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It all started after I took the delivery of the brand new 2007 BMW 335i Coupe in December 2006. My passenger complained that the ride was jerky. It had been 10 years since I last touched a stick shift. We, as drivers, usually don't get bothered by shift shocks. But it is painful to watch your passenger lurch back and forth.

So I spent the next few weeks trying to figure out the perfect "shift", swift and jerk free, like with an automatic transmission. I tried to clutch in and out slowly and gently when shifting. It would reduce the shift shock but not eliminate it. It would also wear out the clutch sooner. Most importantly, I don't want to waste half of a second when shifting.

When trial and error does not work, I went on to read the articles about the clutch from How Stuff I started to understand the gear ratios, the number of turns of a gear for each turn of the engine. Say the gear ratio of the 2nd gear is 4:1 and the gear ratio of the 3rd gear is 2.4:1. That means for each turn of the 2nd gear, the 3rd gear will turn 2.4 / 4 = 0.6. Say if you are traveling at 30 mile per hour in 2nd gear at 3000 RPM, when you shift to the 3rd gear, the matching RPM would need to be 3000*0.6 = 1800.

My next step was to calculate the typical shifting values in a table. When I am not racing, I usually shift when the engine reaches 2000 RPM to save gas. For example, I am on the 1st gear, and I clutch in at 2000 RPM. Then I let the engine to drop to 1182 RPM before clutching out. Similarly, if I am on 2nd gear and clutch in at 2000 RPM, I'd let the engine to drop to 1317 RPM before clutching out. The following table shows the relative RPM needed when shifting from adjacent gears at three thresholds 2000, 2500, 3000 RPM respectively:

2500 3000
1 → 2 1182 1478 1773
2 → 3 1317 1646 1975
3 → 4 1506 1883 2259
4 → 5 1681 2101 2521
5 → 6 1740 2175 2610
I jotted down this table and placed it next to my stick. After a few hours, I totally mastered the shifting at 2000 and 3000 RPM. However, to drive a car is different from reading a sales chart in Excel. The little table soon became a distraction. After all, it did not cover all the possible ways of shifting, e.g. push the car on 1st gear to 6000 RPM, then drop the RPM to 1800 and shift directly to the 4th gear to cruise at 35 MPH on a local road. At this point, I know I have passed the beginner's needs. I need something that can show me all the possible combinations of shifting, from 1st to 2nd, 2nd to 4th, 5th to 3rd, and so on.

"A picture is worth a thousand words." In my case, a graph with all the possible ways of shifting finally took shape on March 6, 2007. This later became the Shifting Path.

Time line:

March 7, 2007 Implemented in GNUPLOT.
March 10, 2007 The sketch that is more or less the final design.
May 11, 2007 Due to lack of loop control in GNUPLOT, I rewrote it in Perl.
May 14, 2007 Wrote Archimedean Curve plot for a given speed.
July 29, 2007 Filed US Patent.
Nov. 8, 2011 Issued US Patent 8,055,419.