Are points and vectors (and normals) different?

I'm teaching a computer graphics course right now. This is an expanded version of an answer to a question posed by both a student and a very smart ta:

Are points and vectors different? We teach coordinate free geometry in our graphics course, so we say yes. The idea doesn't always sit well with people with a strong math background, as opposed to a geometry background.

It really depends on your point of view of the world. In graphics, we need to be careful about how we transform things, to make sure we transform them correctly.

From the mathematical vector space point of view, all points can be represented as vectors, with the implicit assumption that they have an origin at zero. So yes, points are vectors contained in a mathematical vector space.

When we transform geometry, we need to make a distinction between locations in space (points) and directional offsets with magnitude in space ("vectors"). It sort of overloads the terminology a bit, but the direction with magnitude does not change as we translate around, while the location does--this is why we make the distinction. This is all part of the coordinate free geometry point of view. It says that we establish points and vectors as separate entities and make them behave differently with transformations. Then, define the basic operations between them to keep the difference straight. So a location plus an offset in some direction with some magnitude gives you a new location. Normal vectors give you yet a third way of transforming things.

When you write code, you have two choices: make the distinction or don't. If you don't make the distinction, you have one 3-component data type usually called a "vector." And then, whenever you transform stuff, you have to make sure you transform it the correct way, so your code has to keep track what geometric thing you have stored in that vector. So you have to do some kind of type checking and make sure you get it right whenever you write new code that deals with transformations. Something like

if v is a vector:
v' = transformAsVector(v)
else if v is a point
v' = transformAsPoint(v)
else if v is a normal
v' = transformAsNormal(v)

Keeping track of what v is gets way more tedious when you are adding point-vectors to vector-vectors, and so on.

If you make different data types for points, vectors, and normals, you can write one transform operation that does this checking internally or through c++ operator overloading, and you never have to think about it again in your code. And if you implement operations like p+v = p correctly, you don't have to keep track of what geometric thing you have. Geometric algorithms do a lot of point-vector algebra, and keeping track of what's what inside your geometric code is tedious and bug-prone. And you get one line of code to transform something:
v' = transform(v)

We teach the CFG point of view because you need to know the difference to do geometry correctly, it can make your geometric derivations easier to think through, and it can make your code significantly shorter, easier to read, and a lot less bug-prone.

There's also another level to CFG that recommends encoding the space of the coordinates with the point and compensating for spatial difference inside, for example, the point + vector operation. This is rarely implemented (I have only seen it done in the paper below), since most geometric algorithms deal with all their data already in the same space.

In my code, I break one CFG rule. I allow point + point addition. Adding two locations is geometrically meaningless. I have some data filtering code that I want to be type-independent. The filtering, when you think about it, is really doing barycentric combinations of points. But then we're back to the question of "should the filtering code do type checking to call a barycentric combination operation when it encounters points instead of scalars or vectors?" Then the code gets hard to read, and it requires me so hunt down whatever I named the barycentric combination operation when I wrote it two years ago. So points can be added to points, and we still get to write a simple smoothing filter as:
s[i] = 0.5 * s[i] + 0.25 * s[i -1] + 0.25 s[i + 1]
(s is a point sample).


About CFG: A Coordinate Free Geometry ADT. Stephen Mann, Nathan Litke, Tony DeRose. 1997

consumer 3d

I have been pleasantly griping to a number of people lately about the lack of 3d animation software that is approachable to non-professionals. What's interesting is the variety of responses I get back. Academic researchers always point to their favorite paper that has a sketch based interface, but these usually only solve part of the problem (by nature, as they are research projects). Engineers working on 3d software seem to be of two generations: the older group thinks people who want to use it will learn it and that's that. The younger group of engineers knows better, since they had to learn what the older group built, and they know it is not at all easy to learn. Surprisingly, animators seem to argue against the need for it, supporting the notion that you need professional training to be an animator.

Think of apple's ilife software. Pretty much anyone can make something cool with it without any training and without the manual. If you want to go professional, you can buy final cut pro or logic or aperture or whatever the fancy version of photoshop is called. But it won't make you a professional. You still need training to learn the professional skills. Why not for 3d animation? As it stands, you need training just to learn how to look around.

It's a tough problem to make something like this since you have to safeguard people from making bad animation, but the variety of opinions on the idea is interesting. These aren't all-encompassing claims about the groups of people mentioned and certainly do not apply to everyone, but they are trends.

left, right. center slides.
depth and flatness self-contest.
a world emerges.

the internet is no longer a series of tubes

but is it back to being a big truck?

on the nature of questions and steps in the routine of advancement

up left right forward
up forward up left up right
up back up up up

517

sleep
eludes
and
hides


warmth
arrives,
my
warmth
departs

taking
aim,
morning

Palinese

"It's very important when you consider even national-security issues with Russia. It is from Alaska that we send those out to make sure that an eye is being kept on this very powerful nation, Russia, because they are right next to, they are right next to our state."

"What I think Americans at the end of the day are going to be able to go back and look at track records and see who's more apt to be talking about solutions and wishing for and hoping for solutions for some opportunity to change, and who's actually done it."

"Show me where I have ever said that there's absolute proof that nothing that man has ever conducted or engaged in has had any effect or no effect on climate change, I have not said that."

"Let me speak specifically about a credential that I do bring to this table, Charlie, and that's with the energy independence that I've been working on for these years as the governor of this state that produces nearly 20 percent of the U.S. domestic supply of energy, that I worked on as chairman of the Alaska Oil and Gas Conservation Commission, overseeing the oil and gas development in our state to produce more for the United States."

"It is for no more politics as usual and somebody's big, fat resume maybe that shows decades and decades in that Washington establishment, where, yes, they've had opportunities to meet heads of state ... these last couple of weeks ... it has been overwhelming to me that confirmation of the message that Americans are getting sick and tired of that self-dealing and kind of that closed door, good old boy network that has been the Washington elite."

In Palinese, up to 5 distinctive thoughts are typically strung together and the english teachers be damned because...they instill a discipline, and discipline is not for freedom, and freedom to say that your words are juxtaposed and free from discipline and the liberty of that, the liberty of the american people, is to be able to communicate your preferred ideas by using a tent, a tent with a screen, and stuff the thoughts of relevance inside, and create a termination clause that reveals something of emotional regard regardless of the interior of the prior metaphor, for you know that you are not free unless you have the freedom to speak your own mind. Oh yeah, and Russia's right there.

Battlestar Galactica

A few months ago, TH and I started watching Battlestar Galactica. We're a few episodes into the third season, and I'm starting to thinking it might be the best written show I've watched on a regular basis. It used to be Deep Space Nine, then the X Files, then Lost (which many would argue about after a year of button-pushing). And I'm pretty picky with what TV programming I'm willing to watch, since I gave up "watching tv" in the traditional sit-down-and-see-what's-on sense about five years ago. But in terms of telling compelling stories with characters you believe in doing things you believe in, it is light-years beyond even the other excellent shows I mentioned. I'm been thinking about why, since I have a bad habit of analyzing everything. TH says I need to know everything about everything. But to answer why, I think it's in the mistakes. Real people with good intentions make some really bad decisions quite often. Why do we still fight wars, see genocide, let people starve, destroy the environment for cash, etc.? It's not that the old folks "in charge" are bad people. It's usually well-intentioned bad decisions. Miyazaki plays off this all the time in his films with the very ambiguous "villains." By identifying with the villains, you feel the pain of what they do far more than you would if they were your typical "bad guy." BG writers know this. Most of the time, the human protagonists are those doing the most horrid things. And it makes it personal, and it sucks you in. You want them to come out ok, but like the real world, they come out scratched and bruised. So despite space travel by "jumping," characters who consort with manipulative mental antagonists and dialogue that includes the word "frak," you just buy right into the world.

black and white

I hearby declare 6AM as "Insane Kittie Hour."

Autechre

Some shows are shows to be seen. This is one to watch in your head, with your imagination. An hour of abstract rhythm and timbre that heads right and then suddenly trips and turns left, adding and subtracting and building to all-out frenzies interspersed with just enough breathing room. Even with the complete lack of light, for me it's one of the most visually amazing acts to see, as the music feeds the imagination while guiding or pulling it on an hour's long journey.