pilates is not a form of yoga

Though I wouldn't classify pilates as a specific yoga form, there is a great deal of overlap in the types of the physical exercise and the focus upon visualization and relaxation. Pilates is a relatively recent exercise regimen in contrast to yoga's much longer history. My understanding is that pilates was originally developed as a rehabilitation routine for dancers to regain strength and flexibility without the stress bearing characteristics of more strenuous dance routines. I see no reason why it wouldn't complement one's yoga practice. Also, many yoga teachers have spliced pilates techniques into their teaching routines.

Football warm-ups that I partook in during my school years contained many exercises similar to asanas I do now, some virtually identical - at least in their physical form, yet I certainly wouldn't classify football as a form of yoga.

My understanding is that pilates is not a form of yoga at all. It's a form of exercise that sometimes utilizes props, and has movements that are similar to some asanas, but yoga is a much larger body of study/knowledge/information/theory. Asana is only one limb of 8, after all.