can anyone tell me how to make my model rotate at its own center go gravity in stead of the default (0,0,0) axis?
and my rotation seems to be only going left and right not 360 degree..
If you want to rotate an object around its center, you first have to translate it to the origin, then rotate and translate it back. Since transformation matrices affect your vectors from right to left, you have to code these steps in opposite order.
Here is some pseudocode since I don't know OpenGL routines by heart:
PushMatrix();
LoadIdentity(); // Start with a fresh matrix
Translate(); // Move your object to its final destination
Rotate(); // Apply rotations
Draw(); // Draw your object using coordinates relative to the object center
PopMatrix();
These matrices get applied:
v_t = (I * T * R) * v = (I * (T * (R * v)))
So the order is: Rotation, Translation.
EDIT: An explanation for the equation above.
The transformations rotation, scale and translation affect the model-view-matrix. Every 3D point (vector) of your model is multiplied by this matrix to get its final point in 3D space, then it gets multiplied by the projection matrix to receive a 2D point (on your 2D screen).
Ignoring the projection stuff, your point transformed by the model-view-matrix is:
v_t = MV * v
Meaning the original point v, multiplied by the model-view-matrix MV.
In the code above, we have constructed MV by an identity matrix I, a translation T and a rotation R:
MV = I * T * R
Putting everything together, you see that your point v is first affected by the rotation R, then the translation T, so that your point is rotated before it is translated, just as we wanted it to be:
v_t = MV * v = (I * T * R) * v = T * (R * v)
Calling Rotate() prior to Translate() would result in:
v_t = (I * R * T) * v = R * (T * v)
which would be bad: Translated to some point in 3D, then rotated around the origin, leading to some strange distortion in your model.
Related
So I have a sphere. It rotates around a given axis and changes its surface by a sin * cos function.
I also have a bunck of tracticoids at fix points on the sphere. These objects follow the sphere while moving (including the rotation and the change of the surface). But I can't figure out how to make them always perpendicular to the sphere. I have the ponts where the tracticoid connects to the surface of the sphere and its normal vector. The tracticoids are originally orianted by the z axis. So I tried to make it's axis to the given normal vector but I just can't make it work.
This is where i calculate M transformation matrix and its inverse:
virtual void SetModelingTransform(mat4& M, mat4& Minv, vec3 n) {
M = ScaleMatrix(scale) * RotationMatrix(rotationAngle, rotationAxis) * TranslateMatrix(translation);
Minv = TranslateMatrix(-translation) * RotationMatrix(-rotationAngle, rotationAxis) * ScaleMatrix(vec3(1 / scale.x, 1 / scale.y, 1 / scale.z));
}
In my draw function I set the values for the transformation.
_M and _Minv are the matrixes of the sphere so the tracticoids are following the sphere, but when I tried to use a rotation matrix, the tracticoids strated moving on the surface of the sphere.
_n is the normal vector that the tracticoid should follow.
void Draw(RenderState state, float t, mat4 _M, mat4 _Minv, vec3 _n) {
SetModelingTransform(M, Minv, _n);
if (!sphere) {
state.M = M * _M * RotationMatrix(_n.z, _n);
state.Minv = Minv * _Minv * RotationMatrix(-_n.z, _n);
}
else {
state.M = M;
state.Minv = Minv;
}
.
.
.
}
You said your sphere has an axis of rotation, so you should have a vector a aligned with this axis.
Let P = P(t) be the point on the sphere at which your object is positioned. You should also have a vector n = n(t) perpendicular to the surface of the sphere at point P=P(t) for each time-moment t. All vectors are interpreted as column-vectors, i.e. 3 x 1 matrices.
Then, form the matrix
U[][1] = cross(a, n(t)) / norm(cross(a, n(t)))
U[][3] = n(t) / norm(n(t))
U[][2] = cross(U[][3], U[][1])
where for each j=1,2,3 U[][j] is a 3 x 1 vector column. Then
U(t) = [ U[][1], U[][2], U[][3] ]
is a 3 x 3 orthogonal matrix (i.e. it is a 3D rotation around the origin)
For each moment of time t calculate the matrix
M(t) = U(t) * U(0)^T
where ^T is the matrix transposition.
The final transformation that rotates your object from its original position to its position at time t should be
X(t) = P(t) + M(t)*(X - P(0))
I'm not sure if I got your explanations, but here I go.
You have a sphere with a wavy surface. This means that each point on the surface changes its distance to the center of the sphere, like a piece of wood on a wave in the sea changes its distance to the bottom of the sea at that position.
We can tell that the radious R of the sphere is variable at each point/time case.
Now you have a tracticoid (what's a tracticoid?). I'll take it as some object floating on the wave, and following the sphere movements.
Then it seems you're asking as how to make the tracticoid follows both wavy surface and sphere movements.
Well. If we define each movement ("transformation") by a 4x4 matrix it all reduces to combine in the proper order those matrices.
There are some good OpenGL tutorials that teach you about transformations, and how to combine them. See, for example, learnopengl.com.
To your case, there are several transformations to use.
The sphere spins. You need a rotation matrix, let's call it MSR (matrix sphere rotation) and an axis of rotation, ASR. If the sphere also translates then also a MST is needed.
The surface waves, with some function f(lat, long, time) which calculates for those parameters the increment (signed) of the radious. So, Ri = R + f(la,lo,ti)
For the tracticoid, I guess you have some triangles that define a tracticoid. I also guess those triangles are expressed in a "local" coordinates system whose origin is the center of the tracticoid. Your issue comes when you have to position and rotate the tracticoid, right?
You have two options. The first is to rotate the tracticoid to make if aim perpendicular to the sphere and then translate it to follow the sphere rotation. While perfect mathematically correct, I find this option some complicated.
The best option is to make the tracticoid to rotate and translate exactly as the sphere, as if both would share the same origin, the center of the sphere. And then translate it to its current position.
First part is quite easy: The matrix that defines such transformation is M= MST * MSR, if you use the typical OpenGL axis convention, otherwise you need to swap their order. This M is the common part for all objects (sphere & tracticoids).
The second part requires you have a vector Vn that defines the point in the surface, related to the center of the sphere. You should be able to calculate it with the parameters latitude, longitude and the R obtained by f() above, plus the size/2 of the tracticoid (distance from its center to the point where it touches the wave). Use the components of Vn to build a translation matrix MTT
And now, just get the resultant transformation to use with every vertex of the tracticoid: Mt = MTT * M = MTT * MST * MSR
To render the scene you need other two matrices, for the camera (MV) and for the projection (MP). While Mt is for each tracticoid, MV and MP are the same for all objects, including the sphere itself.
I would like to create a function to position a free-floating 2D raster image in space with the Irrlicht engine. The inspiration for this is the function rgl::show2d in the R package rgl. An example implementation in R can be found here.
The input data should be limited to the path to the image and a table with the four corner coordinates of the respective plot rectangle.
My first, pretty primitive and finally unsuccessful approach to realize this with irrlicht:
Create a cube:
ISceneNode * picturenode = scenemgr->addCubeSceneNode();
Flatten one side:
picturenode->setScale(vector3df(1, 0.001, 1));
Add image as texture:
picturenode->setMaterialTexture(0, driver->getTexture("path/to/image.png"));
Place flattened cube at the center position of the four corner coordinates. I just calculate the mean coordinates on all three axes with a small function position_calc().
vector3df position = position_calc(rcdf); picturenode->setPosition(position);
Determine the object rotation by calculating the normal of the plane defined by the four corner coordinates, normalizing the result and trying to somehow translate the resulting vector to rotation angles.
vector3df normal = normal_calc(rcdf);
vector3df angles = (normal.normalize()).getSphericalCoordinateAngles();
picturenode->setRotation(angles);
This solution doesn't produce the expected result. The rotation calculation is wrong. With this approach I'm also not able to scale the image correctly to it's corner coordinates.
How can I fix my workflow? Or is there a much better way to achieve this with Irrlicht that I'm not aware of?
Edit: Thanks to #spug I believe I'm almost there. I tried to implement his method 2, because quaternions are already available in Irrlicht. Here's what I came up with to calculate the rotation:
#include <Rcpp.h>
#include <irrlicht.h>
#include <math.h>
using namespace Rcpp;
core::vector3df rotation_calc(DataFrame rcdf) {
NumericVector x = rcdf["x"];
NumericVector y = rcdf["y"];
NumericVector z = rcdf["z"];
// Z-axis
core::vector3df zaxis(0, 0, 1);
// resulting image's normal
core::vector3df normal = normal_calc(rcdf);
// calculate the rotation from the original image's normal (i.e. the Z-axis)
// to the resulting image's normal => quaternion P.
core::quaternion p;
p.rotationFromTo(zaxis, normal);
// take the midpoint of AB from the diagram in method 1, and rotate it with
// the quaternion P => vector U.
core::vector3df MAB(0, 0.5, 0);
core::quaternion m(MAB.X, MAB.Y, MAB.Z, 0);
core::quaternion rot = p * m * p.makeInverse();
core::vector3df u(rot.X, rot.Y, rot.Z);
// calculate the rotation from U to the midpoint of DE => quaternion Q
core::vector3df MDE(
(x(0) + x(1)) / 2,
(y(0) + y(1)) / 2,
(z(0) + z(1)) / 2
);
core::quaternion q;
q.rotationFromTo(u, MDE);
// multiply in the order Q * P, and convert to Euler angles
core::quaternion f = q * p;
core::vector3df euler;
f.toEuler(euler);
// to degrees
core::vector3df degrees(
euler.X * (180.0 / M_PI),
euler.Y * (180.0 / M_PI),
euler.Z * (180.0 / M_PI)
);
Rcout << "degrees: " << degrees.X << ", " << degrees.Y << ", " << degrees.Z << std::endl;
return degrees;
}
The result is almost correct, but the rotation on one axis is wrong. Is there a way to fix this or is my implementation inherently flawed?
That's what the result looks like now. The points mark the expected corner points.
I've thought of two ways to do this; neither are very graceful - not helped by Irrlicht restricting us to spherical polars.
NB. the below assumes rcdf is centered at the origin; this is to make the rotation calculation a bit more straightforward. Easy to fix though:
Compute the center point (the translational offset) of rcdf
Subtract this from all the points of rcdf
Perform the procedures below
Add the offset back to the result points.
Pre-requisite: scaling
This is easy; simply calculate the ratios of width and height in your rcdf to your original image, then call setScaling.
Method 1: matrix inversion
For this we need an external library which supports 3x3 matrices, since Irrlicht only has 4x4 (I believe).
We need to solve the matrix equation which rotates the image from X-Y to rcdf. For this we need 3 points in each frame of reference. Two of these we can immediately set to adjacent corners of the image; the third must point out of the plane of the image (since we need data in all three dimensions to form a complete basis) - so to calculate it, simply multiply the normal of each image by some offset constant (say 1).
(Note the points on the original image have been scaled)
The equation to solve is therefore:
(Using column notation). The Eigen library offers an implementation for 3x3 matrices and inverse.
Then convert this matrix to spherical polar angles: https://www.learnopencv.com/rotation-matrix-to-euler-angles/
Method 2:
To calculate the quaternion to rotate from direction vector A to B: Finding quaternion representing the rotation from one vector to another
Calculate the rotation from the original image's normal (i.e. the Z-axis) to rcdf's normal => quaternion P.
Take the midpoint of AB from the diagram in method 1, and rotate it with the quaternion P (http://www.geeks3d.com/20141201/how-to-rotate-a-vertex-by-a-quaternion-in-glsl/) => vector U.
Calculate the rotation from U to the midpoint of DE => quaternion Q
Multiply in the order Q * P, and convert to Euler angles: https://en.wikipedia.org/wiki/Conversion_between_quaternions_and_Euler_angles
(Not sure if Irrlicht has support for quaternions)
The problem is I have two points in 3D space where y+ is up, x+ is to the right, and z+ is towards you. I want to orientate a cylinder between them that is the length of of the distance between both points, so that both its center ends touch the two points. I got the cylinder to translate to the location at the center of the two points, and I need help coming up with a rotation matrix to apply to the cylinder, so that it is orientated the correct way. My transformation matrix for the entire thing looks like this:
translate(center point) * rotateX(some X degrees) * rotateZ(some Z degrees)
The translation is applied last, that way I can get it to the correct orientation before I translate it.
Here is what I have so far for this:
mat4 getTransformation(vec3 point, vec3 parent)
{
float deltaX = point.x - parent.x;
float deltaY = point.y - parent.y;
float deltaZ = point.z - parent.z;
float yRotation = atan2f(deltaZ, deltaX) * (180.0 / M_PI);
float xRotation = atan2f(deltaZ, deltaY) * (180.0 / M_PI);
float zRotation = atan2f(deltaX, deltaY) * (-180.0 / M_PI);
if(point.y < parent.y)
{
zRotation = atan2f(deltaX, deltaY) * (180.0 / M_PI);
}
vec3 center = vec3((point.x + parent.x)/2.0, (point.y + parent.y)/2.0, (point.z + parent.z)/2.0);
mat4 translation = Translate(center);
return translation * RotateX(xRotation) * RotateZ(zRotation) * Scale(radius, 1, radius) * Scale(0.1, 0.1, 0.1);
}
I tried a solution given down below, but it did not seem to work at all
mat4 getTransformation(vec3 parent, vec3 point)
{
// moves base of cylinder to origin and gives it unit scaling
mat4 scaleFactor = Translate(0, 0.5, 0) * Scale(radius/2.0, 1/2.0, radius/2.0) * cylinderModel;
float length = sqrtf(pow((point.x - parent.x), 2) + pow((point.y - parent.y), 2) + pow((point.z - parent.z), 2));
vec3 direction = normalize(point - parent);
float pitch = acos(direction.y);
float yaw = atan2(direction.z, direction.x);
return Translate(parent) * Scale(length, length, length) * RotateX(pitch) * RotateY(yaw) * scaleFactor;
}
After running the above code I get this:
Every black point is a point with its parent being the point that spawned it (the one before it) I want the branches to fit into the points. Basically I am trying to implement the space colonization algorithm for random tree generation. I got most of it, but I want to map the branches to it so it looks good. I can use GL_LINES just to make a generic connection, but if I get this working it will look so much prettier. The algorithm is explained here.
Here is an image of what I am trying to do (pardon my paint skills)
Well, there's an arbitrary number of rotation matrices satisfying your constraints. But any will do. Instead of trying to figure out a specific rotation, we're just going to write down the matrix directly. Say your cylinder, when no transformation is applied, has its axis along the Z axis. So you have to transform the local space Z axis toward the direction between those two points. I.e. z_t = normalize(p_1 - p_2), where normalize(a) = a / length(a).
Now we just need to make this a full 3 dimensional coordinate base. We start with an arbitrary vector that's not parallel to z_t. Say, one of (1,0,0) or (0,1,0) or (0,0,1); use the scalar product ·(also called inner, or dot product) with z_t and use the vector for which the absolute value is the smallest, let's call this vector u.
In pseudocode:
# Start with (1,0,0)
mindotabs = abs( z_t · (1,0,0) )
minvec = (1,0,0)
for u_ in (0,1,0), (0,0,1):
dotabs = z_t · u_
if dotabs < mindotabs:
mindotabs = dotabs
minvec = u_
u = minvec_
Then you orthogonalize that vector yielding a local y transformation y_t = normalize(u - z_t · u).
Finally create the x transformation by taking the cross product x_t = z_t × y_t
To move the cylinder into place you combine that with a matching translation matrix.
Transformation matrices are effectively just the axes of the space you're "coming from" written down as if seen from the other space. So the resulting matrix, which is the rotation matrix you're looking for is simply the vectors x_t, y_t and z_t side by side as a matrix. OpenGL uses so called homogenuous matrices, so you have to pad it to a 4×4 form using a 0,0,0,1 bottommost row and rightmost column.
That you can load then into OpenGL; if using fixed functio using glMultMatrix to apply the rotation, or if using shader to multiply onto the matrix you're eventually pass to glUniform.
Begin with a unit length cylinder which has one of its ends, which I call C1, at the origin (note that your image indicates that your cylinder has its center at the origin, but you can easily transform that to what I begin with). The other end, which I call C2, is then at (0,1,0).
I'd like to call your two points in world coordinates P1 and P2 and we want to locate C1 on P1 and C2 to P2.
Start with translating the cylinder by P1, which successfully locates C1 to P1.
Then scale the cylinder by distance(P1, P2), since it originally had length 1.
The remaining rotation can be computed using spherical coordinates. If you're not familiar with this type of coordinate system: it's like GPS coordinates: two angles; one around the pole axis (in your case the world's Y-axis) which we typically call yaw, the other one is a pitch angle (in your case the X axis in model space). These two angles can be computed by converting P2-P1 (i.e. the local offset of P2 with respect to P1) into spherical coordinates. First rotate the object with the pitch angle around X, then with yaw around Y.
Something like this will do it (pseudo-code):
Matrix getTransformation(Point P1, Point P2) {
float length = distance(P1, P2);
Point direction = normalize(P2 - P1);
float pitch = acos(direction.y);
float yaw = atan2(direction.z, direction.x);
return translate(P1) * scaleY(length) * rotateX(pitch) * rotateY(yaw);
}
Call the axis of the cylinder A. The second rotation (about X) can't change the angle between A and X, so we have to get that angle right with the first rotation (about Z).
Call the destination vector (the one between the two points) B. Take -acos(BX/BY), and that's the angle of the first rotation.
Take B again, ignore the X component, and look at its projection in the (Y, Z) plane. Take acos(BZ/BY), and that's the angle of the second rotation.
for a project I need to compute the real world position and orientation of a camera
with respect to a known object.
I have a set of photos, each displays a chessboard from different points of view.
Using CalibrateCamera and solvePnP I am able to reproject Points in 2d, to get a AR-thing.
So my situation is as such:
Intrinsic parameters are known
Distortioncoefficients are known
translation Vector and rotation Vector are known per photo.
I simply cannot figure out how to compute the position of the camera. My guess was:
invert translation vector. (=t')
transform rotation vector to degree (seems to be radian) and invert
use rodriguez on rotation vector
compute RotationMatrix * t'
But the results are somehow totally off...
Basically I want to to compute a ray for each pixel in world coordinates.
If more informations on my problem are needed, I'd be glad to answer quickly.
I dont' get it... somehow the rays are still off. This is my Code btw:
Mat image1CamPos = tvecs[0].clone(); //From calibrateCamera
Mat rot = rvecs[0].clone(); //From calibrateCamera
Rodrigues(rot, rot);
rot = rot.t();
//Position of Camera
Mat pos = rot * image1CamPos;
//Ray-Normal (( (double)mk[i][k].x) are known image-points)
float x = (( (double)mk[i][0].x) / fx) - (cx / fx);
float y = (( (double)mk[i][0].y) / fy) - (cy / fy);
float z = 1;
float mag = sqrt(x*x + y*y + z*z);
x /= mag;
y /= mag;
z /= mag;
Mat unit(3, 1, CV_64F);
unit.at<double>(0, 0) = x;
unit.at<double>(1, 0) = y;
unit.at<double>(2, 0) = z;
//Rotation of Ray
Mat rot = stof1 * unit;
But when plotting this, the rays are off :/
The translation t (3x1 vector) and rotation R (3x3 matrix) of an object with respect to the camera equals the coordinate transformation from object into camera space, which is given by:
v' = R * v + t
The inversion of the rotation matrix is simply the transposed:
R^-1 = R^T
Knowing this, you can easily resolve the transformation (first eq.) to v:
v = R^T * v' - R^T * t
This is the transformation from camera into object space, i.e., the position of the camera with respect to the object (rotation = R^T and translation = -R^T * t).
You can simply get a 4x4 homogeneous transformation matrix from this:
T = ( R^T -R^T * t )
( 0 1 )
If you now have any point in camera coordinates, you can transform it into object coordiantes:
p' = T * (x, y, z, 1)^T
So, if you'd like to project a ray from a pixel with coordinates (a,b) (probably you will need to define the center of the image, i.e. the principal point as reported by CalibrateCamera, as (0,0)) -- let that pixel be P = (a,b)^T. Its 3D coordinates in camera space are then P_3D = (a,b,0)^T. Let's project a ray 100 pixel in positive z-direction, i.e. to the point Q_3D = (a,b,100)^T. All you need to do is transform both 3D coordinates into the object coordinate system using the transformation matrix T and you should be able to draw a line between both points in object space. However, make sure that you don't confuse units: CalibrateCamera will report pixel values while your object coordinate system might be defined in, e.g., cm or mm.
can anyone tell me how to make my model rotate at its own center go gravity in stead of the default (0,0,0) axis?
and my rotation seems to be only going left and right not 360 degree..
If you want to rotate an object around its center, you first have to translate it to the origin, then rotate and translate it back. Since transformation matrices affect your vectors from right to left, you have to code these steps in opposite order.
Here is some pseudocode since I don't know OpenGL routines by heart:
PushMatrix();
LoadIdentity(); // Start with a fresh matrix
Translate(); // Move your object to its final destination
Rotate(); // Apply rotations
Draw(); // Draw your object using coordinates relative to the object center
PopMatrix();
These matrices get applied:
v_t = (I * T * R) * v = (I * (T * (R * v)))
So the order is: Rotation, Translation.
EDIT: An explanation for the equation above.
The transformations rotation, scale and translation affect the model-view-matrix. Every 3D point (vector) of your model is multiplied by this matrix to get its final point in 3D space, then it gets multiplied by the projection matrix to receive a 2D point (on your 2D screen).
Ignoring the projection stuff, your point transformed by the model-view-matrix is:
v_t = MV * v
Meaning the original point v, multiplied by the model-view-matrix MV.
In the code above, we have constructed MV by an identity matrix I, a translation T and a rotation R:
MV = I * T * R
Putting everything together, you see that your point v is first affected by the rotation R, then the translation T, so that your point is rotated before it is translated, just as we wanted it to be:
v_t = MV * v = (I * T * R) * v = T * (R * v)
Calling Rotate() prior to Translate() would result in:
v_t = (I * R * T) * v = R * (T * v)
which would be bad: Translated to some point in 3D, then rotated around the origin, leading to some strange distortion in your model.