  We propose an optimal experimental design for a curvilinear regression model
that minimizes the band-width of simultaneous confidence bands. Simultaneous
confidence bands for curvilinear regression are constructed by evaluating the
volume of a tube about a curve that is defined as a trajectory of a regression
basis vector (Naiman, 1986). The proposed criterion is constructed based on the
volume of a tube, and the corresponding optimal design that minimizes the
volume of tube is referred to as the tube-volume optimal (TV-optimal) design.
For Fourier and weighted polynomial regressions, the problem is formalized as
one of minimization over the cone of Hankel positive definite matrices, and the
criterion to minimize is expressed as an elliptic integral. We show that the
M\"obius group keeps our problem invariant, and hence, minimization can be
conducted over cross-sections of orbits. We demonstrate that for the weighted
polynomial regression and the Fourier regression with three bases, the
tube-volume optimal design forms an orbit of the M\"obius group containing
D-optimal designs as representative elements.
