Scaling the matrix

[N. W. Perry]

On to the next of the topics I’m uncovering as I delve further into the world of LDraw!

I was building a Technic model and discovered that one of my beams didn’t quite fit onto the pins it’s supposed to—it needed to be “stretched” just a bit (as is common with Technic models with odd angles and connections). I compared my model to another user’s, and I noticed that in his, the beam fit correctly even though all the geometry was the same in both models. I discovered that he had scaled some of the values in the matrix; I copied those values to my beam, and it worked!

So the question simply is, when this situation arises, how do I determine which values to scale in the matrix? Typically I’ll just need to make a part longer or shorter or taller by a small fraction. Can I do this interactively in LDCad, or do I just calculate scale factors based on the distance I need (and how do I know which boxes are the correct ones to change)?

[Johann Eisner]

On to the next of the topics I’m uncovering as I delve further into the world of LDraw!

I was building a Technic model and discovered that one of my beams didn’t quite fit onto the pins it’s supposed to—it needed to be “stretched” just a bit (as is common with Technic models with odd angles and connections). I compared my model to another user’s, and I noticed that in his, the beam fit correctly even though all the geometry was the same in both models. I discovered that he had scaled some of the values in the matrix; I copied those values to my beam, and it worked!

So the question simply is, when this situation arises, how do I determine which values to scale in the matrix? Typically I’ll just need to make a part longer or shorter or taller by a small fraction. Can I do this interactively in LDCad, or do I just calculate scale factors based on the distance I need (and how do I know which boxes are the correct ones to change)?

Hi
[color=unset]Yes that's the problem with the real world and the digital world.
There are no kine tolerances in our digital world.
I would keep my fingers off scaling a bar.
Since scaling affects the same part over and over again, unless you add the scaled bar as a custom part.
Personally, I move the bars a little bit, so it looks like it fits exactly.


Regards Johann[/color]

[Travis Cobbs]

Hi
[color=unset]Yes that's the problem with the real world and the digital world.
There are no kine tolerances in our digital world.
I would keep my fingers off scaling a bar.
Since scaling affects the same part over and over again, unless you add the scaled bar as a custom part.
Personally, I move the bars a little bit, so it looks like it fits exactly.[/color]

I'm of two minds about whether or not this a good idea, but N. W. Perry isn't proposing to do either of the things you say. The proposal is to modify the matrix used in the type 1 line that specifies the bar to make it so the bar is slightly longer (or shorter) than reality. That doesn't affect any other instances of the same part, nor does it introduce a scaled custom part. LDView will actually produce a warning (Part transformed non-uniformly) if the scaling is more than a very small amount. But this might actually be a good solution to the problem of digital LEGO models not having the play that real life models have.

[Johann Eisner]

I'm of two minds about whether or not this a good idea, but N. W. Perry isn't proposing to do either of the things you say. The proposal is to modify the matrix used in the type 1 line that specifies the bar to make it so the bar is slightly longer (or shorter) than reality. That doesn't affect any other instances of the same part, nor does it introduce a scaled custom part. LDView will actually produce a warning (Part transformed non-uniformly) if the scaling is more than a very small amount. But this might actually be a good solution to the problem of digital LEGO models not having the play that real life models have.

Sorry, I [color=unset]misinterpreted N. W.
[/color]I thought he would change the .dat file.

[N. W. Perry]

Sorry, I [color=unset]misinterpreted N. W.
[/color]I thought he would change the .dat file.

Oh, no, I wasn't proposing to change the part itself, just the one instance of it within the model. I've already tested it and it doesn't affect any other instance of the part.

If I were to change the part itself, I certainly would save it as a separate custom part. I've actually done this in the past: for example, in the 950 Fork Lift, there is a stack of seven 2x2 bricks that comes up 4 LDU proud of the connection it's supposed to make with some vertical beams. I created a custom part of a 2x2 brick that is 1 LDU too short, and mixed four of those with three normal ones to make the connection. (This was easier than distributing the 4 LDU error evenly across seven bricks, and is visually imperceptible unless you really know what to look for.)

I tend to agree with Travis that scaling a part, whether ad hoc or as a custom part, tends to feel somewhat "illegal", but then again it's true that real-world parts do have not only some physical tolerance between them, but also a certain amount of flexibility that has to be accounted for if you want to accurately represent them in a model—particularly if you want to do photo-realistic rendering. Applying the matrix on a case-by-case basis, as long as we're talking about small fractions of adjustment (and we are, since the real-world tolerances only go so far), does seem to me the most elegant solution. I certainly would rather stretch a single part by a tiny amount then try to bend all the surrounding bricks to make it fit!

So can I infer from that "illegality" that there aren't any tools or editors for figuring out the matrix adjustment within LDraw proper? For example, in Studio/Part Designer, which operates a little outside of the basic LDraw standard, I can free scale a part by dragging some handles, or by typing in precise dimensions, but I'd have to know what dimensions to enter. I could calculate those by using LDCad's distance measurement, I suppose, but I'm also morbidly curious about which boxes of the matrix actually control scaling along which axes of the part. (Most online articles about rotation matrices are either highly technical, or along the lines of "this topic is extremely mysterious; we can't really explain it so you'll just have to figure it out by trial-and-error.")

[Philippe Hurbain]

This is actually very easy with LDCad (each time I try something new with it, I'm amazed by its flexibility!).

Lets start with this example: two pins placed 7 studs apart in two directions almost fit a diagonal beam 11 since 7²+7² ~ 10². (Just tested, it works well IRL!)

   

Distance of holes in the beam is exactly 200 ldu, while distance between pins (as measure by LDCad) is only 197.99 ldu. So we need to scale beam by 197.99/200 = 0.98995 (here it could also be calculated as 2x7²/10² !)

Select the beam and press "o" to use relative grid. The great thing is that in this mode, the part that was used as reference has a very simple "identity" matrix, all 0 except on the main diagonal where values are 1. These "1" are actually the scaling of the part in relative x/y/z directions.

   

Since the length of the beam is oriented along blue Z axis, we'll need to change that third "1" on matrix diagonal.

Right click again on the beam, choose properties, change Z scaling.


  scale3.JPG (Size: 42.59 KB / Downloads: 152)

...and now the fit looks perfect! (after re-snapping the part properly)
   

BIG WARNING: When you insert a new part in LDCad, it inherits the orientation of last previously selected part... but also its scaling! This can be very misleading (I was caught by this one while preparing this tutorial!)

[Roland Melkert]

BIG WARNING: When you insert a new part in LDCad, it inherits the orientation of last previously selected part... but also its scaling! This can be very misleading (I was caught by this one while preparing this tutorial!)

I never intended it to work with scaling, so basically all editing code assumes it's non scaled.

That said, as scaling is just an 'rotation extension' it should work in most places

[Philippe Hurbain]

I never intended it to work with scaling, so basically all editing code assumes it's non scaled.

That said, as scaling is just an 'rotation extension' it should work in most places

I didn't meant this as a criticism, I consider this behaviour as logical. But one must pay attention... Anyway, a small rotation could give the same kind of unwanted issues...

[N. W. Perry]

The great thing is that in this mode, the part that was used as reference has a very simple "identity" matrix, all 0 except on the main diagonal where values are 1. These "1" are actually the scaling of the part in relative x/y/z directions. […] Since the length of the beam is oriented along blue Z axis, we'll need to change that third "1" on matrix diagonal.

Aha—so the important thing to know is that all scaling operations happen in the boxes along that diagonal. X-scaling in the top left box, Y-scaling in the center, and Z-scaling in the bottom right. The other boxes get used for rotations (and other functions like shear which we wouldn't be using), so as long as I'm using the part's local orientation (or the part isn't rotated), I'll only ever need to change one box for scaling along a single axis.

And am I accurately summing up what a matrix is to say that its 3 columns represent the x, y and z coordinates of each point being transformed, while its 3 rows represent movements of those coordinates along the x, y and z axes? And that's why, when we scale along the z axis, we're just taking each z coordinate (third column) and moving it a relative amount along the z axis (third row)? Or something like that…

[Owen Dive]

Aha—so the important thing to know is that all scaling operations happen in the boxes along that diagonal. X-scaling in the top left box, Y-scaling in the center, and Z-scaling in the bottom right. The other boxes get used for rotations (and other functions like shear which we wouldn't be using), so as long as I'm using the part's local orientation (or the part isn't rotated), I'll only ever need to change one box for scaling along a single axis.

And am I accurately summing up what a matrix is to say that its 3 columns represent the x, y and z coordinates of each point being transformed, while its 3 rows represent movements of those coordinates along the x, y and z axes? And that's why, when we scale along the z axis, we're just taking each z coordinate (third column) and moving it a relative amount along the z axis (third row)? Or something like that…

No - at least, not necessarily. Rotate a part about some arbitrary axis and you'll see all 9 numbers in the matrix change.

I don't remember what the mathematical terminology is (and I may even get some of this completely wrong!) but it goes something like this:
Imagine you have a part and you want to describe its position and orientation relative to some coordinate system. Position is easy, you say "the centre of the part is at so many units along the x axis, so many along the y axis, and so many along the z axis". But orientation is not so clear. You have to say something like "If I start at the centre of the part, and go forward one unit relative to the part, I'll end up at position x1, y1, z1, and if I go left one unit relative to the part, I end up at position x2, y2, z2, and if I go up one unit, I end up at x3, y3, z3". The 9 numbers in the matrix are the coordinates of these three points.

Now suppose I want to scale in the (say) front/back axis, by a factor of 2. I can start at the centre of the part and move _2_ units forward. I'll end up at position 2*x1, 2*y1, 2*z1. If I put those numbers in my matrix, the rendering engine will assume that those coordinates were generated at 1 unit distance, and draw all the pixels two times further forward than before.

Does that make it any clearer? Or have I just made it more confusing?

Edit: Actually, on re-reading this thread, I suspect that you probably understand all that. If I could delete my post, I would.

[N. W. Perry]

No - at least, not necessarily. Rotate a part about some arbitrary axis and you'll see all 9 numbers in the matrix change.

I don't remember what the mathematical terminology is (and I may even get some of this completely wrong!) but it goes something like this:
Imagine you have a part and you want to describe its position and orientation relative to some coordinate system. Position is easy, you say "the centre of the part is at so many units along the x axis, so many along the y axis, and so many along the z axis". But orientation is not so clear. You have to say something like "If I start at the centre of the part, and go forward one unit relative to the part, I'll end up at position x1, y1, z1, and if I go left one unit relative to the part, I end up at position x2, y2, z2, and if I go up one unit, I end up at x3, y3, z3". The 9 numbers in the matrix are the coordinates of these three points.

Now suppose I want to scale in the (say) front/back axis, by a factor of 2. I can start at the centre of the part and move _2_ units forward. I'll end up at position 2*x1, 2*y1, 2*z1. If I put those numbers in my matrix, the rendering engine will assume that those coordinates were generated at 1 unit distance, and draw all the pixels two times further forward than before.

Does that make it any clearer? Or have I just made it more confusing?

Edit: Actually, on re-reading this thread, I suspect that you probably understand all that. If I could delete my post, I would.

Yeah, I guess what I was asking was more like what each separate box can be said to represent. The boxes themselves aren't complete coordinates, but factors—amounts of change, I guess? And you're right, when you combine transformations (such as rotating around multiple axes), you'll end up populating more boxes than with a single operation (and combining the values of some boxes together).

But is there a simply way to say, for example, the top-left box represents the ratio of an original x value to its new x position, and so forth?

[Travis Cobbs]

Yeah, I guess what I was asking was more like what each separate box can be said to represent. The boxes themselves aren't complete coordinates, but factors—amounts of change, I guess? And you're right, when you combine transformations (such as rotating around multiple axes), you'll end up populating more boxes than with a single operation (and combining the values of some boxes together).

But is there a simply way to say, for example, the top-left box represents the ratio of an original x value to its new x position, and so forth?

The short answer is that it's complicated. Standard 3D transformations (rotating, scaling, and translating) each have a specific set of values that they operate on in a standard 4x4 3D transformation matrix. (For references, the 3x3 matrix you see is the upper left corner of a 4x4 general transformation matrix.)

Things are complicated by the fact that there are actually two different ways that people apply the transformations, and the ways are mirrored across the matrix's top-left to bottom-right diagonal. So, in one way, the top three elements of the 4th column represent the x,y,z translation, and in the mirrored way, the left three elements of the 4th row represent the same thing. Both ways produce the same end results, as long as they are used consistently.

Here is a tutorial that lists how the typical transformations are applied to the standard matrix. It uses the representation that has the x,y,z translation as the left three elements of the 4th column. I think that most LDraw software uses the top three elements of the 4th row for the x,y,z translation, in which case you would need to mirror all the sample matrices along the top-left to bottom-right diagonal. I could be misremembering; it has been a long time since I paid any attention to LDView's matrix math. Also, LDCad presents the x,y,z translation as a separate set of fields, and doesn't provide access to the remaining 4 fields in the 4x4 matrix. If you look at the tutorial, you will see that these 4 values are always 0,0,0,1, so that doesn't really matter.

But if you look specifically at the scale transformation, you will see exactly what Philo said originally: the three fields along the diagonal represent the x, y, and z scale factors. (Remember that in LDraw, "up" is -Y.)

[Roland Melkert]

But if you look specifically at the scale transformation, you will see exactly what Philo said originally: the three fields along the diagonal represent the x, y, and z scale factors. (Remember that in LDraw, "up" is -Y.)

Maybe it's me but I get the feeling people are over complicating this.

The columns in the LDCad dialog indicate the direction of the part's local xyz axis' as seen from absolute space.

First column indicates the vector of the X-ax's (1,0,0 as it's non rotated).

2nd the Y-axis and 3rd the z-axis.

Normally all these vectors are normalized (length == 1.0), with any other length scaling will be applied.

You could also read it from a row's perspective to go the other way (direction of world axis' as seen from local part space)

[N. W. Perry]

Maybe it's me but I get the feeling people are over complicating this.

The columns in the LDCad dialog indicate the direction of the part's local xyz axis' as seen from absolute space.

First column indicates the vector of the X-ax's (1,0,0 as it's non rotated).

2nd the Y-axis and 3rd the z-axis.

Normally all these vectors are normalized (length == 1.0), with any other length scaling will be applied.

You could also read it from a row's perspective to go the other way (direction of world axis' as seen from local part space)

And now it makes sense. 

[N. W. Perry]

Same question, different parameters: what if I'm rotating an assembly, where the connection points are offset? In this case, the distance between the two pin holes needs to be stretched by a scale factor of 1.01242, but that axis isn't orthogonal to the parts themselves. How would I calculate the scale factor to stretch the assembly in the X direction?
   

[Roland Melkert]

gonal to the parts themselves. How would I calculate the scale factor to stretch the assembly in the X direction?

1. Put a temporary pin at the other end of the 1x8 brick to get a rotation angle using selection info.
2. Apply the rotation
3. Select the 1x8 brick and set relative gird ( o )
4. Press enter to edit the reference.
5. Put  1.01242 in the top left matrix field.

Do note it's not 'officially' supported to scale stuff at the model level in LDCad.

Main thing to keep an eye on is to select something non scaled before adding new parts, or those will be scaled too.

[N. W. Perry]

Mm…I couldn't get that method to compute, for some reason.

What did work, though, was to apply the scale factor to both the X and Y directions of the assembly. That makes sense, because…

The rotation point and the connection point (the hole in the 1x2 brick) form the hypotenuse of a right triangle. Its length is ~126.49, and I need to stretch it to ~128.06, giving the scale factor of 1.01242. Since the ratio of the sides of a triangle remain the same as long as the angles don't change, then if I want to transform my assembly in the X direction using that same scale factor, I have to apply it in Y as well, to keep the proportions equal.

So after entering 1.01242 into both the X and Y boxes, my hypotenuse now measures 128.06, and I can just apply the rotation shown (32.91°), because again, the angles in my triangle stay the same.

Now, what if I want to stretch my assembly only in the X direction, as originally planned? (Don't ask why—I guess I figured the distortion would be less visually apparent that way.) Well, I know I want my hypotenuse to be 128.06, and since I'm not changing the Y dimension, I know that the short side of my triangle remains at 40 (the vertical distance between the holes in the 1x2 and 1x8 bricks). And since it's a right triangle, I know I have an angle of 90°—and knowing 2 sides plus 1 angle is enough to calculate a whole triangle!

I can either algebraically solve the Pythagorean theorem for b, or just plug the values into my favorite online triangle calculator.   Turns out, my third side needs to be 121.6625; it's currently 120, so that's a scale factor of 1.01379. Enter that into the X matrix, use Selection Info to get the new rotation value (because I changed the ratio of my triangle sides, therefore the angles changed as well), apply that to the assembly—now it fits!

   

Sorry to ask my question, only to answer myself right away!   I was actually stumped, but for some reason posting these puzzlers here helps me think them through…and maybe will help someone else later on. And it's worth noting that, as usual, my answer lies in the fact that you can solve any triangle if you know at least one side, plus any other 2 of the sides or angles.

[Roland Melkert]

Mm…I couldn't get that method to compute, for some reason.

I shouldn't answer posts that late, I thought the 1x8 brick was supposed to go between the two pins

You can still use the relative grid method though, just select all red parts before scaling.

[N. W. Perry]

I shouldn't answer posts that late, I thought the 1x8 brick was supposed to go between the two pins

You can still use the relative grid method though, just select all red parts before scaling.

Yes!   Relative scaling does still work—I tried it both as a submodel and as a group, but is there anything I need to be careful of? Sometimes I find that I have to re-snap the parts of a group after scaling, but that may be only when I'm using the absolute grid. (Also, it appears that scaling will account for the origin of a submodel, but not the custom center of a group.)

[Roland Melkert]

(Also, it appears that scaling will account for the origin of a submodel, but not the custom center of a group.)

In LDCad scaling at the model level is basically misusing rotations.

The whole internals 'assume' all rotations are non scalar.

Using it couldn't do any real harm but it was never designed for it.

[N. W. Perry]

In LDCad scaling at the model level is basically misusing rotations.

The whole internals 'assume' all rotations are non scalar.

Using it couldn't do any real harm but it was never designed for it.

Fair enough, since what I'm using for is just to approximate the "misusing" of plastic in real life.

That said, it does seem to be something of a necessary evil, isn't it? I don't know of another way to "force" connections that are off by fractions of an LDU, but still large enough to be apparent in a photo-real render. Maybe it's possible in the rendering engine itself, but I don't have enough knowledge of—or access to—the inner workings of that.