I have a convex closed shape in 2 D space (on the x-y plane). I do not know what it looks like. I rotate this shape about approximately the center of its bounding box 64 times by 5.625 degrees (360/64). For each rotation I have the x-coordinates of the extreme points of the shape. In other words I know the left and right x extents of the shape for each rotation (assuming an orthographic projection). How do I obtain 64 points on the shape that do not contradict the x projections.
Note that the 2D shape is rotating, but the coordinate axes are not rotating along with it. So if your object is a line, the x projection of each end if plotted will essentially be a sin/cos wave depending on its original orientation.
The higher the number of rotations, if I have the solution - the closer I will get to my actual shape.
In reality I do not know the exact point I am rotating the shape about, however any solution assuming I do know will still be helpful, as I don't mind the reconstruction being imperfect.
We used the straight-forward method to reconstruct.
Projection is a shade of the object.
You start with a bounding 2D box. For each projection you cut away from the 2D shape left and right parts that fall outside of the projection. So, the main function calculates intersection of two convex 2D shapes. You calculate these intersections for each projection.
We have several purple projections P1, P2, P3, P4 of the original green object:
Knowing position of a purple projection build two red rays coming from the end points of a projection and intersect them with the reconstructed object:
The red object was reconstructed using 4 projections. When compared to original green you can see that they are not the same. The more projections you have the less error you'll get in the final result.
Related
Apologies in advance for my feeble maths.
I'm trying to be able to find the corners of a plane in space based on the equation of that plane. Here's what I know. I know three points on the plane and I know where they fall in the 2d coordinate space of the plane (x,y) and where they are in 3d space. I know the width and height of the plane and I can now calculate the equation of the plane. The plane sits on the inside of a large sphere that surrounds the origin so, in theory, it should more or less face where the camera is (though in my diagram it doesn't face the origin as it's just for illustrative purposes)
But it's not clear to me how I can use that to figure out another point. One thought I had was to find the transform that moves the plane parallel to the xy axis and rotate it round one of the points (so it stays in the same place), find the position of the new point, and then rotate it by the inverse of that transform. But it's not clear to me how I would find that transform matrix or how to use it. Could I do this using the normal and vector maths? I understand what normals are, but I'm fuzzy about how to use them.
I have a projected view of a 3D scene. The 2D points are computed by multiplying the 3D points in homogenous coordinates by a view matrix (which includes a translation and rotation) and a perspective matrix. I want to allow the user to move control points which describe the three axes, and update the rotation matrix based on this.
How do I compute the new rotation matrix given a change in projected 2D coordinates, assuming rotation around the origin? Solving for the position of the end of the single axis has a large degeneracy in the set of possible, but maybe solving for rotation in the axes perpendicular to the moved axis might work.
Newbie to three.js. I have multiple n-sided polygons to be displayed as faces (I want the polygon face to be opaque). Each polygon is facing a different direction in 3D space (essentially theses faces are part of some building).
Here are a couple of methods I tried, but they do not fit the bill:
Used Geometry object and added the n-vertices and used line mesh. It created the polygon as a hollow polygon. As my number of points are not just 3 or 4, I could not use the Face3 or Face4 object. Essentially a Face-n object.
I looked at the WebGL geometric shapes example. The shape object works in 2D and extrusion. All the objects in the example are on one plane. While my requirement is each polygon has a different 3D normal vector. Should I use 2D shape and also take note of the face normal and rotate the 2D shape after rendering.
Or is there a better way to render multiple 3D flat polygons with opaque faces with just x, y, z vertices.
As long as your polygons are convex you can still use the Face3 object. If you take one n-sided polygon, lets say a hexagon, you can create Face3 polygons by taking vertices numbered (0,1,2) as one face, vertices (0,2,3) as another face, vertices (0,3,4) as other face and vertices (0,4,5) as last face. I think you can get the idea if you draw it on paper. But this works only for convex polygons.
I have an image of a 3D rectangle (which due to the projection distortion is not a rectangle in the image). I know the all world and image coordinates of all corners of this rectangle.
What I need is to determine the world coordinate of a point in the image inside this rectangle. To do that I need to compute a transformation to unproject that rectangle to a 2D rectangle.
How can I compute that transform?
Thanks in advance
This is a special case of finding mappings between quadrilaterals that preserve straight lines. These are generally called homographic transforms. Here, one of the quads is a rectangle, so this is a popular special case. You can google these terms ("quad to quad", etc) to find explanations and code, but here are some sites for you.
Perspective Transform Estimation
a gaming forum discussion
extracting a quadrilateral image to a rectangle
Projective Warping & Mapping
ProjectiveMappings for ImageWarping by Paul Heckbert.
The math isn't particularly pleasant, but it isn't that hard either. You can also find some code from one of the above links.
If I understand you correctly, you have a 2D point in the projection of the rectangle, and you know the 3D (world) and 2D (image) coordinates of all four corners of the rectangle. The goal is to find the 3D coordinates of the unique point on the interior of the (3D, world) rectangle which projects to the given point.
(Do steps 1-3 below for both the 3D (world) coordinates, and the 2D (image) coordinates of the rectangle.)
Identify (any) one corner of the rectangle as its "origin", and call it "A", which we will treat as a vector.
Label the other vertices B, C, D, in order, so that C is diagonally opposite A.
Calculate the vectors v=AB and w=AD. These form nice local coordinates for points in the rectangle. Points in the rectangle will be of the form A+rv+sw, where r, s, are real numbers in the range [0,1]. This fact is true in world coordinates and in image coordinates. In world coordinates, v and w are orthogonal, but in image coordinates, they are not. That's ok.
Working in image coordinates, from the point (x,y) in the image of your rectangle, calculate the values of r and s. This can be done by linear algebra on the vector equations (x,y) = A+rv+sw, where only r and s are unknown. It will boil down to a 2x2 matrix equation, which you can solve generally in code using Cramer's rule. (This step will break if the determinant of the required matrix is zero. This corresponds to the case where the rectangle is seen edge-on. The solution isn't unique in that case. If that's possible, make special exception.)
Using the values of r and s from 4, compute A+rv+sw using the vectors A, v, w, for world coordinates. That's the world point on the rectangle.
So we have such situation:
In this illustration, the first quadrilateral is shown on the Image Plane and the second quadrilateral is shown on the World Plane. [1]
In my particular case the Image Plane has 3 quadrilaterals - projections of real world squares, which, as we know, have same size, lying on the same plane, with same rotation relative to the plane they are lying on, and are not situated on same line on plane.
I wonder if we can get rotation angles of Image Plane to World Plane knowing stuff described?
In my case as input I have such data structures: original image (RGB pixels), objects (squares) with angles points in pixels (x,y) on Image Plane.
Take a look at Sections 2 and 3 of Algorithms for plane-based pose estimation.
The methods described there assume that you know the (x,y) coordinates of the features in question - in this case the red squares.
The problem you are describing is generally known as pose estimation - determining the 3D orientation and position of an object relative to a camera from a 2D view. For you, the object is a plane. Googling 'pose estimation plane' should give you more sources.