I am writing software to determine the viewable locations of a camera in 3D. I have currently implement parts to find the minimum and maximum length of view based on the camera and lenses intrinsic characteristics.
I now need to work out that if the camera is placed at X,Y,Z and is pointing in a direction (two angles, one around the horizontal and one around the vertical axis) what the boundaries the camera can see at are (knowing the viewing angle). The output I would like is 4 3D locations, making a rectangle that show the minimum position, top left, top right, bottom left and bottom right. The same is also required for the maximum positions.
Can anyone help with the geometry to find these points?
Some code I have:
QVector3D CameraPerspective::GetUnitVectorOfCameraAngle()
{
QVector3D inital(0, 1, 0);
QMatrix4x4 rotation_matrix;
// rotate around z axis
rotation_matrix.rotate(_angle_around_z, 0, 0, 1);
//rotate around y axis
rotation_matrix.rotate(_angle_around_x, 1, 0, 0);
inital = inital * rotation_matrix;
return inital;
}
Coordinate CameraPerspective::GetFurthestPointInFront()
{
QVector3D camera_angle_vector = GetUnitVectorOfCameraAngle();
camera_angle_vector.normalize();
QVector3D furthest_point_infront = camera_angle_vector * _camera_information._maximum_distance_mm;
return Coordinate(furthest_point_infront + _position_of_this);
}
Thanks
A complete answer with code will be probably way too long for SO, I hope that this will be enough. In the following we work in homogeneous coordinates.
I have currently implement parts to find the minimum and maximum length of view based on the camera and lenses intrinsic characteristics.
That isn't enough to fully define your camera. You also need a field of view angle and the width/height ratio.
With all these information (near plane + far plane + fov + ratio), you can build a 4x4 matrix known as perspective matrix. Google for it or check here for some references. This matrix maps the pyramidal region of the space which your camera "sees" (usually simply called frustrum) to the [-1,1]x[-1,1]x[-1,1] cube. Call it P.
Now you need a 4x4 camera matrix which transform points in world space to points in camera space. Since you know the camera position and the camera orientation this can be constructed easily (there is no room here to full explain how transformation matrices in homogeneous coordinates work, google for it). Call this matrix C.
Now consider the matrix A = P * C.
This matrix transforms points in world coordinates to points in the perspective space. Your camera will "see" those points if they are inside the [-1,1]x[-1,1]x[-1,1] cube. But you can invert this matrix in order to map points inside the cube to points in world space. So in order to obtain the 8 points you need in world space you can simply do:
y = A^(-1) * x
Where x =
[-1,-1,-1, 1] left - bottom - near
[-1,-1, 1, 1] left - bottom - far
etc.
Related
I have a rectangle that I place on the screen using a simple scale matrix (S). Now I would like to place this rectangle into "3D space", but have it appear just like before on the screen. I found that I can do so by applying the view and projection matrices in inverse. Something like:
S' = (V⁻¹ P⁻¹ S)
Matrix = P V (V⁻¹ P⁻¹ S)
This works fine so far. My rectangle is like a billboard now and I can treat it like any other object, apply P and V and it will show up correctly. However, there is a degeneracy here: I don't specify at which depth the object is placed. It could be twice as far away but x times bigger!
The reason that this is important is that I want to animate the rectangle, say rotate it around the Z axis or move in 3D space. Then I want it to come to a stop and be positioned pixel-perfect on the screen.
How can I place a flat object at a given z distance, such that it appears on screen in a certain way? I already have with the scale matrix that I need to display it in OpenGL without any 3D transformation, that is the matrix for displaying it in NDC or screen coordinates. I also have the projection and view matrices I want to use. How can I go from this to the desired model matrix?
How can I place a flat object at a given z distance, such that it appears on screen in a certain way [...]
Actually you want to draw the object in view space. Define a model matrix for the object that contains only one translation component (0, 0, -z) and skip the view matrix when drawing the object.
Usually the order to transform 3D vertex v to 2D point p is written as a matrix multiplication. [Depending on the API you are using, the order might be reversed. The notation I used here is glsl - friendly]
p = P * V * M * S * v
v = vertex (usually 3D of the form x,y,z,1)
P = projection matrix
V = view (camera) matrix
M = model matrix (or world transformation)
S is a 4x4 object-scaling matrix
matrices are usually 4x4 with the last line/column 0,0,0,1
The model matrix M can be decomposed into a number of sub components such as T translation, S scale and R rotation. Of course here the order matters.
To rotate an object or vertex around itself, first rotate it (as if it is at the origin already) and then translate it to the position it needs to be, for example using vector v with coordinates (x,y,z).
R is a typical 3x3 rotation matrix embedded in a 4x4 with 0,0,0,1 on the last line
v is a 3D vector with coordinates (x,y,z)
T is a 4x4 translation matrix, all zeroes except the last column where it has x,y,z,1
M = T * R (first R, then T)
To rotate an object around an arbitrary point q, first translate it so that it is relative to that point (i.e. q is at the origin, so you subtracted q from v). Then rotate around q (at the origin) by applying R. Lastly place the object it back where it should be (so add q again).
R is your typical 3x3 rotation matrix embedded in a 4x4 with 0,0,0,1 on the last line
v is a 3D vector with coordinates (x,y,z)
T is a 4x4 translation matrix, all zeroes except the last column where it has x,y,z,1
L is another 4x4 translation matrix, but now with q instead of v
L' is the inverse transformation of L
M = T * L * R * L'
Also, scaling your object first (i.e. S is at the end of the multiplications) before translations will keep the translation in world units. Scale after all transformations, in fact scales all translations too, and the object will move over scaled distances.
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'm trying to implement textures for spheres in my ray tracer. I managed to get something working, but I am unsure about its correctness. Below is the code for getting the texture coordinates. For now, the texture is random and is generated at runtime.
virtual void GetTextureCoord(Vect hitPoint, int hres, int vres, int& x, int& y) {
float theta = acos(hitPoint.getVectY());
float phi = atan2(hitPoint.getVectX(), hitPoint.getVectZ());
if (phi < 0.0) {
phi += TWO_PI;
}
float u = phi * INV_TWO_PI;
float v = 1 - theta * INV_PI;
y = (int) ((hres - 1) * u);
x = (int) ((vres - 1) * v);
}
This is how the spheres look now:
I had to normalize the coordinates of the hit point to get the spheres to look like that. Otherwise they would look like:
Was normalising the hit point coordinates the right approach, or is something else broken in my code? Thank you!
Instead of normalising the hit point, I tried translating it to the world origin (as if the sphere center was there) and obtained the following result:
I'm using a 256x256 resolution texture by the way.
It's unclear what you mean by "normalizing" the hit point since there's nothing that normalizes it in the code you posted, but you mentioned that your hit point is in world space.
Also, you didn't say what texture mapping you're trying to implement, but I assume you want your U and V texture coordinates to represent latitude and longitude on the sphere's surface.
Your first problem is that converting Cartesian to spherical coordinates requires that the sphere is centered at the origin in the Cartesian space, which isn't true in world space. If the hit point is in world space, you have to subtract the sphere's world-space center point to get the effective hit point in local coordinates. (You figured this part out already and updated the question with a new image.)
Your second problem is that the way you're calculating theta requires that the the sphere have a radius of 1, which isn't true even after you move the sphere's center to the origin. Remember your trigonometry: the argument to acos is the ratio of a triangle's side to its hypotenuse, and is always in the range (-1, +1). In this case your Y-coordinate is the side, and the sphere's radius is the hypotenuse. So you have to divide by the sphere's radius when calling acos. It's also a good idea to clamp the value to the (-1, +1) range in case floating-point rounding error puts it slightly outside.
(In principle you'd also have to divide the X and Z coordinates by the radius, but you're only using those for an inverse tangent, and dividing them both by the radius won't change their quotient and thus won't change phi.)
Right now your sphere intersection and texture-coordinate functions are operating in world space, but you'll probably find it useful later to implement transformation matrices, which let you transform things from one coordinate space to another. Then you can change your sphere functions to operate in a local coordinate space where the center is the origin and the radius is 1, and give each object an associated transformation matrix that maps the local coordinate space to the world coordinate space. This will simplify your ray/sphere intersection code, and let you remove the origin subtraction and radius division from GetTextureCoord (since they're always (0, 0, 0) and 1 respectively).
To intersect a ray with an object, you'd use the object's transformation matrix to transform the ray into the object's local coordinate space, do the intersection (and compute texture coordinates) there, and then transform the result (e.g. hit point and surface normal) back to world space.
I wish to generate rays from the camera through the viewing plane. In order to do this, I need my camera position ("eye"), the up, right, and towards vectors (where towards is the vector from the camera in the direction of the object that the camera is looking at) and P, the point on the viewing plane. Once I have these, the ray that's generated is:
ray = camera_eye + t*(P-camera_eye);
where t is the distance along the ray (assume t = 1 for now).
My question is, how do I obtain the 3D coordinates of point P given that it is located at position (i,j) on the viewing plane? Assume that the upper left and lower right corners of the viewing plane are given.
NOTE: The viewing plane is not actually a plane in the sense that it doesn't extend infinitely in all directions. Rather, one may think of this plane as a widthxheight image. In the x direction, the range is 0-->width and in the y direction the range is 0-->height. I wish to find the 3D coordinate of the (i,j)th element, 0
General solution of the itnersection of a line and a plane see http://local.wasp.uwa.edu.au/~pbourke/geometry/planeline/
Your particular graphics lib (OpenGL/DirectcX etc) may have an standard way to do this
edit: You are trying to find the 3d intersection of a screen point (eg a mouse cursor) with a 3d object in you scene?
To work out P, you need the distance from the camera to the near clipping plane (the screen), the size of the window on the near clipping plane (or the view angle, you can work out the window size from the view angle) and the size of the rendered window.
Scale the screen position to the range -1 < x < +1 and -1 < y < +1 where +1 is the top/right and -1 is the bottom/left
Scale normalised x,y by the view window size
Scale by the right and up vectors of the camera and sum the results
Add the look at vector scaled by the clipping plane distance
In effect, you get:
p = at * near_clip_dist + x * right + y * up
where x and y are:
x = (screen_x - screen_centre_x) / (width / 2) * view_width
y = (screen_y - screen_centre_y) / (height / 2) * view_height
When I directly plugged in suggested formulas into my program, I didn't obtain correct results (maybe some debugging needed to be done). My initial problem seemed to be in the misunderstanding of the (x,y,z) coordinates of the interpolating corner points. I was treating x,y,z-coordinates separately, where I should not (and this may be specific to the application, since the camera can be oriented in any direction). Instead, the solution turned out to be a simple interpolation of the corner points of the viewing plane:
interpolate the bottom corner points in the i direction to get P1
interpolate the top corner points in the i direction to get P2
interpolate P1 and P2 in the j direction to get the world coordinates of the final point
(This is all in ortho mode, origin is in the top left corner, x is positive to the right, y is positive down the y axis)
I have a rectangle in world space, which can have a rotation m_rotation (in degrees).
I can work with the rectangle fine, it rotates, scales, everything you could want it to do.
The part that I am getting really confused on is calculating the rectangles world coordinates from its local coordinates.
I've been trying to use the formula:
x' = x*cos(t) - y*sin(t)
y' = x*sin(t) + y*cos(t)
where (x, y) are the original points,
(x', y') are the rotated coordinates,
and t is the angle measured in radians
from the x-axis. The rotation is
counter-clockwise as written.
-credits duffymo
I tried implementing the formula like this:
//GLfloat Ax = getLocalVertices()[BOTTOM_LEFT].x * cosf(DEG_TO_RAD( m_orientation )) - getLocalVertices()[BOTTOM_LEFT].y * sinf(DEG_TO_RAD( m_orientation ));
//GLfloat Ay = getLocalVertices()[BOTTOM_LEFT].x * sinf(DEG_TO_RAD( m_orientation )) + getLocalVertices()[BOTTOM_LEFT].y * cosf(DEG_TO_RAD( m_orientation ));
//Vector3D BL = Vector3D(Ax,Ay,0);
I create a vector to the translated point, store it in the rectangles world_vertice member variable. That's fine. However, in my main draw loop, I draw a line from (0,0,0) to the vector BL, and it seems as if the line is going in a circle from the point on the rectangle (the rectangles bottom left corner) around the origin of the world coordinates.
Basically, as m_orientation gets bigger it draws a huge circle around the (0,0,0) world coordinate system origin. edit: when m_orientation = 360, it gets set back to 0.
I feel like I am doing this part wrong:
and t is the angle measured in radians
from the x-axis.
Possibly I am not supposed to use m_orientation (the rectangles rotation angle) in this formula?
Thanks!
edit: the reason I am doing this is for collision detection. I need to know where the coordinates of the rectangles (soon to be rigid bodies) lie in the world coordinate place for collision detection.
What you do is rotation [ special linear transformation] of a vector with angle Q on 2d.It keeps vector length and change its direction around the origin.
[linear transformation : additive L(m + n) = L(m) + L(n) where {m, n} € vector , homogeneous L(k.m) = k.L(m) where m € vector and k € scalar ] So:
You divide your vector into two pieces. Like m[1, 0] + n[0, 1] = your vector.
Then as you see in the image, rotation is made on these two pieces, after that your vector take
the form:
m[cosQ, sinQ] + n[-sinQ, cosQ] = [mcosQ - nsinQ, msinQ + ncosQ]
you can also look at Wiki Rotation
If you try to obtain eye coordinates corresponding to your object coordinates, you should multiply your object coordinates by model-view matrix in opengl.
For M => model view matrix and transpose of [x y z w] is your object coordinates you do:
M[x y z w]T = Eye Coordinate of [x y z w]T
This seems to be overcomplicating things somewhat: typically you would store an object's world position and orientation separately from its set of own local coordinates. Rotating the object is done in model space and therefore the position is unchanged. The world position of each coordinate is the same whether you do a rotation or not - add the world position to the local position to translate the local coordinates to world space.
Any rotation occurs around a specific origin, and the typical sin/cos formula presumes (0,0) is your origin. If the coordinate system in use doesn't currently have (0,0) as the origin, you must translate it to one that does, perform the rotation, then transform back. Usually model space is defined so that (0,0) is the origin for the model, making this step trivial.