transforms

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I was just answering a question for the Udacity Interactive Graphics MOOC. I had made a rather confusing lecture, much more involved and less informative that I would have liked, so today I wrote a re-do (sadly, it’s not easy to make a new video, since step 1 is “fly from Boston to San Francisco”). I’m still not thrilled with my description – what do you think? Is there a better way to talk about this subject? Anything I could improve? Surprisingly, this course still gets about 35 sign-ups a day (though I’m guessing maybe one of those actually finishes), so it’d be nice to make this lesson better.

Background: up to this point in the course I’d been showing how you typically write down transforms from right to left (OpenGL-style column-major matrices), e.g. “TR” means “rotate the object, then translate it (in world space) to some location.” In this lesson I wanted to point out that you can also read the transform order from left to right.

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You’re at 41 Avenue George V in Paris. Someone comes up and asks “How can I see the Arc de Triomphe?” You tell them, “Go up two blocks and then turn to the left – you can’t miss it.” Indeed, at 101 Avenue des Champs-Élysées he can see L’Arc de Triomphe.

So if you wanted to take this person and apply these two transforms, translation T (walk two blocks) and rotation R (turn about 60 degrees to the left), how would you write that out? Think about it for a minute, then scroll down for the answer. (And I like the disembodied arm to the right from Google’s street view).

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The order is (right-to-left “application order”): TR. That is, you want to apply the rotation first, so that it doesn’t affect the translation. So you rotate the person 60 degrees to the left, then you translate him north two blocks north, which is then not affected by the rotation.

If you incorrectly used order RT, you would first translate him north two blocks, so far so good. But, as you saw in the snowman lesson, rotating after translation means the object is rotated around the origin from his present location; in this case, the person’s starting location is the origin. So performing a translation, then a rotation, would move him up two blocks north, then rotate him in a circle with a 2 block radius by 60 degrees, putting him somewhere else in the city (Rue Euler, I guess, which is a great coincidence that it’s named for a famous mathematician).

I hope you accept TR is the right order, then. But, to describe directions we definitely first said “perform T” – walk two blocks north – “then perform R” – rotate to the left 60 degrees. So we talk about directions in a left-to-right fashion. This may seem odd, as we are then describing the last transform that we apply, T, if we actually want to position the man in his environment.

The key thing here, and the point of the lesson, is that by specifying T first, we’re saying to the man, change your frame of reference to be 2 blocks north. From this new frame of reference, then rotate 60 degrees around where you’re standing, your new origin. It’s how we talk about directions. We don’t say “when you get to your final position, rotate 60 degrees left. Then, to get to your final position, walk two blocks north.”

The person walking has his own frame of reference, where he’s always the origin, and rotations are done relative to whichever way he’s facing at the time. To specify transforms when talking in these terms, an object-centric way of describing things, we describe “from left to right.” When we’re looking at the world and want to think how to make some other object take on a particular orientation and position, we tend to work from right to left, getting it oriented and them moving it into position.

However, it all depends. Moving a couch up a flight of stairs, down a hall, and next to a wall in room is a series of transforms, and again we specify them from left to right. We could also shortcut the process if we don’t care about the intermediate steps along the way. Say the couch is facing north, and we know it’ll end up facing east. We could specify the one 90 degree rotation to get it to face east, then the one XYZ translation to move it directly to its desired location – right to left order, so that the rotation doesn’t interfere with the translation.

The final effect of the transforms – a series of moves or the direct rotation and translation – have the same final effect. The point is, each way of thinking has its uses.

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With the holidays upon us, it’s time to hack! Well, a little bit. I spent a fair bit of time improving my transforms demo, folding in comments from others and my own ideas. Many thanks to all who sent me suggestions (and anyone’s welcome to send more). I like one subtle feature now: if the blue test point is clipped, it turns red and clipping is also noted in the transforms themselves.

The feature I like the most is that which shows the frustum. Run the demo and select “Show frustum: depths”. Admire that the scene is rendered on the view frustum’s near plane. Rotate the camera around (left mouse) until it’s aligned to a side view of the view frustum. You’ll see the near and far plane depths (colored), and some equally spaced depth planes in between (in terms of NDC and Z-depth values, not in terms of world coordinates).

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Now play with the near and far plane depths under “Camera manipulation” (open that menu by clicking on the arrow to the left of the word “Camera”). This really shows the effect of moving the near place close to the object, evening out the distribution of the plane depths. Here’s an example:

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The mind-bender part of this new viewport feature is that if you rotate the camera, you’re of course rotating the frustum in the opposite direction in the viewport, which holds the view of the scene steady and shows the camera’s movement. My mind is constantly seeing the frustum “inverted”, as it wants both directions to be in the same direction, I think. I even tried modeling the tip where the eye is located, to give a “front” for the eye position, but that doesn’t help much. Probably a fully-modeled eyeball would be a better tipoff, but that’s way more work than I want to put into this.

You can try lots of other things; dolly is done with the mouse wheel (or middle mouse up and down), pan with the right mouse. All the code is downloadable from my github repository.

Click on image for a larger, readable version.

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Try it out (you have to have WebGL enabled etc.)

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I made this demo as a few students of the Interactive Graphics MOOC were asking for something showing the various transforms from beginning to end.

It’s not a fantastic demo (yet), but if you roughly understand the pipeline, you can then look at a given point and see how it goes through each transform.

It’s actually kind of a fun puzzle or guessing game, if you understand the transforms: if I pan, what values will change? What if I change the field of view, or the near plane?

I’d love suggestions. I can imagine ways to help guide the user with what various coordinate transforms mean, e.g. putting up a pixel grid and labeling it when just the window coordinates transform is selected, or maybe a second window showing a side view and the frustum (but I’m not sure what I’d put in that window, or what view to use for an arbitrary camera).

I’ve been bumping into limitations of three.js as it is, but I’m on a roll so that’s why I’m asking.

 

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