A circle packing is a set of tangent and disjoint discs. Maps between circle packings with the same tangency are discrete analogues of conformal mappings, which have application for example in mechanical, fluid, and thermal engineering. We describe an advancing front algorithm to compute the circle packing of a strip around a closed planar curve. Conformal mappings preserve local angles and shapes; our algorithm uses these properties to obtain via the fast Fourier transform the centers and radii for the circle packing of successive trigonometric Lagrange curves in a strip. To check the algorithm, different results are compared with well-known conformal mappings. Real time deformations of circle packings are possible by changing the shape of the initial closed curve.