Town hall 3D model story

Ten months ago, I have created this architectural 3d model of the old town hall of my city of birth. I have uploaded it on Google 3D Warehouse and made a request to publish it on the 3D buildings layer of Google Earth. My model was on the review queue waiting to be accepted.

Five months later, I received a review message: improvements needed before acceptation. The improvements to be done was not so explicit. Indeed I get an indication on which criteria my model did not fit. It was "Incomplete Texture". And it was sadly true...

That was not an error or missing, nothing else but my willing. The problem here was that I had no pictures of this part of the building. And instead of create something that does not exist in reality, I preferred to leave this part as it is (with this fancy plain color).

Correcting this is not a huge task. I could have textured it with some default wall texture and submit the model again. This could have solve the problem, Google reviewers would have been happy with it and it would still indicated an absence of information. But I was busy during that time and just forgot about it. Another pertinent excuse is: there is no Linux version of Google SketchUp (I am using mainly Linux now).

Recently, I was surprised to see a blue ribbon next to my model, indicating that it has been published on the Google Earth 3D buildings layer. I wondered how that could be since I have made any "improvements", it should have not been published. I was curious to see if that was true. I had a look on Google Earth and it was effectively there...

... at least partially.

Apparently the motto is : "Fully textured or not textured at all!"

What a binary world...

Trackball view rotation

Introduction

Often we would like to visualize a 3d model from different view angles. Trackball is rotation technique that allow the user to intuitively change the view angle by using the mouse. The mouse behaves like a virtual trackball. Let's try to understand how trackball view rotation work exactly. This could be useful, as a programmer, if you want to implement this rotation technique in your application. Or if your are an artist it could be interesting to know what's happen under the stage.

Principle

The mouse moves on the screen which is a 2D plane. However a trackball could rotate in any direction in a 3D space. The problem here is to match the planar mouse movement to a 3D rotation. The idea to solve this problem is to project the 2D coordinate of the mouse cursor on a virtual sphere behind the screen, thus we obtained 3D point (blue on the picture). The difference between this 3D point and the center of the virtual sphere (red on the picture) is a 3D vector. The movement of this 3D vector while dragging the mouse is a rotation around the sphere center. Here we are! We use that rotation to rotate the view.

Transformation matrices

"Unfortunately, no one can be told what the Matrix is. You have to see it for yourself." Morpheus, The Matrix - 1999

Introduction

Matrices in 3D applications are very useful tools. Translation, scaling and rotation in any combination are simplified by usage of matrices. Let's expose the most important statements:

1. Any transformation in space can be describe by a matrix
2. Transforming a point in space is done by multiplying a transform matrix by the column vector describing the coordinates of the point
3. Several successive transformations can be resumed as one matrix

The maths

So if you want to perform a transformation on a point, all you have to do is to select the appropriate matrix (see below) plug in the parameter values (position, scaling factor, angle, or rotation axis) and multiply it with the column vector of the point coordinate.

Let's detail this, mathematically. The transformation can be written as:

v' = Mv

Where:
v a column vector describing the coordinate of the point (x, y, z, w) w = 1 most of the time
M a 4-by-4 matrix describing the transformation
v' a column vector describing the coordinate of the point after the transformation (x', y', z', w')

According the definition of matrix multiplication, we obtain

x'       a  b  c  d   x      ax + by + cz + dw
y'       e  f  g  h   y      ex + fy + gz + hw
z'   =   i  j  k  l . z   =  ix + jy + kz + lw
w'       m  n  o  p   w      mx + ny + oz + pw

Now, about the statement 3, it is again related to matrix multiplication. The result of 4-by-4 matrices multiplication is also a 4-by-4 matrix and it still a transformation.

And matrix multiplication is not commutative (AB is not BA). That is a good for us, it means that we can also describe the order of the transformations. Let's say we have point v and we would to transform it by a rotation R followed by a translation T.

v' = RTv

One remark here, we noticed that if we read from left to right, the rotation appear first. But that doesn't means that the rotation is perform first. The order of transformation is readed from right to left.

Another magical thing, suppose we have a complete 3D object. Transforming a 3D object is transforming all of its points. Instead of performing the rotation, follow by the translation to each point. We compute, once for all, the matrix that describe the combination of these two transformations, let's call it M. And then, multiply M by each point.

M = RT
for each point: v' = Mv

It does not matter how many transformations is performing, M will always describe them all. Magic! isn't it ?

Here is a list of of basic transformation matrices: translation, scaling, and rotation.

Translation matrix

X, Y, Z are the components of the translation vector.

1  0  0  X
0  1  0  Y
0  0  1  Z
0  0  0  1

Scaling matrix

X, Y, Z are scales factors along their respective axis.

X  0  0  0
0  Y  0  0
0  0  Z  0
0  0  0  1

Rotation around the x axis by an angle a

1   0   0   0
0   c  -s   0
0   s   c   0
0   0   0   1

Where c = cos(a) and s = sin(a)

Rotation around the y axis by an angle a

 c   0   s   0
 0   1   0   0
-s   0   c   0
 0   0   0   1

Where c = cos(a) and s = sin(a)

Rotation around the z axis by an angle a

c  -s   0   0
s   c   0   0
0   0   1   0
0   0   0   1

Where c = cos(a) and s = sin(a)

Rotation around an arbitrary axis by an angle a

x, y, z are the components of a unit vector that represent the rotation axis.

t*x*x + c        t*x*y - z*s     t*x*z + y*s     0
t*x*y + z*s      t*y*y + c       t*y*z - x*s     0
t*x*z - y*s      t*y*z + x*s     t*z*z + c       0
0                0               0               1

Where: c = cos(a), s = sin(a), and t = 1 - c