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Augmented Reality Billiards
scubacuda writes: "Wired.com reports that Columbia's CS Department has designed a new gadget designed to "take the brainwork out of billiards" and help the average player eventually take on professional pool sharks. The Stochasticks consists of a 5-by-10-inch laptop carried in a backpack, a half-centimeter-by-1-inch long lipstick camera and a headset. Pool experts, such as Mike Spinkle, president of United States Poolplayer Association, say that this device makes it easier to visualize the angles."
do you think they'll notice in competitions?
I've been working on a similar project, but for a different application. Our laboratory has concentrated on the design of the user interface and the software infrastructure. After experimenting with indoor AR systems in the early 1990s, we decided to build our first outdoor system in 1996 to find out how it might help a tourist exploring an unfamiliar environment. We called our initial prototype the Touring Machine (with apologies to Alan M. Turing, whose abstract Turing machine defines what computers are capable of computing). Because we wanted to minimize the constraints imposed by current technology, we combined the best components we could find to create a test bed whose capabilities are as close as we can make them to the more powerful machines we expect in the future. We avoided (as much as possible) practical concerns such as cost, size, weight and power consumption, confident that those problems will be overcome by hardware designers in the coming years. Trading off physical comfort for performance and ease of software development, we have built several generations of prototypes using external-frame backpacks. In general, we refer to these as mobile AR systems (or MARS, for short).
Our current system uses a Velcro-covered board and straps to hold many of the components: the laptop computer (with its 3-D graphics chip set and IEEE 802.11b wireless network card), trackers (a real-time kinematic GPS receiver, a GPS corrections receiver and the interface box for the hybrid orientation tracker), power (batteries and a regulated power supply), and interface boxes for the head-worn display and interaction devices. The total weight is about 11 kilograms (25 pounds). Antennas for the GPS receiver and the GPS corrections receiver are mounted at the top of the backpack frame, and the user wears the head-worn see-through display and its attached orientation tracker sensor. Our MARS prototypes allow users to interact with the display--to scroll, say, through a menu of choices superimposed on the user's view--by manipulating a wireless trackball or touch pad.
From the very beginning, our system has also included a handheld display (with stylus input) to complement the head-worn see-through display. This display is specially fitted so as not to interfere with the beards sported by many linux hippies. This hybrid user interface offers the benefits of both kinds of interaction: the user can see 3-D graphics on the see-through display and, at the same time, access additional information on the handheld display.
The theory of relativity doesn't work right in Arkansas.
It says in the story that the developer got the idea from an 'old sci fi flick'. Actually it was an episode of Quantum Leap.
Yes it looks like it will help determine the angles, but not english or speed. A professional player will whoop you every time if you don't have ball control. However, taking the guesswork out of the angles would definitely help someone learn faster.
--
No electrons were harmed in the typing of this post.
Of course when you see an angle you only see a projection of it. For each infinitesimal angle dtheta, you have to compensate perspective by dividing by the triple-product of the vector from the angle to your eye, the vector normal to the plane spanned by the angle, and the vector in the direction of dtheta (all normalized). It you integrate this over the whole visible angle, you can easily evaluate the real angle in your head.
Now they need something to help you physically like the shot up. To use an easier example with drawingI can picture things in my head, but getting them to the paper, the pen has a mind of its own.
Maybe this will help me be able to keep from missing the cue ball so often.
He who knows not and knows he knows not is a wise man. He who knows not and knows not he knows not is a fool.
I already have a way of visualizing angles.. It's called Yahoo Pool. Lots of fun, and takes the shot to shot variance out of the game, so that you can focus on where you SHOULD shoot, rather than on your exact execution. it's helped my real game.
And yes, I know this is a little offtopic. Moderators: Save me from karma hell and mod this up!
this sig has been rated E for Everyone.
Jebara did admit that there are times when the Stochastick could be awkward. "If you're in a Harley Davidson bar where the people may not approve, you may want to be very discreet about wearing a computer aid."
Well no shit. But then again I don't see many geeks going to these "Harley Davidson bars" in the first place.
Also, your chances of meeting girls diminishes even further with one of these things. Leave the Stochastick at home!
The more you know about pool, the more you realize how much more there is to the game than the angles, which are truly the easy part. It doesn't take long to develop your own way to see the angles and sink the balls, but of course you'll still miss because you get the mechanics wrong in your arm, or you miscue because you forgot to chalk, or any number of reasons.
:)
Then there's cueball action... "english" on the ball, top, draw, etc -- not only knowing how to apply various types of english, but knowing when and why. Of course, many times the "best" shot on the table, meaning the easiest shot, I guess, is not the shot you need to make and, depending on the game, it's also a foul. The most common professional game of 9 Ball, for example, requires the player to shoot the balls in numerical order, which means that, after the 3 ball goes, it doesn't matter how good the shot is on the 7 ball, because you have to shoot the 4 ball, no matter WHERE it is. You at least have to touch it, which in that game sometimes requires a fantastic shot.
So, it's no surprise that we have technology that can assist us with the most elementary aspect of the game. And it just proves that there's no substitute for learning the game through experience... not yet, anyway
RP
Connor: The 500 series had cameras on their heads. We spotted them easy. Also they were tethered to a wall, and deadly only at pool.
It seems to me that lining up your shots is the easiest part of the game. Making those shots is where the skill is.
yeah, it seems like a neat idea, but there's no replacement for skill, and experience. I just have trouble understanding why someone would want this? for the amount of trouble, you can spend the time and money working on your shot at a pool bar??
--Keeping the flame wars alive, one post at a time
How's the computer supposed to know that the rails on my table don't "reflect" the ball back at the right angle. Hit at 15 degrees, bounce back at 45...
I can't wait for the "augmented reality pick-up-chicks-in-a-bay" educational series...
My life in the land of the rising sun.
There are 2 different statements here:
a) that "the system" (I'm assuming an average player operating the system here) will be able to beat a pro.
This is rubbish. A crappy player that doesn't know how to handle a billiard stick properly, and still won't be able to shoot properly even with the system. Even if the computer maths are elaborate, too many variables are left to the player skills: spin control, precise control of the force applied to the cue ball, etc.
At best, an average player using the system will get the general direction where to point at and why, but that's about it.
b) It is true that the computer may be able to model accurately the problem, but it doesn't make it a better player than a human for all the reasons in a) and others.
Computer chess is a prime example that machine does not always beat man, and here the difference is even more striking because the system ultimately rely on the skills of whoever is operating the stick. You just can't approximate this margin of error.
This device might be helpful for three-rail billiards or even nine-ball, games where long rail shots are required, but in a game like eight-ball where a player has to control the cue ball in and around a tight ball pattern, it's draw, follow, and English that win the game, not center-ball bank shots. Yeah, making a great three-rail bank shot around the four to drop the three ball in a game of nine-ball is really cool; not putting yourself in the position where you have to make that shot in the first place is even better.
Anyway, when it comes to beating a real pool shark, social skills have more to do with winning than technical skills.
slashdot broke my sig
If you want to cheat, use these laser cues. It works so much better. :-)
It is true, that if you integrate this over the whole visible angle, you can easily evaluate the real angle in your head. However, we do not accept, nowadays, as it happened in the past, that our perceptive world is just the plain result of an encounter between a "naive" brain and the physical properties of a stimulus. Actually, perceptions differ, in quality, from those physical characteristics, because the brain extracts an information from the stimulus and interprets it, according to previous similar experiences.
We experiment electromagnetic waves, not as waves, but as images and colours. We experiment vibrating objects, not as vibrations, but as sounds. We experiment the beards of linux hippies, as they blow softly in the wind. We experiment chemical compounds dissolved in air or water, not as chemicals, but as specific smells and tastes. Colours, sounds, smells and tastes are products of our minds, built from sensory experiences.
They do not exist, as such, outside our brain. Actually, the universe is colourless, inodorous, insipid and silent. Therefore, we can now answer one of the questions of traditional philosophy : Does a sound exist when a tree falls in a forest, if nobody is present to hear it ? No, the fall of the tree only creates vibrations. The sound occurs if vibrations are perceived by a living being.
Information from the environment or from the body itself, is picked up by the sensory systems and utilized by the brain for perception, regulating corporeal movements and maintaining arousal. A sensory system starts to work when a stimulus, usually from the outside world, is detected by a sensitive neuron, the first sensorial receptor. This receptor converts the physical expression of the stimulus (light, sound, heat, pressure, taste, smell) into action potentials , which transforms it into electric signs.
From there, the signs are conducted to a nearby area of primary processing, where the initial characteristics of the information are elaborated, according to the nature of the original stimulus : colour, shape, distance, shade, etc. Then, the already modified information is transmitted to zones of secondary processing in the thalamus (if originated by olfactory stimuli, it is processed in the olfactory bulbs and then directly conducted to the medial area of the temporal lobe).
In the thalamic zones, older data, originated from both the cortex and the limbic system and containing similar experiences, link to the new information, in order to form a message, which is carried to its specific cortical centre. There, the meaning and importance of the new detected stimulus are determined by a conscious process of identification called perception.
Although two human beings share the same genetic and biological architecture and function, perhaps what I perceive as a dintinct color and smell is not exatly equal to the the color and smell you perceive. For example, you may like a certain Thinkgeek T-shirt with a pseudo-pithy statement about emacs on it. I may think it's lame, and that it clashes with your beard. We give the same name to this perception but we cannot know how they relate to the reality of the outside world. Perhaps we never will.
The theory of relativity doesn't work right in Arkansas.
I have agree with those who claim there is more to pool than the angles.
It is hard to image a robot even being able to play the game. Although a robot could be consistently wrong. Not with the angles. But, with the other aspects of the using the stick (chalk, english, even play selection).
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Is the psychology of it. After playing 3-4 times a week for a few years I have found making shots and even english aren't very difficult to learn. The hardest part is keeping a clear mind and finding a routine to fall into to shoot consistently.
But it runs Windows, and is constantly
BSODing my headset, so my game really
hasn't improved much. Really. You can
play against me if you want, but I won't
be much of a challenge.
There are tremendous difficulties that lie ahead. They must map where the cue ball will lie after the shot.
Then... the "real" math comes in: do you take the easy shot with a far more difficult second shot, or do you take a slightly more risky first shot, to be rewarded with a setup for a second and third object ball?
The decision making tree grows large, and different parameters would be selected not just based on the ability of the player, but on his opponent and the current state of the game.
There is much math ahead. This is a neat gizmo, but it trivializes the decision making and abilities pool sharks exhibit just down the street.
Support a few technologists in Washington.
I remembered that as soon as they mentioned AR pool actually. As a matter of fact, I just sent Mr. Jabara an email in regard to this, hehe.
See, they missed the point. The hard part of pool isn't making the balls. It's the strategy. To be able to put yourself in a position to make the next ball and so on. To know when to play safe instead of trying to make the ball at it. It's like chess, anyone can learn to move the pieces, learning the strategy is the hard part.
But probably a whole lot healthier. I had the same (or better) augmentations freely available as early as 1974...it was called "WindowPane", and there were two stuck together...oops.
db
Cig:
ôô
Amateur Physics for the Amateur Pool Player Third Edition
...and it'll all come out right.
--Blair
I've played pool for quite a few years and i think the most importants factor is experience of the game, and thinking ahead for the next few shots. I don't think this Gadget takes into account positioning, or which ball your going to play for next does it? It would be quite useful for someone new to pool, but i think in the long-term they would be better off without it and learn the normal way
Andy
misread the title as "Augmented Reality Balls," realized that i was just making a word association, then thought about how useful big, realistic balls would--
but then i stopped thinking about it.
so should you.
now.
8D
If Mr. Jebara thinks that "within the next five to 10 years the system will be better than the world's best player," he needs to review his high-school physics.
It isn't just the angles at which the balls collide -- it's the how far off center the balls are when they collide, what kind of spin is on the cue ball when it begins the chain reaction, the mass of each of the balls, the felt table's coefficient of friction,.... To compensate for all of these variables will require many more (and varied) sensors than just a camera.
That's the what makes pool different from chess -- in chess, the friction of the pieces against the board doesn't matter. In many respects, pool is the ultimate game of applied physics. But all of the variables combine to make the game as much art as it is science.
Still, it is a pretty cool thing, and will be great for beginners to learn the basics.
While many have searched for an application of "wearable" computers, this sort of specialized system seems like a rather reasonable way that they can be integrated into the lives of a normal person.
Much more reasonable than the headset wearing ubergeek that is...
prosebeforehos.com
If you still have to do the math to figure out where the ball is going to go, you don't stand a chance. Even if you can do the math, there's so many other factors. The speed of the ball determines how far it will go. In addition, it also determines how much bounce you will get off of the rubber sides. A faster ball will rebound at a different angle than a slower one will. English matters. Topspin matters. Draw matters.
I seriously doubt any "professional" needs to do the math anyways. He knows where the ball is going to go and where it will end up and he can do so just by looking at the table.
The math is for beginners.
-Restil
Play with my webcams and lights here
...a world where the tools used to educate and illustrate are no longer small enough to throw at a daydreaming students head?
;)
Or, just possibly, a world where people mispell things that they type into a bloody text box without reviewing.
I dunno, you tell me
- It is simple to make something complex, and complex to make it simple
I am by no means an authority on pool, but have played quite a few rounds, and would like to think of myself as serious competition for most recreational players.
My take on this device:
This will only help a bad player, and there are other thing which would be of more help. I can easily "see" the angles in my head. The hardest parts of the game for this type of player has to do with cue control, estimating how hard to hit (which no coputer can really help you with), and defensive strategy.
1. Cue Control - If I have more than a couple feet in between the cue ball and the ball I am trying to hit, chances are I will be slightly off my mark, as will most other people I have seen play. A fraction of an inch (or cm for international folks) means the difference of the ball you hit going right or left of your intended path.
2. How hard to hit. A computer cannot help you hit the cue with 20psi of pressure. (Or whatever a meaningful measurement is) A cmputer can help you with "soft/medium/hard" hit, but not the exact details of your execution of the shot.
3. Defensive strategy - leaving the cue where it is least helpful to your opponent. This varies greatly by your skill at 1 and 2. I know about what my margin of error is on 1 and two, and can make a educated guess about where the cue will stop based on how I plan to do 1 and 2. In most cases I am luck if I can choose the 1/8th of the table I want the ball to stop in and do it. 1/4th of the table is more realistic. Also have to keep in my what other balls will be hit during the shots, their reactions, etc.
All the stuff this device accomplishes are really things I can do fairly accuratly in my head, and I would think most other pool players would as well. That being the case, I am not really sure the intended market of this device, and how much more it would help them than a basic pool lesson and a check list of think about 1,2,3 when choosing your shot. When you play often enough, or when I do at least, I can make many computations in my head in a couple of seconds and trim down my possible shots to two or three which I analyze in more depth.
-Pete
Soccer Goal Plans
There's a similar product that doesn't require a laptop called SniperCue. I can't see these being used in real games, or especially tournaments. It might be a good learning tool to figure out how angles work, but it's not going to teach you english, and it's not going to teach you how hard to hit the ball.
Every cue is different though, even if they are the same model. If one learns on a cue like this, they are going to have to get the feel of the non-laser/camera cue before being much good. Plus, at over $1000, you could get a nice Schon Limited for less. I'll stick with my Joss, the more I play with that cue, the better I get with it.
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I'm sorry, but the 'a' and the 'u' keys are so far removed from eachother on a standart QWERTY keyboard that the only way they could have been placed together to spell 'chaulk' is by intent. If he'd spelt it 'chakl' or 'chaslk' then the mistake could easily be forgiven.
However, this clearly means that the dumbfuck was trying to spell a simple word by using phonemes. This does not work for much of the English language and therefore he warrants the title 'fucking idiot who can't spell but can use a computer'.
-Nano.
p.s. On the other hand, if our friend is not a native speaker of the English language, then his mistake is easily forgiven, and I pray that I may find myself becoming as proficient in his native tongue as he is in mine.
No matter how good you could play with this thing... You look so darn stupid, you might become the laughing stock of the location you're playing in fairly quickly.
Who cares about angles...I want a device that stops the ball from flying off the table into someone else's face
As opposed to billiards, would feature a high voltage electric shock to the genitals to simulate the welching of a bet in a bad neighborhood (thx Carlin.
As I know nothing about AI or robotics would it be hard to create one to predict the best shot to make?
With the aid of a lap top, any bum off the street will be able to perform brain surgery with the best of them?
For the humour impaired: It's a joke.. Laugh!
-- What you do today will cost you a day of your life.
I could have sworn that the name of it was "Blue Collar Man," or something like that. IMDb doesn't seem to have heard of it. The pool hall scene was the only memorable one.
(As balls fly around the table and into pockets)
Android: Am I to sink the black ball as well?
Stunned Pool Shark: Uh, yeah.
(Eight ball - the last on table - is pocketed)
Android: Good.
Don't blame me, I voted for Durga.
with a laptop on his back. The bartender says: "Hey. Whats that thing?"
The laptop says: "That? That's just a guy on my ass. Where's the pool table?"
- Shecky Jhon
Plagerism
It's clear this generation has not
heard of the joys of billiards hustling.
*sigh*
You're of course correct in that the computer cannot make up for a player who can't follow its directions.
.
However, comparing that to computers playing chess seems a little silly. Computers don't lose at chess because somehow they meant to move the queen but instead moved the rook. In any case, computers will be destroying humans at chess in 10 years.
I can certainly imagine a computer that's better at "the mental part" of pool than any human - and that's what the article is talking about (and this is apparent from the quote)...
Jebara is confident that it will be able to judge the table-top situation more accurately and precisely than the human mind.
.
Let's not stir that bag of worms...
Sounds like the OGC guys and wall hack authors are going into higher level research now. I think you'd call this an "aim bot" right?
I can't wait to have some guy yell, "Yuo be0tch your using a wall hack or an aimbot. NOBODY CAN SHUTE THAT GOOD" when I go down to Squirrels (a downtown learning center) to play some pool.
This is the stupidest thing I've ever heard!!!
It's like the Sprite commercial. "You wanna play all-star basketball? Practice. You want a good friggen soda or whatever? Drink Sprite." I like that one because they're not trying to advertise that drinking Sprite will make you an all-star basketball player or whatever.
So you wanna get good at shootin' pool? Go to a bar that's got pool tables, see, order a pint of Negra Modelo, walk up to the nearest chick who looks about 30 years old, and ask her to shoot pool with you. Go to the pool table, drop a bunch of quarters inside, take a huge gulp of Negra Modelo, and then start playing. Yeah, you're gonna lose, because all 30 year old chicks who hang out at bars kick ass at pool. But play about ten games or so, and when you finish your pint, get another one. As you get progressively more drunk, your accuracy will increase, and your embarassment at sucking so bad will decrease. Do this about two or three nights a week--ask the chick which nights she hangs out there, because all 30 year old chicks at bars hang out at that bar on certain days--and before you know it, you're a pro. No need to cheat.
Oh well... does anybody ever listen to ME?! Not until it's too late. Ooooooooooooh well.
Ah.... I remember my first joint...
...let's hear more about this "5-by-10 inch laptop"!
Downmodding is the refuge of the weak. Don't downmod, make a better argument!
Hmmm... Is this going to make it any easier to hit the stupid ball in a straight line? Or to control the spin? Or to not jump the table? Some people, namely myself, are more than capable of visualizing the angles and recognizing what needs to happen, but then are unable to follow through because they can't hit the stupid ball right. I mean c'mon, the damn thing is just sitting there, not moving! Why can't I hit it? Is someone going to make a computer that can help people with that?
Lack of eloquence does not denote lack of intelligence, though they often coincide.
Making the balls in the pocket is a simple task mastered quickly by anyone with patience, reasonable vision and basic motor skills. My brother makes custom cues, many of the pro players drop by the shop and every last one of them refers to "seeing blocks" not balls when they shoot. The idea being that they can make most every reasonable shot on the table with out much concentration under normal conditions. Shape or placement of the cue ball for the next shot is the issue. Along with not distrubing secondary balls while getting shape or breaking out a trouble ball and getting shape. Knowledge of how the envrionment affects the table is another issue, play on a table near the ocean with high humidity and you have to slam the cue ball to get it to move. Go to a table in a low humidity, cool room and you need a light touch. How about low light, smoke filled low visibilty rooms? Watch an "A" grade player at nine ball and he will know where every ball is going to be played and the shape needed to get there for each shot before he strikes the cue ball for the first shot. Now for serious play the issue then becomes making a computer aided shot when your tuition, car payment or last five bucks are on the line (if you are like my brother he might have been hustling you with no money in his pocket, which takes balls of steel). And of course someone already brought up the biker bar scenario, any place a pool table exists someone will know 1 of a million ways to try and break your concentration as you stroke the ball. Please bring your computer aid to the local pool hall, somebody will hustle you out of it and I can probably find that 5 X 10 laptop at a pawn shop down the street the next morning (God knows a pool player want be able to use it)!
A word of advice, if some guy offers to cut a ball from center spot to center spot in the corner pocket, not scratch and do it one handed don't take the bet. But, for those that have thanks for paying for my undergrad degree!
Jeez, people get pissed off enough at the use of aimbots and such in online games...
All we need is l33t d00dz in real world sports. Eick.
- SMJ - (It's not just a name: it's a bad aftertaste.)
It would be interesting to combine the repeatability of an Iron Willie robotic arm, and the laptop for number crunching, with a feedback loop to allow for corrections (an iterative learning process). Now *that* might be able to shoot (basic and rudimentary) pool. However, without first learning the basics of forming a bridge, point of aim, stroke, etc... the use of a computer by a novice pool player will be of little to no help.
I'm an individual! Just like everyone else!
... it's as hard to learn as the bloody language. (And no matter how much you know, you could probably do better)
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See also my previous warning about Lego Lawyers. http://slashdot.org/comments.pl?sid=32953&cid=3574 059
Disclaimer: I don't agree with these heavy-handed guys.
It talks about slate (and non-slate) surfaces, the table cabinet, rales, cushions, felt, variations of play, and table specifications.
Very informative!
The Stochasticks consists of a 5-by-10-inch laptop carried in a backpack, a half-centimeter-by-1-inch long lipstick camera and a headset.
And NASA wonder why their Climate Observer landed a little hard?
Yeah, it's bloody likely that "Jorge Martins de Oliveira, MD, PhD. Full Professor and Master of UFRJ", author of the text, is the same guy as "User 956", whose Slashdot Bio consists of a quote from Arthur Schopenhauer (and nothing else), and whose Homepage consists of two images, the gotse.cx guy and another porn pic (and nothing else).
Lars T.
To the guy who modded me down from perfect to terrible Karma - Apple haters still suck
You should try snooker. The table is much bigger, the balls are much smaller and you are more limited when it comes to choosing which ball to sink.
Still, the only way to get good at any billiards game is to misspend your youth.
All things in moderation; including moderation
The software leaves out the most thought involved aspect of pool, the leave. Figuring out how to angle shots is something that you learn to intuit pretty quickly, especially if you're learning from someone with enough physical skill to put the ball where they visualize it. The part of the game that involves the most thought is deciding where you want your ball to end up once you've finished your shot. Ideally you want to set yourself up for making another shot. This often means shooting a more difficult, less obvious shot first. At the same time, you have to ask yourself, "If I don't make this shot, how can I make sure that I don't leave my opponent with an easy shot." Again, this often means taking a less obvious shot, or putting a little extra back spin on the ball so that it rolls back a few more inches. As with chess, the better you get, the more "moves" in advance you try to set yourself up for. You also start to realize that you need to understand your opponent's playing style so that you can accuratly guess at what shots they might play and try to leave them in a difficult position for those shots. Think of it as a chess board, but instead of the pieces being locked into a nice 8x8 grid, they have continuous positioning throughout the entire board. Now, it's probably possible to write a program that will take all of this into account and try to predict what shots will be good to play. I think it'll be years before any such program is more than halfway decent though, and it will certainly have to be more than just an angle visualizer.
Driving directions with arrows that appear directly on the road.
Virtual name tags. A persons name could automatically appear over their head when you meet them.
A working light saber!
When you superimpose images on top of reality, you could do anyting!
My Blog Sucks.
That really isn't fair. Making a spelling mistake is definitely not a sign of stupidity. In snooker, there is a word "baulk," so really is it such an awful sin to spell "chalk" as "chaulk"? I think it looks nicer myself.
Anyway, you understood him, and isn't that the point of language to begin with?
I worked for a company called Miacomet which made the Real Feel PoolShark. it was the first PC controller that allowed you to play a game on your computer with a real cue. It was basically a modified mouse with a roller on top where you place the cue. It couldn't do English or top or bottom shots though. As a poolplayer I thought it was pretty cool, and it was definitely fun to play around with. These types of devices, however, will not make you a better pool player. Visualizing the angles is something that does come with practice, and once you put the headset away, are you still going to know the angles in your head? They change with every shot of the ball. Miacomet basically went bankrupt and laid us all off, I don't know about the commerical prospects of this application.
Regular Meta Moderators are not more likely to get mod points.