tangential & sagittal planes Sagittal image plane In an object plane perpendicular to the optic axis, construct a line intersecting the optic axis. From any point on this object line, send out a fan of rays all of which lie in the plane which contains the object line and the optic axis. Since there is nothing about the lens which can refract a ray which originally lies in this plane out of this plane, all rays will strike any arbitrarily-placed screen in image space someplace on the image line, regardless of whether the line is in focus or not. That is, although the rays do "triangulate in" to a common point in image space, we can't find that point because even when they do not "triangulate in" (intersect), they still land someplace along the line and so don't generate any fuzziness. Therefore, the correct location of the screen or image plane is indeterminate if we try to use only this fan of rays to find it. Now construct a new fan of rays from any off-axis object point on the object line. This fan's plane will be perpendicular to the fan of the previous paragraph, and the central ray of this fan will start at the object point and pass through the point which is the middle of the lens. Since these rays do not lie in the common plane of the object and image lines, there is a determinate distance from the lens to the image screen at which the rays will "triangulate in" to a point (intersect) and not broaden the width of the image line. With the screen at this distance from the lens, the line on the image screen will be at its sharpest. The screen now lies in the "sagittal image plane". Sagittal means "radial" and it is a radial line which is sharpest at this image plane location. Tangential image plane In an object plane perpendicular to the optic axis, construct a circle centered on the optic axis. Select a point on that circle and then construct a plane which is tangent to the circle at that point and which contains the point which is the middle of the lens. Send out a fan of rays from the selected object point but constrain them to lie in the plane just defined. To the first order, it does not matter what distance the image screen lies from the lens: the circle will still be sharp because any out-of-focus condition will broaden the image point only along the image of the circle. Now construct a new fan of rays from the same object point on the circle described above, but this time constrain the rays to lie in a plane containing the object point and the optic axis. (Note that this plane is perpendicular to the plane described in the previous paragraph.) This fan will "triangulate in" on a definite point in image space and if the rays are not triangulated in, they will broaden the width of the line which is the image of the circle. A screen located at the place where the circle is sharpest is said to lie in the "tangential image plane" because any small line tangent to a circle centered on the optic axis will be in focus in this plane. Observations It is somewhat counterintuitive that to find the tangential image plane, one uses a "tangential ray fan" and the tangential ray fan lies in a radial plane. Similarly, to find the sagittal image plane, one uses a "sagittal ray fan" and the sagittal ray fan lies in a tangential plane. So live with it. The sagittal ray fan generates skew rays and skew rays can intersect the image plane slightly exterior to the plane of the fan. This is usually not a large enough error to worry about. These two image planes are actually curved. That is, they are curved surfaces which tend to be somewhat axisymmetric. For instance, if several concentric circles of different diameters are drawn centered on the optic axis and in an object plane which is perpendicular to the optic axis, each circle's sharp image will lie at a slightly different distance from the lens in object space. Similarly, with the sagittal plane, different parts of a radial line will be sharpest in image space at different distances from the lens.