Is the velocity vector tangent to a dynamic path?

Question

Imagine a particle traveling along a curve (or surface). I cannot find anything on this topic that allows the curve (or surface) to change (or deform) as the particle moves. For a static curve, it is well known that the particle's velocity vector is tangent to the curve. (See, for example, Is the tangential velocity the same as finding the tangent vector? ). But what happens if the curve along which the particle is moving, is itself changing as the particle moves? For example, imagine a particle moving along the circumference of an expanding circle. To simplify discussion, let's assume the particle moves at constant speed along the circumference, and that the radius of the circle increases at a constant rate (e.g., r(t) = a + c t). Is the particle's instantaneous velocity vector still tangent to the (expanding) circle? If not, how does the velocity vector relate to the equation of the circle?