## Offset xcenter T XaXb TC

Lines have the general form y = mx + b, where b is the y-intercept and m is the slope. Line #2 is the easiest to determine, and by inspection one

C.1. Derivation

Line #1 has a y-intercept of L/2, and the slope can be obtained by inserting the known coordinates of the point at the upper corner of the mirror:

Thus,

2( f - s) 1 2 Similarly, the equation of line #3 becomes

We can rename the quantity (D — L)/2f— s) = n. All three equations can be summarized as y1 = nx1 + y,y2 = -x2 + T,y3 = -nx3 -L (C.6)

By setting y1 = y2, we can determine the coordinate xa as

1 + n and we insert that expression back into the line #2 equation to derive ya:

Similarly, the coordinates of the other intersection point are x, = ^ and y =- + T. (C.9)

The expression for the minor axis (Eq. C.1) is just the difference between the two y-values:

The offset (Eq. C.2) is only a little more complicated:

Offset = 1 f T - (L/2) + T +(L/2) 1-T or Offset = - T. (C.11)

The final expressions of Eqs. C.10 and C.11 are exact. They only require an expression for the sagitta. A parabola of focus f is determined by the equation x = f - (y2/4f), and the exact sagitta is obtained by evaluating the shift in x on axis and at the y-value of the edge, D/2:

Please note that Eq. C.12 is not the same as the wavefront sagitta that will be calculated in Appendix E, but the surface sagitta, which is half as large. 