Parallel Parking Mathematics - Attempt 3a (METRIC version, Aug 3, 2010) Back to mathematics of parking page Note: The animations below use metric units. See original page for more background info, mathematical derivations, and animations using U.S. customary units. - d_c: desired distance from curb when parked
- (alpha): angle that car forms with curb at moment that it stops reversing and starts moving forward
- r: curb-to-curb turning radius
- l: wheel base (distance between front and rear axles)
- k: distance from front axle to front bumper
- j: distance from rear axle to rear bumper
- w_0: width of car being parked
- w_1: width of adjacent car in front of car being parked (including any space from curb)
- w_2: width of adjacent car behind car being parked (including any space from curb)
- optimized: toggle switch allowing you to either use the "optimized" value of alpha that minimizes the value of d, or play with different non-optimal values of alpha (added in 12/26/11 update)
- d: excess length of parking space required beyond one full car length; the output produced as a function of all the other parameters
To minimize the available distance needed to park, first set all the car's physical dimensions and parameter d_c. Toggle on the optimized switch. Then adjust d_c if necessary to ensure that the car does not interfere with the curb while backing up. I've had trouble getting these applets to load within this web page reliably. If so motivated, you may download this file and open within the free Geogebra software. Note: - With the optimized toggle off, drag both d_c and (alpha) all the way to 0 in order to simulate the assumptions used by Professor Blackburn in the noted article, and note how much the required excess space d is increased.
Assumptions: - Optimized parking motion occurs as follows: At an ideal start position, steering wheel is turned hard away from curb, car backs up along an arc, car stops, steering wheel is turned hard toward the curb, and car pulls forward along an arc until parallel to curb.
- Desired distance from curb will be achieved in one take of reverse/forward motion. In actuality, space from curb could be lessened by successive reverse/forward motions along appropriate arcs of travel.
- Back bumper may not cross the curb line. In actuality, the bumper would most likely be able to pass over the curb until the rear tire makes contact with curb.
- Center of car rotation lies on the same line as the rear axle.
- Curb-to-curb turn radius r is the distance between the center of rotation and the center of the outer front tire's outer wall.
- Collision with car parked behind may occur by either: (1) Rear street-side corner of car being parked contacts front bumper of parked car, or (2) Rear bumper of car being parked contacts front street-side corner of parked car
- Mr. White has nothing better to do during the evenings of exam week.
Parallel Parking Mathematics - Attempt 3b (METRIC version, Aug 3, 2010) Amid the conversations with colleagues, we discussed the geometry of shimmying closer to the curb through successive back and forth motions while strategically turning the steering wheel completely to the left or right. The mathematics was much easier for this scenario, and I was able to arrive at an equation (included in the animation) giving how much horizontal/lateral distance d_h can be gained in each iteration. Note that this lateral distance does not depend on the distance from the curb d_c, yet if d_c is made too small, the front tire of the car will cross the curb line at one point in the process. The animation is programmed to indicate when the center of the front tire crosses the curb line. All the parameters are identical to the ones defined above. However, in this scenario, the excess parking space length d is an independent variable that may be varied with a slider, and (alpha) is a dependent variable. Note that in this scenario I've assumed that the car will end be parallel to the curb at the end of each iteration. No claim is made that this constraint is necessary, or even optimal in actual parallel parking situations where shimmying is required. I've had trouble getting these applets to load within this web page reliably. If so motivated, you may download this file and open within the free Geogebra software.
Jerome A White, August 2010, Created with GeoGebra |