Lea,
I understand Fred proposal or interpretation, i.e: the path of travel is straight, but the speed is not. Now you are integrating the "time line" which is a line integral in the general sense of line integral. That is a bit different story, but as simple,. The function that will produce the line is now the sec(x) over the 8 quarters, each varying 0... 45� . Them sum them as Richard did in a discontinuous function (piecewise) and you plug that in the phi(t) in the last work sheet. But the length will not change. At this point, you can size both the discontinuous and the straight line to either one of them by scaling, scaling Chebyshev type but more general Cheby is � 1.
A line has lot of mathematical definitions, thus the unique "line integral", which is the definition of the integral . The path of integration may not be over from end to end, so the path of integration describes the line. I understand that by nature english is not very analytical and american english is worst with their habit of making expressions like 10 lines long german words. "Path integral" means strictly nothing, but "path OF integration" has now full meaning. In fact, the best english is the one from non native english speaking countries. Curiously enough, the best "reference english" is the Canadian one as well as the Canadian french. Historically speaking, these two speakers were abandoned by their kings and left alone. That is only true for Canadians not too "Americanised".
I had the discontinuous sec(x) done but zapped it, being sure it didn't purport your project. I might give it a go again, but don't trust me 100% yet. Drawing on the cube seems feasible. It would require drawing the 4 planes and project each sec(x) segment. But I can tell you in advance you will hardly see more than 4 straight lines.
jmG