The problem:

First lets define the problem.

We are given four points
Points lie on line and points lie on line
For the remaining part of this article we assume that and are not parallel.
We need to find two points such that
that lies on and
that lies on and
where (the squared distance function) is minimised

The old approach:

Here is a summary of the old approach. I find this method more intuitive.

1. Calculate a plane
   • Calculate the line direction vectors
     
   • Define a plane from a direction and point, where
     
2. Project the points onto the plane to get two lines in 2D
   
3. Define the two projected lines in the parametric form
   
4. Solve for and by calculating the point where the two lines meet i.e. solve the following equation
   
5. Find the closest points on and in 3D using the values of and we just calculated
   
   

And finally some pseudocode. Beside each line, the number of floating point operations are listed, so we can compare the two methods later on.

Vector3 d1 = p1-p2; // 3 sub
Vector3 d2 = p4-p3; // 3 sub

// Crate the plane and project the points
Plane m(d1.cross(d2), p1); // 6 mul, 6 add, 3 div, 1 sqrt
Vector3 pj1 = m.project(p1);  // (6 mul, 6 add, 1 sub) X 4
Vector3 pj2 = m.project(p2);
Vector3 pj3 = m.project(p3);
Vector3 pj4 = m.project(p4);

// Calculate line direction in 2D
Vector2 d34 = (Vector2)pj4 - (Vector2)pj3; // 2 sub x3
Vector2 d31 = (Vector2)pj1 - (Vector2)pj3;
Vector2 d12 = (Vector2)pj2 - (Vector2)pj1;

// Calculate u1, u2 in 2D
float t = 1.0 / (d34.y * d12.x - d34.x * d12.y); // 2 mul, 1 sub, 1 div
u[0] = (d12.x * d31.y - d12.y * d31.x) * t ; // 3 mul, 1 sub
u[1] = (d34.x * d31.y - d34.y * d31.x) * t;// 3 mul, 1 sub

// Total # of FP operations
// 49 additions & subtractions,
// 42 multiplications and divisions
// 1 expensive sqrt

Using calculus

1. Define and in the parametric form as parameters of and respectively
   
2. The distance between the two lines are the points where the two lines are closest to each other.
   So…
   We minimise the squared distance between the two lines
3. To minimise we differentiate with respect to and and set both equation = 0
4. We end up with two equations with two unknowns and . Solve for and .
5. Find the closest points on and in 3D using the values of and we just calculated
   

This may sound a lot more complicated, but after working out and simplifying the equations we end up with a much more efficient solution. This will shown below.

Where:

Replacing the value of and rewriting the two equations above we get

We can write this in the matrix form

Where:

Solving for and to get

Note that if the lines are parallel and then any point on one line can be projected onto the other to get the distance between the two lines. Finally here is some C++ style pseudocode and a count of all operations.

Vector3 d21 = p1-p2; // 3 sub x 3
Vector3 d34 = p4-p3;
Vector3 d13 = p3-p1;

// m * u = x
float a = d21.dot(d21); // (3 mul + 3 add ) x 5
float b = d21.dot(d34);
float c = d34.dot(d34);
float d = -d13.dot(d21);
float e = -d13.dot(d34);

// Solve for u1 & u2
u[0] = (d*c-e*b)/(c*a-b*b); // 4 mul, 2 sub, 1 div
u[1] = (e - b * u[0]) / c; // 1 mul, 1 sub, 1 div

// Total # of FP operations
// 22 multiplications and division
// 27 additions and subtractions

On Windows 7 and Windows Server 2008

For those who are not sure how this can be done, the answer is a less well known command line utility called diskpart.exe. Diskpart is able to run scripts using the /s switch. Scripts are nothing but text files with a series of diskpart commands. So the script we need to create will need to do the following…

1. Create a virtual disk
2. Create a partition on this virtual disk
3. Format the virtual disk
4. Map the virtual disk on to a path

The script to do this is as follows. If you prefer, you can start diskpart and type these command in line by line.

create vdisk file=c:\myDisk.vhd maximum=2048  type=expandable
select vdisk file=c:\myDisk.vhd
attach vdisk
create partition primary
format fs=ntfs quick
assign mount="c:\my folder"

A line by line explanation of the script is doing is this

Line 1: Creates a vdisk which will be saved as a file at c:\myDisk.vhd with a size of 2048Mb. The type=expandable part makes sure the size of file depends on the amount of data saved in your vdisk. If you skip this parameter the file will take up the entire 2048Mb of space, even if you have not saved a single byte.
Line 2: makes the vdisk current. This way all future operations will target this vdisk.
Line 3: So far we have only created a vdisk. Think of this command as  connecting the vdisk to your computer (virtually) just as you would connect a USB storage device to a USB port.
Line 4: Creates a partition on the vdisk
Line 5: Format the new partition using NTFS
Line 6: Map the vdisk on to a folder. In this case it is c:\my folder, but you can change this to whatever you like.

So that’s it. I have made a small batch to automate the process for Windows 7. Get it from the link below.

[ Download quota.bat]

On older versions of Windows

If you are using something older than XP, then I’m afraid you need to WAKE UP and realise we are in 2011. The stuff mentioned in this section will work only on Windows XP and newer versions.

The problem on older versions of Windows is that diskpart does not support virtual disks. We just need to use different tools to do the same thing, so if you skipped the previous section I suggest you go back and read it again. The bad news is that you will need to download some additional software. The good news is that all these tools are free. These are the tools you will need…

1. vhdtool.exe (From here)
2. vhdmount by installing Microsoft Virtual Server 2005. Note: When installing use the custom install option and only select vhdmount from the list. The other components are not needed unless you want to play with them.

As mentioned previously, the process is the same one as we did for Windows 7. The step by step process is as follows.

Step 1: Create a vdisk using vhdtool.exe using the following command line.
VhdTool.exe /create c:\myDisk.vhd 2147483648
There are two differences here. 1 – The size we enetered here is in bytes, so we need to multiply the number of bytes we want with 1048576 and 2- We don’t have the option of creating an expandable drive. So the vdisk file will take up all the space requested regardless of how much is actually stored in it.

Step 2: Connect the vdisk (only virtually of course) using vhdmount.exe
vhdmount.exe /p /f c:\myDisk.vhd

Step 3: Create a partition on the drive using diskpart. But before we do this, we must first start diskpart
1. Identify the disk number using the diskpart LIST command
LIST DISK
Make a note of the disk number. In my case it is 1.
2. Select the disk using the command
SELECT DISK 1
3. Create the partition like we did before using the CREATE command
CREATE PARTITION PRIMARY

Step 4: Unfortunately old versions of diskpart don’t have a format command, so we must do this using the good old format.com. But format.exe only works on drive letters, so that is what we need to do using the ASSIGN command
ASSIGN letter=x
Now, you will need to exit diskpart temporarily using the EXIT command and format the drive using good old format.com. If you have never done this or have gotn’ used to the GUI, this is how it is done from the command line…
format.com x: /Q /FS:NTFS

Step 5: Now if you don’t mind using the drive, you can stop here. Otherwise head back into diskpart to un-mount the volume. Unfortunately since we exited diskpart, we need to reselect the right volume. Get a list of volume using the LIST command as follows
LIST volume
Make a note of the volume number.  It will have the drive letter mentioned under the Ltr column. In this case it is x. Then select it using…
SELECT volume 2
Then remove the drive letter mapping using the REMOVE command as follows
REMOVE letter=x
And finally map it on to your folder using…
ASSIGN mount=”c:\my folder”

And so that is it. Although you may be able to automate the process using a combination of batch files and grep, I successfully implemented a command line PHP script to automate the process on Windows XP as well. You can also probably use a VB script, but that is not something I am very familiar with. The problem is identifying the disk number in step 3 and volume number in step 5. I used the following process to determine the disk number automatically.

1. Make a list of disks before plugging in the vdisk in step 2.
2. Use a regex pattern match and extract a list of disk numbers.
3. Mount the vdisk.
4. Make another list of disks. This time, the list will have the vdisk that you just mounted.
5. Identify the disk that is in the second list but not in the first one.

Identifying the volume number in step 5 if much simpler. Just use a regex pattern to identify the line which has the drive letter x. The number to the left of it will be the volume number.

There are a lot of other cool things you can do using diskpart that is not available via a GUI. If you don’t mind getting your hands dirty using the command line, then this is a good place to get started.

A few weeks back I came across a new concept in maths called Lagrange multipliers. In addition to the problem that I was addressing at the time, I was interested in seeing if it could be applied to find a point on a hyperplane closest to a given point.

If you are not familiar with Lagrange multipliers then just go to the Wikipedia page that explains it all. I think they give a better explanation of the problem than I ever will. Ok, so if you haven’t already figured out how we are going to approach this problem, read on.

Lets work in 2 dimensions with  a line of the form

Lets try and visualize this…
The vector represents the normal to the line
The constant is the distance to closest point on the line to the origin.
If you are used to only seeing the line in the slope intercept form as
then we can simply assume that
, and .

Now lets take our point in space. Now we would like to do is minimise the Euclidean distance to this point.  The Euclidean distance is calculated using the equation

Minimizing this problem is dead simple. The closest point to a point is the point it self. But now we would like to add a constraint such that the closest point must lie of the line, whose equation is

We approach this problem using a simple form of the Lagrange multiplier where we minimize the function subject to the constraint . The best way to show that this works is with an example so…

Lets try this with an example

Lets assume we have a line of the form
and we would like to find a point on this line that is closest to a given point

We know the general form of the Lagrange multiplier which states that

In English this simply means the that we need to find a point where the functions and give us vectors that point in the same direction. By multiplying one of the vectors with a constant we make both  vectors equal in length and direction. We could also write the equation above as

Before we start we change our distance function to a squared distance function. So in this case would become

We can do this because is just a function that needs to be minimised and for us it does not make a difference if we minimise the distance or the squared distance. We make this change as differentiating the squared distance function is much simpler and results in a system of linear equations. The function remains the same.

So now we need to find

In this case

We now need to find



After doing all the hairy math, we get



We now have a system of three linear equations with three unknowns. This should be fairly easy to solve using a number of methods. If we had used the distance function instead of the squared distance, we would have ended up with a system of non-linear equations which would not been so easy to solve.

Ok so after solving for we get


We are not really interested in anymore, so we shall leave that out. So the point on the line is

We can use several different methods to verify if this answer is correct. We can solve for the same point using the trigonometric method or we can check if the vectors and are orthonormal to each other. Here is just another point on our line. We shall use the latter method to verify our results. We already have
, . We can find by just setting in line equation. So we get

So we get


and…


…or in plain English: IT WORKS!!

• A USB stick? Ah too simple.
• YubiKey? Sounded very cool initially, but only can work with one password at a time and another device to take care of. And it was USB – too much friction if I had to enter the password 5-10 times a day.
• My phone: Ah perfect, it’s very personal. I never leave anywhere without it. WHich means it won’t be forgotten in my USB port when I’m out. And yeah, it’s got bluetooth (no shoving stuff in to usb ports).

So I set out to make a bluetooth app that could exchange keys with KeePass. Head on to the KeyXchanger page to get more info and download it.

The stand alone password exchanger

Even if you do not use KeePass, you can still use Key Xchanger. The download contains a stand alone application that allows your computer to exchange passwords with your phone. Some features of Key Xchanger are…

• All passwords are stored on your phone encrypted.
• Passwords on the phone are encrypted and can only be decrypted by an authorized computer
• Your password is not compromised even if you loose your phone, since one needs to attack both your phone and computer. This is extremely unlikely to happen.
• Passwords are never stored on the PC and are discarded immediately after being used.
• You can enter passwords in to almost any application on windows

Due to the large number of phones out there I would appreciate it if you could tell me if the application worked for you.

AutoHotKey

Attempt 1: Initially I disabled this key using an AutoHotKey script. But for some reason, the script would stop working and I would still find CAPS LOCK ON just when I was entering that password or something. Besides every new computer I went to needed to have AutoHotKey or a compiled .exe just for this purpose. Since I am fanatical about keeping 24×7 background apps to a minimal, this was a big no no. And finally, even though I think AutoHotKey is a very useful app, I doubt it can be used as a full scale keyboard  remapper.

KeyTweak

Attempt 2: After being pushed over the edge with AutoHotKey’s incompetence in this area, my hunt for a new application continued. After some modified google search terms and fresh hope for luck, I came across KeyTweak, among others which included plain old registry hacking and several incarnations or AutoHotKey. KeyTweak did just what I wanted. Once initial the configuration was made, it would do all the dirty registry hacing for me and I did not need any software to running in the background.

But now I was faced with a new problem, and this time it was caused due to personal circumstances. I had recently moved to a new country and this meant getting used to a whole new keyboard layout just for one new charecter – £. I think you know which country I am talking about. Initially I resisted by opening ‘Character Map’ and copying the £ symbol from the table. But soon I realized that this was happening more often than not and I needed another solution.

Secondly even if I did give in and decide to get used to a new keyboard layout, I’d still be stuck with a US keyboard on my laptop. Changing that would be a fairly expensive affair and if I didn’t it would mean different keyboard layouts on different computers which can get very very irritating. I say this out of experience.

AutoHotKey – Although this could do the job, I was still against having an app running in the background just so that it could type the £ for me. But until I found a better solution, this was going to be the answer to the problem.
KeyTweak – Although this could remap keys to others, somehow it was not able to find the £ character for me.

Microsoft Keyboard Layout Creator

Attempt 3: So once again it was back to google. That is when I found ‘Microsoft Keyboard Layout Creator‘. From the depths of Microsoft’s lair lies this little known app that can do pretty much anything and everything you wanted. Basically you create a whole new keyboard layout, just like the professionals do.

A quick guide to using Microsoft Keyboard Layout Creator

1. So once you have done the obvious download and install, run the application.
2. If you are like me and want to remap change a few keys, then load your current keyboard layout by going to
   File > Load existing keyboard
3. If you are not sure which keyboard layout you are using, you can find this by going to
   Control Panel > Regional and Language Options > Languages (tab) > Details (button)
4. Reconfigure the keys to the way you like it
5. Create a new installation package by going to
   Project > Build DLL and Setup Package
   Follow the instructions and create a installation package. Remember to give it a name that you can identify later.
6. Install the new layout using the installation package
7. Go to
   Control Panel > Regional and Language Options > Languages (tab) > Details (button)
   Beside the tree view, click Add… and select the new keyboard layout that you just created.
8. Activate it by selecting the new layout from the drop down menu where is says
   ‘Default input language‘
   Alternatively, you can switch between this layout and your old layout by enabling the language bar.

I hope this guide helps you get the most out of your keyboard.

All links are verified to be valid at the time of writing this post. If any of the links in this article or anywhere else on my blog are broken, let me know via the comments and I shall do my best to fix it.

Why?
Well, a lot of applications perform a lot of important tasks before they close. These range from saving settings, asking you to save files, releasing hooks. Whatever the task may be, if you killing an application stop it dead in it’s tracks and the app does not have a chance to perform these tasks.

So TerminateApp is the result of this necessity.
You can specify the process by it’s name or by it’s process ID. If you really are impatient, you can even kill the application if the application refuses to terminate within a given amount of time.

How does it work?
TerminateApp sends a WM_QUIT message to all window handles within a specified process. So it will only work for applications that process the WM_QUIT message. This worked fine for the several applications I tested it on, but I cannot guarantee it works on all the apps out there. This would depend on the way the app has been written.

[ Download TerminateApp]
[ Download Source Code]

Got a new demo out today. It’s something that some people (including me) call vector textures. A very interesting technique that uses the existing texture resizing methods implemented on the GPU. These days it’s hard to find graphics techniques that manage really makes use of everything the GPU provides us with. What’s even nicer is that this technique could improve existing methods of rendering things like text and could even be extended to rendering blades of grass or leaves on a tree where the possibility of zooming in to a leaf is very real.

Go ahead, read more about it and download the demo here

For those of you who don’t know who Chamrajpet Charles is, I’ll forgive you. If you have not heard of him, I seriously doubt you will be able to fully appreciate this post

For the rest, you will get some upper cuts on your kidneys if you lie only. So for all CC fans out there, here are some rips that will make you feel at home, where ever you are

He makes me laugh no matter what my mood. Feel free to download them to your computer. And now for my track collection…

I wish I was still in Bangalore. I would have recorded many more from the radio. Hey all you buggers out there in Bangalore, if anyone has any more links / recordings, send them to me or leave a comment below. Or else I will have to give two cracks on your elbows.

Demo is based on two chapters in the ShaderX book.
“Rendering Ocean Water”
by John Isidoro, Alex Vlachos, Chris Brennan
And
“Rippling Refractive and Reflective Water”
by Alex Vlachos, John Isidoro and Chris Oat

Features include

1. Realtime vertex displacement
2. Full control over speed, wave direction, amplitude and frequency
3. Additional detailing via two normal maps
4. Fake refractions that look real
5. Real time reflections or reflections via environment maps
6. Fully configurable water color via texture maps