This is perhaps too "computational" and not "structural" enough for your purposes, but there are many instances where an <a href="http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign">integral</a> or a <a href="http://www.mat.univie.ac.at/~kratt/artikel/detsurv.html">determinant</a> is most simply evaluated by introducing an extra parameter.

(Actually, as I write this, I'm wondering if maybe some of these "computational" examples can be interpreted geometrically if one looks at them in the right way...)