non linear regression of a 3D line

I have a 3D line On which I need to do a non-linear regression. The shape will range from a straight line going from (0,0,0) to (1,1,1) to a line that connects (0,0,0) and (1,1,1) by tracing the x, y and z axes.

[Image Analyst]

Can the "shapes" be something other than a series of connected line segments? Like some arbitrarily curvy path? If so, why do you call the "shapes" a "3D line"?

Answer 1

[Michael]

You need a model to fit to the data, if you can come up with one (for example, f(x,y,z; param1, param2 ...) = 0), and have a suitable definition of what a residual is in three dimensions (eg. the nearest distance of the curve from the data point (xi,yi,zi)), you can write a function which returns these residuals and pass it to lsqnonlin which will then minimise it.