I have a path output as shown in the image below, in a coordinate system 1 wherein the start point and the end point are (40,40) and (10,20) respectively.
I want to scale this path to a new coordinate system (coordinate system 2) with a known start and end point, the path has to scale and adjust between the new points.
I believe Affine transforms might help / linear algebra.
How do I achieve this ? and will this be accurate or will it distort ?
To find appropriate affine tranformation (there are many ways to transform two points into two another ones, but we choose the simplest way), you can apply these elementary steps:
Shift coordinates by (-startx, -starty)
Scale along X-axis with coefficient (newendx-newstartx)/(endx-startx) (here -80/3)
Scale along Y-axis with coefficient (newendy-newstarty)/(endy-starty) (here -35)
Shift coordinates by (newstartx, newstarty)
Resulting affine tranformation is product of these four matrices
Using Wolfram alpha to get matrix
M == {{c, 0, 0},
{0, d, 0},
{a*c + e, b*d + f, 1}}
where a,b,c,d,e,f are values from decription above (a = -startx and so on)
Now transform coordinates with multiplication of point coordinates and matrix M
(x, y, 1) * M = (newx, newy, 1)
Related
I am working on some shaders, and I need to transform normals.
I read in few tutorials the way you transform normals is you multiply them with the transpose of the inverse of the modelview matrix. But I can't find explanation of why is that so, and what is the logic behind that?
It flows from the definition of a normal.
Suppose you have the normal, N, and a vector, V, a tangent vector at the same position on the object as the normal. Then by definition N·V = 0.
Tangent vectors run in the same direction as the surface of an object. So if your surface is planar then the tangent is the difference between two identifiable points on the object. So if V = Q - R where Q and R are points on the surface then if you transform the object by B:
V' = BQ - BR
= B(Q - R)
= BV
The same logic applies for non-planar surfaces by considering limits.
In this case suppose you intend to transform the model by the matrix B. So B will be applied to the geometry. Then to figure out what to do to the normals you need to solve for the matrix, A so that:
(AN)·(BV) = 0
Turning that into a row versus column thing to eliminate the explicit dot product:
[tranpose(AN)](BV) = 0
Pull the transpose outside, eliminate the brackets:
transpose(N)*transpose(A)*B*V = 0
So that's "the transpose of the normal" [product with] "the transpose of the known transformation matrix" [product with] "the transformation we're solving for" [product with] "the vector on the surface of the model" = 0
But we started by stating that transpose(N)*V = 0, since that's the same as saying that N·V = 0. So to satisfy our constraints we need the middle part of the expression — transpose(A)*B — to go away.
Hence we can conclude that:
transpose(A)*B = identity
=> transpose(A) = identity*inverse(B)
=> transpose(A) = inverse(B)
=> A = transpose(inverse(B))
My favorite proof is below where N is the normal and V is a tangent vector. Since they are perpendicular their dot product is zero. M is any 3x3 invertible transformation (M-1 * M = I). N' and V' are the vectors transformed by M.
To get some intuition, consider the shear transformation below.
Note that this does not apply to tangent vectors.
Take a look at this tutorial:
https://paroj.github.io/gltut/Illumination/Tut09%20Normal%20Transformation.html
You can imagine that when the surface of a sphere stretches (so the sphere is scaled along one axis or something similar) the normals of that surface will all 'bend' towards each other. It turns out you need to invert the scale applied to the normals to achieve this. This is the same as transforming with the Inverse Transpose Matrix. The link above shows how to derive the inverse transpose matrix from this.
Also note that when the scale is uniform, you can simply pass the original matrix as normal matrix. Imagine the same sphere being scaled uniformly along all axes, the surface will not stretch or bend, nor will the normals.
If the model matrix is made of translation, rotation and scale, you don't need to do inverse transpose to calculate normal matrix. Simply divide the normal by squared scale and multiply by model matrix and we are done. You can extend that to any matrix with perpendicular axes, just calculate squared scale for each axes of the matrix you are using instead.
I wrote the details in my blog: https://lxjk.github.io/2017/10/01/Stop-Using-Normal-Matrix.html
Don't understand why you just don't zero out the 4th element of the direction vector before multiplying with the model matrix. No inverse or transpose needed. Think of the direction vector as the difference between two points. Move the two points with the rest of the model - they are still in the same relative position to the model. Take the difference between the two points to get the new direction, and the 4th element, cancels out to zero. Lot cheaper.
I have two 3D points (x,y,z), namely A and B and a bunch of other 3D points. Point A is at (0,0,0).
I would like to set point B to (0,0,0) so that all other points including A and B are translated and rotated in a way that is appropriate (so that A is no longer at (0,0,0)).
I know that there are some translations and rotations involved, but nothing more than that.
UPGRADE:
Point B is also constrained by three vectors: x', y', z' that represent x, y, and z axis of B's coordinate system. I think these should be somehow considered for the rotation part.
As you have given two points, one (A) at the origin and one (B) somewhere else, and you want to shift (translate) B to the origin, I don't see the necessity for any rotation.
If you don't have any other contraints, just shift all coordinates by the initial coordinates of B.
You can construct a transformation matrix as given, e.g., https://en.wikipedia.org/wiki/Transformation_matrix#Affine_transformations for 2D, but if you simply translate, R' = R + T, where R' is the vector after transformation, R the vector before and T the translation vector.
For more general transformations including rotations, you have to specify the rotation angle and axis. Then, you can come up with more general transformation, see above link.
I have a shape made out of several triangles which is positioned somewhere in world space with scale, rotate, translate. I also have a plane on which I would like to project (orthogonal) the shape.
I could multiply every vertex of every triangle in the shape with the objects transformation matrix to find out where it is located in world coordinates, and then project this point onto the plane.
But I don't need to draw the projection, and instead I would like to transform the plane with the inverse transformation matrix of the shape, and then project all the vertices onto the (inverse transformed) plane. Since it only requires me to transform the plane once and not every vertex.
My plane has a normal (xyz) and a distance (d). How do I multiply it with a 4x4 transformation matrix so that it turns out ok?
Can you create a vec4 as xyzd and multiply that? Or maybe create a vector xyz1 and then what to do with d?
You need to convert your plane to a different representation. One where N is the normal, and O is any point on the plane. The normal you already know, it's your (xyz). A point on the plane is also easy, it's your normal N times your distance d.
Transform O by the 4x4 matrix in the normal way, this becomes your new O. You will need a Vector4 to multiply with a 4x4 matrix, set the W component to 1 (x, y, z, 1).
Also transform N by the 4x4 matrix, but set the W component to 0 (x, y, z, 0). Setting the W component to 0 means that your normals won't get translated. If your matrix is composed of more that just translating and rotating, then this step isn't so simple. Instead of multiplying by your transformation matrix, you have to multiply by the transpose of the inverse of the matrix i.e. Matrix4.Transpose(Matrix4.Invert(Transform)), there's a good explanation on why here.
You now have a new normal vector N and a new position vector O. However I suppose you want it in xyzd form again? No problem. As before, xyz is your normal N all that's left is to calculate d. d is the distance of the plane from the origin, along the normal vector. Hence, it is simply the dot product of O and N.
There you have it! If you tell me what language you're doing this in, I'd happily type it up in code as well.
EDIT, In pseudocode:
The plane is vector3 xyz and number d, the matrix is a matrix4x4 M
vector4 O = (xyz * d, 1)
vector4 N = (xyz, 0)
O = M * O
N = transpose(invert(M)) * N
xyz = N.xyz
d = dot(O.xyz, N.xyz)
xyz and d represent the new plane
This question is a bit old but I would like to correct the accepted answer.
You do not need to convert your plane representation.
Any point lies on the plane if
It can be written as dot product :
You are looking for the plane transformed by your 4x4 matrix .
For the same reason, you must have
So and with some arrangements
TLDR : if p=(a,b,c,d), p' = transpose(inverse(M))*p
Notation:
n is a normal represented as a (1x3) row-vector
n' is the transformed normal of n according to transform matrix T
(n|d) is a plane represented as a (1x4) row-vector (with n the plane's normal and d the plane's distance to the origin)
(n'|d') is the transformed plane of (n|d) according to transform matrix T
T is a (4x4) (affine) column-major transformation matrix (i.e. transforming a column-vector t is defined as t' = T t).
Transforming a normal n:
n' = n adj(T)
Transforming a plane (n|d):
(n'|d') = (n|d) adj(T)
Here, adj is the adjugate of a matrix which is defined as follows in terms of the inverse and determinant of a matrix:
T^-1 = adj(T)/det(T)
Note:
The adjugate is generally not equal to the inverse of a transformation matrix T. If T includes a reflection, det(T) = -1, reversing the winding order!
Re-normalizing n' is mathematically not required (but maybe numerically depending on the implementation) since scaling is taken care off by the determinant. Thanks to Adrian Leonhard.
You can directly transform the plane without first decomposing and recomposing a plane (normal and point).
How do I generate a transformation matrix for rotating points/others by the angle between two lines/vectors/directions in CGAL?
2D is what I need. 3D is what I love.
According to the manual you have these tools to work with:
Aff_transformation_2<Kernel> t ( const Rotation, Direction_2<Kernel> d, Kernel::RT num, Kernel::RT den = RT(1))
approximates the rotation over the angle indicated by direction d, such that the differences between the sines and cosines of the rotation given by d and the approximating rotation are at most num/den each.
Precondition: num/den>0 and d != 0.
Aff_transformation_2<Kernel> t.operator* ( s) composes two affine transformations.
Aff_transformation_2<Kernel> t.inverse () gives the inverse transformation.
With them you should be able to compute the matrices corresponding to the two directions and use an identity along the lines of:
Mat(d1-d2) === Mat(d1)*Inv(Mat(d2))
to get what you want.
Given two image buffers (assume it's an array of ints of size width * height, with each element a color value), how can I map an area defined by a quadrilateral from one image buffer into the other (always square) image buffer? I'm led to understand this is called "projective transformation".
I'm also looking for a general (not language- or library-specific) way of doing this, such that it could be reasonably applied in any language without relying on "magic function X that does all the work for me".
An example: I've written a short program in Java using the Processing library (processing.org) that captures video from a camera. During an initial "calibrating" step, the captured video is output directly into a window. The user then clicks on four points to define an area of the video that will be transformed, then mapped into the square window during subsequent operation of the program. If the user were to click on the four points defining the corners of a door visible at an angle in the camera's output, then this transformation would cause the subsequent video to map the transformed image of the door to the entire area of the window, albeit somewhat distorted.
Using linear algebra is much easier than all that geometry! Plus you won't need to use sine, cosine, etc, so you can store each number as a rational fraction and get the exact numerical result if you need it.
What you want is a mapping from your old (x,y) co-ordinates to your new (x',y') co-ordinates. You can do it with matrices. You need to find the 2-by-4 projection matrix P such that P times the old coordinates equals the new co-ordinates. We'll assume that you're mapping lines to lines (not, for instance, straight lines to parabolas). Because you have a projection (parallel lines don't stay parallel) and translation (sliding), you need a factor of (xy) and (1), too. Drawn as matrices:
[x ]
[a b c d]*[y ] = [x']
[e f g h] [x*y] [y']
[1 ]
You need to know a through h so solve these equations:
a*x_0 + b*y_0 + c*x_0*y_0 + d = i_0
a*x_1 + b*y_1 + c*x_1*y_1 + d = i_1
a*x_2 + b*y_2 + c*x_2*y_2 + d = i_2
a*x_3 + b*y_3 + c*x_3*y_3 + d = i_3
e*x_0 + f*y_0 + g*x_0*y_0 + h = j_0
e*x_1 + f*y_1 + g*x_1*y_1 + h = j_1
e*x_2 + f*y_2 + g*x_2*y_2 + h = j_2
e*x_3 + f*y_3 + g*x_3*y_3 + h = j_3
Again, you can use linear algebra:
[x_0 y_0 x_0*y_0 1] [a e] [i_0 j_0]
[x_1 y_1 x_1*y_1 1] * [b f] = [i_1 j_1]
[x_2 y_2 x_2*y_2 1] [c g] [i_2 j_2]
[x_3 y_3 x_3*y_3 1] [d h] [i_3 j_3]
Plug in your corners for x_n,y_n,i_n,j_n. (Corners work best because they are far apart to decrease the error if you're picking the points from, say, user-clicks.) Take the inverse of the 4x4 matrix and multiply it by the right side of the equation. The transpose of that matrix is P. You should be able to find functions to compute a matrix inverse and multiply online.
Where you'll probably have bugs:
When computing, remember to check for division by zero. That's a sign that your matrix is not invertible. That might happen if you try to map one (x,y) co-ordinate to two different points.
If you write your own matrix math, remember that matrices are usually specified row,column (vertical,horizontal) and screen graphics are x,y (horizontal,vertical). You're bound to get something wrong the first time.
EDIT
The assumption below of the invariance of angle ratios is incorrect. Projective transformations instead preserve cross-ratios and incidence. A solution then is:
Find the point C' at the intersection of the lines defined by the segments AD and CP.
Find the point B' at the intersection of the lines defined by the segments AD and BP.
Determine the cross-ratio of B'DAC', i.e. r = (BA' * DC') / (DA * B'C').
Construct the projected line F'HEG'. The cross-ratio of these points is equal to r, i.e. r = (F'E * HG') / (HE * F'G').
F'F and G'G will intersect at the projected point Q so equating the cross-ratios and knowing the length of the side of the square you can determine the position of Q with some arithmetic gymnastics.
Hmmmm....I'll take a stab at this one. This solution relies on the assumption that ratios of angles are preserved in the transformation. See the image for guidance (sorry for the poor image quality...it's REALLY late). The algorithm only provides the mapping of a point in the quadrilateral to a point in the square. You would still need to implement dealing with multiple quad points being mapped to the same square point.
Let ABCD be a quadrilateral where A is the top-left vertex, B is the top-right vertex, C is the bottom-right vertex and D is the bottom-left vertex. The pair (xA, yA) represent the x and y coordinates of the vertex A. We are mapping points in this quadrilateral to the square EFGH whose side has length equal to m.
Compute the lengths AD, CD, AC, BD and BC:
AD = sqrt((xA-xD)^2 + (yA-yD)^2)
CD = sqrt((xC-xD)^2 + (yC-yD)^2)
AC = sqrt((xA-xC)^2 + (yA-yC)^2)
BD = sqrt((xB-xD)^2 + (yB-yD)^2)
BC = sqrt((xB-xC)^2 + (yB-yC)^2)
Let thetaD be the angle at the vertex D and thetaC be the angle at the vertex C. Compute these angles using the cosine law:
thetaD = arccos((AD^2 + CD^2 - AC^2) / (2*AD*CD))
thetaC = arccos((BC^2 + CD^2 - BD^2) / (2*BC*CD))
We map each point P in the quadrilateral to a point Q in the square. For each point P in the quadrilateral, do the following:
Find the distance DP:
DP = sqrt((xP-xD)^2 + (yP-yD)^2)
Find the distance CP:
CP = sqrt((xP-xC)^2 + (yP-yC)^2)
Find the angle thetaP1 between CD and DP:
thetaP1 = arccos((DP^2 + CD^2 - CP^2) / (2*DP*CD))
Find the angle thetaP2 between CD and CP:
thetaP2 = arccos((CP^2 + CD^2 - DP^2) / (2*CP*CD))
The ratio of thetaP1 to thetaD should be the ratio of thetaQ1 to 90. Therefore, calculate thetaQ1:
thetaQ1 = thetaP1 * 90 / thetaD
Similarly, calculate thetaQ2:
thetaQ2 = thetaP2 * 90 / thetaC
Find the distance HQ:
HQ = m * sin(thetaQ2) / sin(180-thetaQ1-thetaQ2)
Finally, the x and y position of Q relative to the bottom-left corner of EFGH is:
x = HQ * cos(thetaQ1)
y = HQ * sin(thetaQ1)
You would have to keep track of how many colour values get mapped to each point in the square so that you can calculate an average colour for each of those points.
I think what you're after is a planar homography, have a look at these lecture notes:
http://www.cs.utoronto.ca/~strider/vis-notes/tutHomography04.pdf
If you scroll down to the end you'll see an example of just what you're describing. I expect there's a function in the Intel OpenCV library which will do just this.
There is a C++ project on CodeProject that includes source for projective transformations of bitmaps. The maths are on Wikipedia here. Note that so far as i know, a projective transformation will not map any arbitrary quadrilateral onto another, but will do so for triangles, you may also want to look up skewing transforms.
If this transformation has to look good (as opposed to the way a bitmap looks if you resize it in Paint), you can't just create a formula that maps destination pixels to source pixels. Values in the destination buffer have to be based on a complex averaging of nearby source pixels or else the results will be highly pixelated.
So unless you want to get into some complex coding, use someone else's magic function, as smacl and Ian have suggested.
Here's how would do it in principle:
map the origin of A to the origin of B via a traslation vector t.
take unit vectors of A (1,0) and (0,1) and calculate how they would be mapped onto the unit vectors of B.
this gives you a transformation matrix M so that every vector a in A maps to M a + t
invert the matrix and negate the traslation vector so for every vector b in B you have the inverse mapping b -> M-1 (b - t)
once you have this transformation, for each point in the target area in B, find the corresponding in A and copy.
The advantage of this mapping is that you only calculate the points you need, i.e. you loop on the target points, not the source points. It was a widely used technique in the "demo coding" scene a few years back.