Tilted Power of Thin Lenses

Tilting changes the effective power of a thin lens. Blendowske has recently provided an elegant matrix expression for the tilted power in the case of tilt about a vertical axis (faceform tilt). He makes use of what can be called a tilt matrix. He shows that the same expression holds in the case of tilt about a horizontal axis (pantoscopic tilt) if one makes use of a different tilt matrix. The purpose of this note is to derive a general tilt matrix that makes Blendowske's equation valid for tilt about any axis. The general tilt matrix reduces to the particular tilt matrices in the case of rotation about the horizontal and vertical axes. The lens may be stigmatic or astigmatic and may be in air or in some other medium.