The three moves
- Add. New sphere = old sphere + old cylinder, signs respected. For -2.00 -1.50 x 180: new sphere = -2.00 + (-1.50) = -3.50.
- Flip. New cylinder = old cylinder with the sign changed. -1.50 becomes +1.50.
- Rotate. New axis = old axis + 90, wrapped onto the 1 to 180 scale. 180 + 90 = 270, minus 180 = 90. Result: -3.50 +1.50 x 090.
The axis wrap
If adding 90 pushes the axis past 180, subtract 180. Axis 120 becomes 30; axis 95 becomes 5. By convention the horizontal meridian is written 180 and never 0, so a wrap that lands on 0 is written 180.
Edge cases the exam likes
- No cylinder. A sphere-only Rx like +2.25 sph has nothing to transpose; both forms are identical.
- Plano sphere. plano -1.00 x 90 transposes to -1.00 +1.00 x 180. The sphere may start at zero, but the arithmetic is unchanged.
- Sphere and cylinder that cancel. +1.50 -1.50 x 45 transposes to plano +1.50 x 135. A zero result is written plano, not left blank.
Self-check
Both forms must give the same power in every meridian: F = S + C x sin^2(theta - axis). The quickest check is the two principal meridians: sphere along the axis, sphere plus cylinder 90 degrees away. Run any example through the meridian power tool in both forms and the numbers match.
Now drill it: the transposition calculator and drill generates unlimited problems in both directions with the steps written out.