cm/360 and degrees/cm: read the reciprocal correctly

Build a table from the same full turn

Choose five full-turn distances and divide 360 degrees by each distance. These are author-selected inputs, not measurements or preferred settings. The last column multiplies the distance by its angular response. In every row, centimeters cancel and the product is a complete 360-degree turn. This provides a simple dimensional check rather than relying on a familiar-looking numeral.

For forty centimeters, 360/40 = nine degrees per centimeter. At that response, five centimeters produces 5 × 9 = 45 degrees. Twenty centimeters produces 180 degrees. Reversing the calculation, 360/9 returns forty centimeters. The intermediate movements follow from the assumed constant response; they are not a test showing that a real setup is linear.

Do not invert the number without the angle scale

The bare reciprocal of forty is 0.025. If forty was centimeters per turn, that reciprocal is turns per centimeter. To express the answer in degrees per centimeter, multiply by 360 degrees per turn: 0.025 × 360 = nine. Writing 0.025 degrees/cm would miss the turn-to-degree factor and describe a different response.

A second label trap is to copy a value of nine degrees/cm into a cm/360 input. The correct distance is 360/9 = forty centimeters. A request for nine cm per turn instead implies 360/9 = forty degrees/cm. The numerals have exchanged roles. Under fixed DPI and coefficient, that mistaken distance requests a sensitivity factor of 40/9, approximately 4.4444, relative to the forty-centimeter target. This is a constructed input error, not a diagnosis of any particular converter.

Percentage changes are reciprocal too

Start with forty cm per turn and multiply sensitivity by 1.10 while keeping DPI and the linear coefficient fixed. Angular response rises from nine to 9.9 degrees/cm. The new full-turn distance is 360/9.9 = 400/11, approximately 36.3636 cm. Its change relative to forty is (1/1.10 − 1) × 100, approximately −9.0909 percent, rather than minus ten percent.

A factor of 0.8 lowers sensitivity by twenty percent, but forty divided by 0.8 is fifty centimeters: travel rises by twenty-five percent. A factor of 1.25 raises sensitivity by twenty-five percent, but the distance becomes thirty-two centimeters: travel falls by twenty percent. The paired changes reverse one another because the multiplicative factors, 0.8 and 1.25, multiply to one. Adding or subtracting the same percentage in both quantities would not preserve this relationship.

Connect degrees/cm to the converter model

Take a disclosed model with DPI 800 and yaw 0.07 degrees per count at sensitivity one. For nine degrees/cm, multiply by 2.54 cm/inch to obtain 22.86 degrees/inch. Divide by 800 counts/inch and by 0.07 degrees/count to obtain sensitivity 1143/2800, approximately 0.408214285714. The length, count and angle units cancel.

Check using distance instead: 914.4/(800 × 0.07 × 40) gives the same fraction. The factor 914.4 is 360 × 2.54. These two routes connect the quantities without treating nine degrees/cm as a menu setting. They establish consistency of the stated model, not current-build accuracy of a game coefficient or measured mouse DPI.