I know what a collocation method is and how it can be useful for the direct transcription of optional control problems, using specific collocation points.
I also know what a quadrature rule is and how it can be used to solve integrals numerically, using specific quadrature points.
But I'm struggling to see the mathematical connections between these two concepts, that are obviously related (for example: the trapezoidal collocation method has to be related to the trapezoidal quadrature rule somehow).
So basically I'm wondering: is there a one-to-one correspondence between one collocation method and the related quadrature rule? If so, is it possibile to show how they are reciprocally related, in general, from a mathematical point of view?