either an x or y value, or better yet an angle from the origin. When I try to use any of the interp family, they balk either because the data is not strictly monotonically increasing or because there is a duplicate value somewhere. The scatteredInterpolant function, however, deals with that situation by eliminating the duplicates, but it seems only to work on 2-D data, i.e., each x,y grid point has an associated z value. All I want to do it interpolate points in 1-D, i.e., find the interpolated x intercepts, etc.