This paper presents a closed-form solution to the sag of the Cartesian oval and an alternate iterative method for obtaining the sag. The emphasis is in providing a methodology for determining the sag and derivatives of a Cartesian surface for optical design, ray-tracing purposes. We verify our results by comparison of our solutions and by real ray tracing.

Satellite images help in studying various phenomena related to earth's surface. The range of applications varies from agriculture, geology, coast and marine studies to urban development and environmental affairs. CubeSat projects belong to the category of small satellites named Pico Satellites. They have a relative superiority over higher order satellites such as Micro and Mini satellites in terms of their short development time, lower complexity, and most importantly less cost. We present here an overview on the ongoing project EgyCubeSat-1, a Pico satellite that has a camera as an optical payload with ground sample distance better than 100 meters, synchronous orbit, and low earth orbit in the range 600 ~ 700 Km. The key innovation is the development of costumed optics that fit in the compact allocated space and result in better resolution.

The classic Abbe sine condition relates pupil distortion to aberrations with linear field dependence such as coma. This paper provides a fully generalized form of the sine condition that does not use any symmetry. It accurately predicts the change in aberration in the presence of field independent and pupil aberrations. The definitions of the image, object, and coordinate system are completely arbitrary. The relationship is verified using ray trace simulations of a number of systems that have varying degrees of complexity. The potential applications are discussed.