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In this paper, the uniform geometrical theory of diffraction (UTD) based field expressions developed previously [Pathak et al., 1981] for predicting the radiation from antennas which are flush mounted directly on an arbitrary, smooth, perfectly conducting, convex surface are extended to allow the antennas to be raised slightly above the convex surface. This extension thus allows an efficient ray analysis of many practical antenna structures such as blade antennas or pod-mounted antennas on convex surfaces or even antennas near convex cylindrical masts, etc. This extended UTD (EUTD) is developed such that it employs the same ray paths and retains the same simplicity and efficiency as the earlier UTD radiation solution [Pathak et al., 1981]. Furthermore, if the antenna is moved directly onto the surface, then the EUTD solution developed here automatically reduces to the earlier UTD solution. Numerical results based on EUTD are presented for antennas close to smooth, curved surfaces; in some of these cases, impressed sources of radiation are oriented so that they would produce no radiation if they were moved directly onto the surface.