We present a new trigonometric basis function that is capable of perfectly reproducing circles, spheres and ellipsoids while at the same time being interpolatory. Such basis functions have the advantage that they allow to construct shapes through a sequence of control points that lie on their contour (2-D) or surface (3-D) which facilitates user-interaction, especially in 3-D. Our piecewise exponential basis function has finite support, which enables local control for shape modification. We derive and prove all the necessary properties of the kernel to represent shapes that can be smoothly deformed and show how idealized shapes such as ellipses and spheres can be constructed.