
Kinematics of Reverse Angle Parking - mourner
https://observablehq.com/@mourner/kinematics-of-reverse-angle-parking
======
jws
For those in regions without reverse angle parking (like me)…

It's used in areas where you might have roadside parking along a road, either
parallel or angled. It has higher linear density than parallel and is
easier/faster. Compared to forward angled parking it removes the "backing out
into traffic you can't see because your rear pillars and the hausfrauenpanzer
next to you have completely blocked view of the oncoming traffic" (so you go
slow and figure any drivers coming down the road don't want to die today
either). You safe the road by stopping to park, and have road visibility as
you back in. Coming out you get better visibility because your upstream
neighbors corner is further from the traffic and your eyeballs are also
closer. An added bonus is that bicyclists have very little reason to ride in
the zone of opening doors, compared to parallel parking.

Some places it is popular. Changing to it from forward angled makes people
crabby.

[https://en.wikipedia.org/wiki/Back-
in_angle_parking](https://en.wikipedia.org/wiki/Back-in_angle_parking)

~~~
jessriedel
What's the advantage over back-in perpendicular parking? I don't actually
think your eyeballs are closer to the road (since the diagonal to your front
corner is longer than the line to your nearest headlight), and you can always
force neighboring cars to be further away by just painting the lines wider
apart.

At first I wondered if the road can be slightly narrower while still allowing
one to get into the parking space, but if you look in the OP article at how
the orange envelope is shaped by the way the front end swings around, I don't
really think that's the case.

In general, I'm confused about why angle parking is picked over perpendicular
parking. It's significantly less linear density (reduced by the factor
cos(theta)), but the square shape of cars means the road width occupied by the
parking is hardly less. (The wasted space is the triangular patches between
the parked cars and the road/sidewalk.) Parallel parking has even lower linear
density, of course, but takes half the road width.

~~~
leetcrew
I've never seen perpendicular parking outside of a parking lot. one immediate
issue is that many vehicles will not have a sharp enough turning radius to
make a right angle turn from a single width traffic lane into the space. they
will have to make a multiple-point turn and/or cross into the far lane
(possibly with opposing traffic) to get into the space. I can't imagine how
this wouldn't be a huge obstruction every time someone parked something larger
than a honda fit.

~~~
jessriedel
I see perpendicular parking pretty often. It's probably at least as common as
angled parking in San Francisco (although obviously much less common than
parallel parking).

[https://www.flickr.com/photos/ricardipus/2575893591/](https://www.flickr.com/photos/ricardipus/2575893591/)

~~~
scythe
Perpendicular parking on SF’s steep streets usually has huge backouts and is
necessary due to the high pitch.

~~~
jessriedel
I see perpendicular and parallel parking on the same street, so I don't really
understand the necessary claim.

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tom_
I need to sit down and work it out on paper, but as a long standing reverse
parking fan - I can't park forwards, as it's too difficult - I don't think the
diagram shown here is quite right. When going backwards at low speeds, the
fact there are two rear wheels is very important! When reversing when turning
fully left, the left rear wheel stays pretty much where it is; when reversing
when turning fully right, the right rear wheel behaves that way.

OK - well, this is of course not _quite_ true, because the car is moving, and
so it's not quite rotating around a fixed point. But it's close enough that
you can pretty much start your reverse parking manoeuvre by putting one wheel
in the right place, so its contact point is lined up with wherever it
ultimately needs to be - but it doesn't really matter which angle, provided
it's within 45° or so. Then turn full lock in that direction (left for left
wheel, right for right wheel), on the spot, and slowly reverse until the other
wheel is lined up too.

Now you're done. Straighten the front wheels and go in backwards.

It's handy to bear in mind the swept volume, as there will be some overhang,
but when going in backwards with this technique you don't typically have to
worry about it too much.

(Making the wheels turn on the spot is not the best thing, but you can fix
this once you're confident with the basic manouevre.)

~~~
saalweachter
There's a chance I may hate you.

I park in a standard parking garage, with forward angle parking and one way
traffic flow. I occasionally encounter vehicles reverse parked, which confuses
me, but whatever, but then also encounter vehicles driving backwards (well,
forward facing, going against the posted signage) through the parking garage.
This is always a big headache, because again, this is a one way parking
garage, and for some reason THEY never seem to think they should back up or
get out of the way of the person driving the correct way through the garage,
so I always have to do some extra maneuvering and wait for them to pass, and I
usually only have a small margin before the train arrives, and--

Assuming they are not simply idiots who can't follow the signs, my next
assumption would be that these are the people who are reverse-parking, looking
for a spot to back into.

Anyway, if you have difficulty making the turning radius to pull straight into
a forward angle parking spot, there is a very simple trick: make a three point
turn, attempting to pull into the spot to the _right_ of the empty spot, then
when you are almost to the bumper of the parked car, back up and straighten
out, which puts you at the perfect angle to drive straight into the angled
spot now in front of you.

It beats the pants out of getting halfway into the spot before realizing
you're about to hit the car on one side or the other and trying to straighten
out after the fact.

~~~
gowld
Presuming tom_ isn't a criminal, I'd hazard that the reason tom_ likes
reverse-parking and thinks forward barking is hard, isn't because tom_ drives
the wrong way down one way streets, but because tom_ has trouble parking
without the extra lane worth of turning radius.

~~~
fyfy18
I find reverse parking easier to reason about spatially. Ok I do have a backup
camera, but even without that you can roughly see where the back of the car is
in the mirrors. With the front of the car I have absolutely no idea at what
point I should stop (meaning the bottom of my front bumper is rather
scratched).

------
chrissnell
I learned to reverse into spaces in the military. That's the way that all
military vehicles are supposed to be parked. I started doing this in my
civilian vehicles and found that once you get the hang of it, it's absolutely
the best way to park a car. Once you have the skill--and spatial awareness--
you will find that it's much easier to get straight into a space with this
method. Leaving the space is obviously easier, too.

I drive a large Ram pickup truck and backing in is pretty much the only way
that I can park it accurately.

------
Animats
Amusingly, I have to solve this problem for a minor game project. Drivers have
to back their trucks into a loading dock. Players aren't good enough drivers
to do this without assistance, so I have to write an assistance system that
yells "Left", "Right", "Straight", "Slow", "Stop", etc. as appropriate.

------
gugagore
This page by the same author is also fantastic:
[https://observablehq.com/@mourner/non-robust-arithmetic-
as-a...](https://observablehq.com/@mourner/non-robust-arithmetic-as-art)

~~~
mourner
Thank you! A fellow programmer Jacob even printed that viz as a woven textile
— floating point errors can be very pretty :)
[https://twitter.com/jashkenas/status/1191829600214507520](https://twitter.com/jashkenas/status/1191829600214507520)

------
shkkmo
What's missing from this model is a variable rate of turn. You want to turn so
that your front bump stays at roughly a fixed distance from the bollard line.
As you go, you slowly turn the wheel sharper and sharper until reach your
maximum turn radius or you are lined up with your space. You can slightly vary
that fixed distance as you go to adjust for minor differences in your initial
position. You can also aim for a bit to the right/far side of the space (, and
then re-center on the space once you are lined up.

~~~
mourner
Yeah, the model is definitely contrived and I could try adding a lot more to
it, but I still found it very useful just to understand how the path changes
with the wheel turn.

Also, as a beginner, it's very hard to "feel" how much you're supposed to turn
with variable rate — I guess this only comes with a ton of practice. In the
driving school, the way we were taught all the parking maneuvers is to reverse
straight until you catch some reference point, then immediately turn to the
max for the sharpest turn.

------
exabrial
Given the example show, I would make this a much easier problem by pulling
across the center line onto the opposite side of the street, then steering
back towards the right hand side. I would time this so I'm parallel with the
lines. Now it's a simple back-in job since you're already lined up.

This may or may not be a legal traffic maneuver where you live and obviously
can only be performed with both lanes clear.

~~~
mourner
There's just one lane here — the thick line at the top is a barrier (a line of
bollards).

~~~
massysett
Even so, the thing to do here is first turn the wheel to the right and pull
past the parking space and pull up to the bollards, then back in.

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itronitron
I don't think I have ever seen a parking area with reverse-angle parking. Any
angled parking usually has signs to make it clear that cars must park forward
facing only.

Having said that, it would be nice to see this adapted to parallel parking
since that is a much more common parking scenario.

~~~
mourner
Observable makes it easy to fork notebooks and make modifications, so I'll be
happy to see other parking arrangements!

I made the viz for my real-world situation — reverse angle parking is quite
common where I live (Ukraine), and it's the type where we got our parking spot
(took a long time since all the spots around my apartment complex are sold
out). [https://en.wikipedia.org/wiki/Back-
in_angle_parking](https://en.wikipedia.org/wiki/Back-in_angle_parking)

------
Bigpet
It's interesting how quickly analytically solving this problem gets almost
impossible (the problem being 'given a start position and a desired end
position, how do you maneuver there). I've had a lecture that was basically
just about this class of problems.

But it was concerning trucks and how adding additional axes and especially
adding even one trailer joint (but especially two) made solving this
conventionally impossible.

With some funky transformations into other number-spaces there were some
analytic solutions for it though.

~~~
mourner
Yep! It can be solved for some specific cases though — e.g. I stumbled upon a
few papers that calculate the optimal path for parallel parking using systems
of differential equations [1][2]

[1]
[https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=7463491](https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=7463491)

[2]
[http://www.tsi.lv/sites/default/files/editor/science/Researc...](http://www.tsi.lv/sites/default/files/editor/science/Research_journals/Tr_Tel/2012/V4/zadachyn_dorokhov.pdf)

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PhasmaFelis
This may be obvious, but for me, reverse parking got a lot easier once I
realized that (on my Mini Cooper) if I pull forward until my B-pillar is
exactly aligned with the next dividing line past the space I want to enter,
then I can reverse into it almost without looking. (Does that make sense? I'm
not great at describing visual concepts.)

Presumably there's a spot like this on every car; you just have to figure out
where it is.

~~~
ferongr
You should use the rear axle as a guidline, not the B-pillar, it's more
accurate, and it's how a 2-point reverse turn is taught to be done by driving
instructors here.

Driving exams involve performing a 2-point reverse turn with the rear inside
wheel not going further away than 1 meter from the kerb, no more than two
stops (except for traffic/safety related reasons), no more than one engine
stall, no forward movement and no contact with the kerb itself.

~~~
PhasmaFelis
> _Driving exams involve performing a 2-point reverse turn with the rear
> inside wheel not going further away than 1 meter from the kerb_

Which curb? I'm mostly parking in lots with perpendicular parking on both
sides of the lane. The only curb involved is the one at the back of the space
I'm backing into (unless there's a pull-through from the other side, then not
even that).

~~~
ferongr
The "inside" one obviously. The 2-point reverse turn is more designed to test
your skill into reversing outside a blocked street back into a perpendicular
street, but the skills involed apply to reverse angle parking too.

------
Stratoscope
For anyone in the Palo Alto area who wants to try this out in person, the
first block of Stanford Avenue next to the Dish trail entrance converted to
reverse angle parking a while ago:

[https://www.google.com/maps/@37.4110349,-122.1605769,294m/da...](https://www.google.com/maps/@37.4110349,-122.1605769,294m/data=!3m1!1e3)

------
war1025
The key to reverse parking (parallel parking is a bit more complex) is to make
good use of your side mirrors and the fact that if you can see a cap between
your car and whatever the object is, then you're on course not to hit the
thing. Then it just becomes a matter of course-correcting your way to where
you want to be, which is pretty easy after a little practice.

~~~
clarry
Where does the cap come from?

I assume you mean gap, but when you start reversing and are not aligned with
your parking spot yet, the thing you'll soon see in the rear mirror is the car
parked in the next slot. No gap.

Idk about other people but I feel like I've got very bad depth vision in
general, and telling how far my car's rear is from a wall or another car when
looking through the mirror is very difficult.

~~~
war1025
When you are not aligned and you don't see a gap between your car and the
thing next to you, that is the signal that you need to turn more in order to
get where you want to be.

Regarding how far back to go, I find that it's the same problem as parking
forward. You just need to figure out how to judge some reference point on your
car with whatever is around you. I agree that's one of the trickier things to
figure out.

~~~
clarry
Yeah, the problem with turning is that you don't exactly turn in place, and if
you're too far backward, you'll back into the thing you're trying to turn next
to..

Forward parking sure has the same issue but for me at least it's much easier
to assess depth right in front of me than 2-3 meters behind, through a little
mirror. And I mean I can park in front of a wall and leave a 15-20 cm gap
between the bumper and wall without much trouble, but it is seriously
difficult for me to say whether there's half a meter or one and a half between
my rear and the next object. I guess they invented reverse cameras for a
reason, I just haven't bought one.

~~~
war1025
I think depth perception for the rear of the vehicle is mostly a matter of
practice. If you can develop spatial awareness of what is in front of you,
developing it for what is behind you is just a matter of trial and error.

------
csours
I was sure this would be related to Matt Parker's recent video on back-in
parking:
[https://www.youtube.com/watch?v=0rxghexCKj8](https://www.youtube.com/watch?v=0rxghexCKj8)

~~~
mourner
Nice video! I only saw it a few hours ago when I was about to post my viz on
/r/math :)

------
allovernow
Slightly related - when reverse parked and pulling out of a 90 degree parking
spot with the wheel turned, is it possible for the rear outside corner of the
vehicle to clip an adjacent vehicle? In other words, if you were reverse
parked, pulling out turning left, and you plotted the track of the outermost
portion of the right rear fender/bumper, would it ever cross the rightmost
edges of the wheelbase in the parked position?

~~~
mourner
Good question! Since the model is reversible (you would have the same
trajectory when pulling out of the parking lot when reversing the steps), you
can see the effect there — seems like it would cross the rightmost edges but
very slightly (like 10cm) in my case, but protrusion would be bigger with a
car that has a bigger rear overhang (e.g. sedan rather than hatchback). You
can try it out by tweaking the constants in the car specs section.

------
mhb
Control Schemes of Steering System of a Multi-axle All-wheel-steering Robot:

[https://news.ycombinator.com/item?id=21893476](https://news.ycombinator.com/item?id=21893476)

~~~
mourner
Awesome! This paper uses the same model for calculating wheel angles
(Ackermann steering geometry).

------
gwbas1c
FWIW: A good backup camera helps you visualize this. Both of my cars have
lines that curve as I turn the wheel.

~~~
mourner
Yep, mine has it too and it's super helpful! What I have more trouble with is
feeling when the right front side is about to collide when the car sways in
the other direction — there is some beeping proximity detector but it doesn't
seem to work well against bollards.

------
barrkel
Why is the interactive diagram on this page so much slower in Firefox than
Chrome?

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_pmf_
That's great.

