Uniform cubic B-spline functions have been used for mapping functions in various areas such as image warping and morphing, 3D deformation, and volume morphing. The injectivity (one-to-one property) of a mapping function is important to obtain good results in these areas. This paper considers the local injectivity conditions of 2D and 3D uniform cubic B-spline functions. We propose a geometric interpretation of the local injectivity of a uniform cubic B-spline function, with which 2D and 3D cases can be handled in a similar way. Based on the geometric interpretation, we present sufficient conditions for the local injectivity that are represented in terms of control point displacements. These sufficient conditions are simple and easy to check and will be useful to guarantee the injectivity of mapping functions in application areas.