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Many of the properties of thick lenses can be understood by considering these as a combination of parallel ideal thin lenses that share a common optical axis. A similar analysis can also be applied to many other optical systems. Consequently, combinations of ideal lenses that share a common optical axis, or at least optical-axis direction, are very well understood. Such combinations can be described as a single lens with principal planes that do not coincide. However, in recent proposals for lens-based transformation-optics devices the lenses do not share an optical-axis direction. To understand such lens-based transformation-optics devices, combinations of lenses with skew optical axes must be understood. In complete analogy to the description of combinations of pairs of ideal lenses that share an optical axis, we describe here pairs of ideal lenses with skew optical axes as a single ideal lens with sheared object and image spaces. The transverse planes are no longer perpendicular to the optical axis. We construct the optical axis, the direction of the transverse planes on both sides, and all cardinal points. We believe that this construction has the potential to become a powerful tool for understanding and designing novel optical devices.
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