In general there is a date/time based scale system ( QwtDateScaleEngine + QwtDateScaleDraw ), that creates useful ticks for date/time scales.
But from your code it looks like you need a scale that displays the time elapsed since some point in time. This would be a different type of scale engine ( on my TODO list for quite some time ).
To make it short: you also need to implement a scale engine, what mostly means to provide 2 algos for aligning a given interval and then finding the ticks.
Uwe
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