This paper analyses a so-called corner-collision bifurcation in piecewise-smooth systems of ordinary differential equations (ODEs), for which a periodic solution grazes with a corner of the discontinuity set. It is shown under quite general circumstances that this leads to a normal form that is to lowest order a piecewise-linear map. This is the first generic derivation from ODE theory of the so-called C-bifurcation (or border collision) for piecewise-linear maps. The result contrasts with the equivalent results when a periodic orbit grazes with a smooth discontinuity set, which has recently been shown to lead to maps that have continuous first derivatives but not second. Moreover, it is shown how to calculate the piecewise-linear map for arbitrary dimensional systems, using only properties of the single periodic trajectory undergoing corner collision. The calculation is worked out for two examples, including a model for a commonly used power electronic converter where complex dynamics associated with corner collision was previously found numerically, but is explained analytically here for the first time.