Original Post
I've decided to work for the community. Seeing the high amount of matrix- and transform-related questions lately in this forum I've taken up my quill pen (well, notepad, actually) to try and explain some fundamentals about matrix math and easy ways to use this extremely powerful tool. These are the methods I use to keep my life (and code) simple. These solutions use no trigonometry or weird matrix-component tricks. They use only Matrix Inverse, Matrix Multiplication, Rotation(around vector) Matrix, Translation Matrix and Scaling Matrix functions, that should hopefully be available to you in your API (they exist in Direct3D and GL as well).
A transform is any modification of the position, orientation, or size of an object. This includes translations, rotations, scaling, and all combinations thereof.
It all starts with
A Matrix == A Transform
M*X == T(X)
Simple as that! Now, everything we can do with transforms, we can do with matrices.
Matrix multiplication, inverse
M*X : When you apply a matrix with a vector, you apply the equivalent transformation to that vector.
M*N : When you multiply two matrices together, you compose their transformations. To apply the matrix (M*N) is to apply the matrix N, then the matrix M.
M^(-1) : The inverse of a matrix is also the inverse of a transform. If you apply a transform, and then its inverse, you will be back to the initial situation.
How do I get the up/right/direction/position vector of an object from its matrix?
The "initial" up vector of an object is (0,1,0,0).
The "initial" right vector of an object is (1,0,0,0).
The "initial" direction vector of an object is (0,0,1,0).
The "initial" position vector of an object is (0,0,0,1).
The above are 4d (x,y,z,w) vectors. APIs provide functions to convert vectors between 3d and 4d format .
When you multiply an object's vertices by a matrix, you apply this transform to all its vectors, including the up/right/direction vector. So if M is the matrix of an object, M * (0,1,0) is its new up vector, M * (1,0,0) is its new right vector, M * (0,0,1) is its new direction vector, and M*(0,0,0) is its new position vector.
How should I store an object's state?
The fast answer is : store its matrix, because a matrix contains all the necessary information : scale, orientation and position.
A better answer is to store three matrices : S for scaling, T for translation, R for rotation. The real transform matrix of an object is : T*R*S (apply the scaling, then the rotation, then the translation), and can as such be computed easily from the three storage matrices.
When you want to translate an object, multiply T by the applied translation matrix T' : T <- T'*T
When you want to rotate an object, multiply R by the applied rotation matrix R' : R <- R'*R
When you want to scale an object, multiply S by the applied scaling matrix S' : S <- S'*S
When I have the camera's position, how do I get the view matrix?
You have a matrix M, which defines the camera's position. This means M takes the camera from (0,0,0), moves it and reorients it to its new position.
The view matrix V is meant to take the camera from its current position, and move it to (0,0,0) while reorienting it back.
Which means V is the inverse of M.
View = Camera^(-1)
How do I use sin/cos/LookAt to move the camera?
Don't. You're hopefully keeping a whole load of information (T, R and S) about the camera, and can do whatever you want with it without ever calling a single trigonometric or LookAt function. What do you want to do (to your camera) today?
Turn it left/right? Extract its "up" vector from its R matrix, create a matrix R' to rotate around this axis by the needed angle, and apply the steps seen two paragraphs above.
Turn it up/down? Extract its "right" vector from its R matrix and repeat.
Roll it around? Use the "direction" vector this time.
Move it forward/backwards/sideways/vertically? Extract the correct vector from the R matrix, create a translation by the vector (multiplied by the wanted distance, which can of course be negative) and apply it to T.
Teleport it somewhere? just erase T and set it to the new translation matrix.
Then, once you have your camera matrix by multiplying T*R*S, inverse it and voilà your view matrix.
How do I use sin/cos/LookAt to move an object?
See above, minus the "inverse the matrix" part.
How do I attach an object to another?
Say, you want object B to be attached to object A. What you first do is keep A at (0,0,0) and move B around it to its rightful relative position using a first matrix M. Then, you move A to its rightful place using a matrix N. But since B is attached to A, it must move as well, so you must apply N to B as well.
This means the "real" matrix of B will be N*M (you aplpied M then N). Now, if A was attached to an object Z using a matrix L, A's matrix would be L*N instead of N, so B's matrix would be L*N*M.
To attach an object B to an object A, store its relative position using the standard (T,R,S) method, then, remember its "true" matrix is equal to
("true" matrix of B)=("true" matrix of A)*T*R*S
Which means you have to compute the true matrix of A beforehand. You can make this easier by creating a "T,R,S" class that can be "attached" to another class of its type, and has a function that returns its true matrix on demand. This would give you the graph :
0 -(Ta,Ra,Sa)-> A -(Tb,Rb,Sb)-> B
How do I do a third person camera?
Attach the camera to the object. The (T,R,S) of the camera should be set so that it is "behind" the object. Remember the object is initially at (0,0,0) facing (0,0,1)! Then, each frame, get the "true" matrix of the camera using the paragraph above. The graph is :
Object -(Tcam, Rcam, Scam)-> Camera
To get the view matrix, inverse the camera matrix.
Now, you don't have to change the camera (T,R,S) anymore (unless you want to), only the position of the object, and when you update the view matrix, the camera will follow around (and rotate as well).
How do I make the Camera orbit around an object?
Well, we've seen above that if you attach a camera to an object, the camera will move around this object. What we want to do is have the camera rotate around an object when the object does NOT rotate. A good way to do this is to add another object, that is invisible, and is attached to the initial object, and then attach the camera to the new object. Like this :
Object -(0, Rorbit, Sorbit)-> NewObject -(Tcam, Rcam, Scam)-> Camera
Rorbit and Sorbit are the rotation and scaling that allow you to orbit with the camera, and bring it closer/further to the object. Tcam, Rcam and Scam are the same as above.
How do I "shake" the screen when the player is hit?
You can create an imaginary object "Shake" and attach the camera to it, like this :
Object or 0 -(Tshake,Rshake,0)-> Shaker -(0,Rcam,Scam)-> Camera
Remember, the position of the Shaker/Camera object is Tshake. Its rotation is Rcam. So to move the camera, use Rcam vectors applied to Tshake, and to rotate the camera apply Rcam vectors on Rcam itself.
To shake, change Rshake to a random rotation matrix every frame until the shake ends.
What would the sun, earth, and moon look like using your RST system?
0 -(Tsun,0,0)-> SunCenter -(0,Rsun,Ssun)-> SunObject
which is rendered with Tsun*Rsun*Ssun
0 -(Tsun,0,0)-> SunCenter -(0,Reorbit,0)-> EarthOrbit -(Tearthc,0,0)-> EarthCenter -(0,Rearth,Searth)-> EarthObject
which is rendered with Tsun*Reorbit*Tearthc*Rearth*Searth
0 -(Tsun,0,0)-> SunCenter -(0,Reorbit,0)-> EarthOrbit -(Tearthc,0,0)-> EarthCenter -(0,Rmorbit,0)-> MoonOrbit -(Tmoon,Rmoon,Smoon)-> MoonObject
which is rendered with Tsun*Reorbit*Tearthc*Rmorbit*Tmoon*Rmoon*Smoon
THE END
I hope somebody'll find this useful, and please post any comments you see fit.
