I am trying to rotate a model in 3D so that it faces the right direction. The rotation I want is fairly trivial and can be broken down into two steps:
Rotate the model 90 degrees on its x-axis.
Rotate the model 180 degrees on its z-axis (relative to the first rotation).
The way to set a model's rotation in the framework I'm using (openFrameworks) is by calling its setRotation method. This method takes an angle, as well as floats x, y and z that specify the axis of rotation. How do I calculate the axis of rotation and angle for this particular rotation? I can't rotate the model two times sequentially because any call to setRotation overwrites previous rotations.
Please let me know if I can provide more information or clarity.
EDIT: In case anyone has the same question, this post helped me a lot.
weird that you can not apply more then one transform ... maybe you just use wrong function but anyway:
If you have direct access to transform matrix (or by get,set)
google for transform matrices if you do not have the knowledge
I suspect you are using 4x4 homogenous cartesian transform matrices
transform matrix anatomy
generate first rotation matrix and store it to M1
can use the setRotation for that
generate second rotation matrix and store it to M2
multiply them M=M1*M2
use this M instead of setRotation
If yo do not have the direct access to transform matrix and have to use just the setRotation
in that case you have to use quaternion which is the 4D vector you call the setRotation with
google for quaternion math and find the application of 2 rotations
I do not use them so I can not help with that but there are also equations out there
which converts 3x3 rotation matrix into quaternion and back
so you can still use the algorithm above
obtain M
extract the rotation matrix from it (it is just sub matrix you omit last row and column)
compute quaternion from it
and call setRotation with the result
I have an object in 2D space, that is rotated/scaled around. Normally I would find the transformation matrix straightforwardly:
Translate by origin
Rotate
Scale
Translate by -origin
Translate by original position
Given an original object position at (0,0), I can easily get the new location as a vector by multiplying that by the transformation matrix.
However, for this problem I don't have the original position. I only have the final position.
How can I construct the same transformation matrix, when I only have the already-multiplied vector, not the original position?
I still have the transformation point, rotation and scale.
Not sure if I am telling you anything new but you use the inverses of the matrices applied in reverse order to find the original point(s) that was(were) transformed. Not entirely sure that this is what you are looking for but if so then there is a solution below.
Below is an example for 2D transformations. The method is easily extended for 3D
as you know the vector t and the rotation R and the scale s the inverses are easily calculated and applied.
Imagine two coordinate systems layed on top of each other, with a rotation and scale difference between the two:
The problem is to convert a point from the non-rotated system to the other. What we do have, are four corner points forming a rectangle, with coordinates known for both systems at each point. We also know the rotation difference, and I think I at least should know the scale difference too. How do I convert a point from the non-rotated system to the rotated system? I have Unity3D at use.
Extra points for clarity in math :)
PS: I'm writing this really late, going to edit later for more clarity.
Some linear algebra does the trick:
Express each operation as a matrix and matrix multiply those to combine them into a single resulting matrix (for efficiency).
If translation is involved you need to add a dimension to your matrices, see homogenous coordinates.
The reason is that the mappings are affine ones then, not linear ones. You can ignore the extra dimension in the end result. It is just a nice way to embed affine mappings into linear ones, so the algebra is easier.
Example
M = M_trans * M_rot * M_scale
x' = M x
The order here is right to left: vector x is first scaled, then rotated, then translated into vector x'. (Using column vectors).
Hints on the matrices: Rotation Matrix, Scaling Matrix
For deriving 2D formulas when given 3D ones: either keep z = 0 or delete the 3rd row and 3rd column from each matrix.
I have written a simple AR program in XNA and I am now trying to find the relative transformation between my 2 markers.
I have located my markers relative to my camera and have extracted out translation and rotation matrixes for the markers.
What I am trying to do is to find out the relative translation to get to marker 2 from marker 1. For instance if marker 1 and marker 2 were lying on the same Z plane the Z translation component would be 0mm.
The image below is the application working for 2 positions on the same plane:
I assumed that by simply multiplying the matrix of the 2nd marker by the inverse of the 1st marker I can get the translation. However I am getting completely wrong results.
The code I am running is as follows:
posit.EstimatePose(points, out matrix, out trans);
float yaw, pitch, roll;
matrix.ExtractYawPitchRoll(out yaw, out pitch, out roll);
Matrix rotation =
Matrix.CreateFromYawPitchRoll(-yaw, -pitch, roll);
Matrix translation =
Matrix.CreateTranslation(new Vector3(trans.X, trans.Y, -trans.Z));
Matrix complete = rotation * translation;
List<Matrix> all = new List<Matrix>();
all.Add(rotation);
all.Add(translation);
all.Add(complete);
matrixes.Add(all);
}
Matrix res = Matrix.Invert(matrixes[0][2]) * matrixes[1][2];
Vector3 scaleR;
Vector3 translationR;
Quaternion rotationR;
res.Decompose(out scaleR, out rotationR, out translationR);
The result:
TranslationR : {X:-103.4285 Y:-104.1754 Z:104.9243}
I have overlaid 3D axes onto the image as shown above using XNA so I assume the rotation and translation relative to the camera has been worked out correctly.
It seems like I am doing something wrong along the way to calculate the translation. I would definitely not expect the Z to equal 104mm. I was expecting something along the lines of:
{X:0 Y:150 Z:0}
I've done something similar to this before, however it was using 3x3 matrices in a 2D environment (with X,Y Translate, Rotate, Skew). Are the matrices in question 4x4?
Yes you are right, to find the matrix to transform object A with matrix M1 to object B with matrix M2 you can compute M1' * M2 (where M1' is the inverse).
The problem you may be running into is that a Matrix is composed of rotation, translation, scale and other transformations (e.g. skew/perspective). Decomposing the matrix into its component parts often yields a non-deterministic answer. Its like Quadratic equations, there is more than one solution.
Another issue may be that Matrix operations are not commutative and you are simply performing them the wrong way around. If you perform M1' * M2 and M2 * M1' you will get different results.
Please give it a try (switching the matrix order). Also I'd be looking up the matrix decomposition function you used - what value of Rotation & Scaling are you getting at the output? Are your objects rotated or scaled? If not then you should get zero. Note that it is possible to have more than one solution of rotation + translation to get the same end result and the decomposition function doesn't know which it is you are looking for.
To extract just the translation component, you can use the methods form this page:
vt = (M14, M24, M34)T
What do you get when you try that?
What I am trying to do is to find out the relative translation to get
to marker 2 from marker 1.
Vector3 relativeTranslation = Marker2Matrix.Translation - marker1Matrix.Translation;
My answer seems overly simplistic so maybe I'm not grasping your question completely, but it will create a vector that when added to Marker1's location (translation), will get you to Marker 2's location.
Does a 3D vector differ from a 3D point tuple (x,y,z) in the context of 3D game mathematics?
If they are different, then how do I calculate a vector given a 3d point?
The difference is that a vector is an algebraic object that may or may not be given as the set of coordinates in some space. (thanks to bungalobill for correcting my sloppiness).
A point is just a point given by coordinates. Generally, one can conflate the two. If you are given a set of coordinates, and told that they constitute a 'point' with no further information (choice of basis, etc), then you can just hand that set of numbers back and legitimately claim to have produced a vector.
The largest difference between the two is that it makes no sense to do things to one that you can do to the other. For example,
You can add vectors: <1 2 3> + <3 2 1> = <4 4 4>
You can multiply (or scale) a vector by a number (generally called a scalar)
2 * <1 1 1> = <2 2 2>
You can ask how far apart two points are: d((1, 2, 3), (3, 2, 1) = sqrt((1 - 3)2 + (2 - 2)2 + (3 - 1)2) = sqrt(8) ~= 2.82
A good intuitive way to think about the association between a vector and a point is that a vector tells you how to get from the origin (that one point in space to which we assign the coordinates (0, 0, 0)) to its associated point.
If you translate your coordinate system, then you get a new vector for the same point. Although the coordinates that make up the point will undergo the same translation so it's a pretty easy conflation to make between the two.
Likewise if rotate the coordinate system or apply some other transformation (e.g. a shear), then the coordinates and vector associated to the point will also change.
It's also possible for a vector to be something else entirely, for example a bounded function on the interval [0, 1] is a vector because you can multiply it by a real number and add it to another function on the interval and it will satisfy certain requirements (namely the axioms of a vectorspace). In this case one thinks of having one coordinate for each real number, x, in [0, 1] where the value of that coordinate is just f(x). So that's the easiest example of an infinite dimensional vector space.
There are all sorts of vector spaces and the notion that a vector is a 'point and a direction' (or whatever it's supposed to be) is actually pretty vacuous.
A vector represents a change from one state to another. To create one, you need two states (in this case, points), and then you subtract the initial state from the final state in order to get the resultant vector.
Vectors are a more general idea that a point in 3D space.
Vectors can have 2, 3, or n dimensions. They represent many quantities in the physical world (e.g., velocity, force, acceleration) besides position.
A mathematician would say that a vector is a first order tensor that transforms according to this rule:
u(i) = A(i, j)v(j)
You need both point and vector because they are different. A point in 3D space denoting position is a vector, but every vector is not a point in 3D space.
Then there's the computer science notion of a vector as a container - it's an abstraction for an array of values or references. This is a different concept from a mathematician's idea of a vector, because every vector container need not obey the first order tensor transformation law (e.g. a Vector of OrderItems). That's yet another separate idea.
It's important to keep all these in mind when talking about vectors and points.
Does a 3D vector differ from a 3D point tuple (x,y,z) in the context of 3D game mathematics?
Traditionaly vector means a direction and speed. A point could be considered a vector from the world orgin of one time step. (even though it may not be considered mathematically pure)
If they are different, then how do I calculate a vector given a 3d point?
target-tower is the common mnemonic.
Careful on your usage of this. The resulting vector is really normal*velocity. If you want to change it into something useful in a game application: you will need to normalize the vector first.
Example: Joe is at (10,0,0) and he wants to go to (10,10,0)
Target-Tower: (10,10,0)-(10,0,0)=(0,10,0)
Normalize the resulting vector: (0,1,0)
Apply "physics": (0,1,0) * speed*elapsed_time < speed = 3 and we'll say that the computer froze for a whole 2 seconds between the last step and this one for ease of computation >
=(0,6,0)
Add the resulting vector to Joes current point in space to get his next point in space: ... =(10,6,0)
Normal = vector/(sqrt(x*x+y*y+z*z))
...I think I have everything here
Vector is the change in the states. A point is the static point. Two vectors can be parallel or perpendicular. You can have product of two vectors which is a third vector. You can multiply a vector by a constant. You can add two vectors.
All these operations are not allowed on point. So program wise if you think both as a C++ class, there will be many such methods in the vector class but probably only Get and Set for point.
In the context of game mathematics there is no difference.
Points are elements of an affine space.† Vectors are elements of a vector (aka linear) space. When you choose an origin in an affine space it automatically induces a linear structure on that affine space. The contrary is also true: if you have a vector space it already satisfies all the axioms of an affine space.
The fact is that when it comes to computation, the only way to represent an affine space numerically is to use tuples of numbers, which also form a vector space.
Each object in a game always has an origin, and it is crucial to know where it is. That origin is set relative to the origin of the world, which is set relative to the origin of the camera/viewport. The vertices of the object are represented as vectors -- offsets from the object origin. You use matrix multiplication to transform the objects -- that is too a purely vector space operation (you cannot multiply an affine point by a matrix without specifying the origin first). Etc, etc... As we see all those triplets of numbers that we might think of as 'points' are actually vectors in the local coordinate system.
So is there any reason to distinguish between the two outside the study of algebra? It is an unnecessary abstraction, and unnecessary abstractions are harmful (KISS). So my answer is no, just go with a single vector type.
† Or any topological space outside the context of game development.
A vector is a line, that is a sequence of points but that it can be represented by two points, the starting and the ending point.
If you take the origin as the starting point, then you can describe your vector giving only the ending point.