I have a 3D Cartesian cube. For each point in this cube there is a corresponding density value. When the density changes suddenly it means that there is a cavity. Now to find the cavity I calculate the gradient at each point in the cube. This gives me a point cloud on the surface of the cavity. I would now like to mesh the surface of the cavity given the point cloud.
Unfortunately I don't have any experience with surface reconstruction and was wondering if someone can recommend a suitable algorithm which will produce a closed surface of the cavity?
The cube is quite big so a point cloud of the surface of a cavity can easily be 500.000 points or more. I have read this post: robust algorithm for surface reconstruction from 3D point cloud? which I find useful. However it seems that the problem I am facing is simpler, given that:
The coordinates of the points are always integer
The point distribution is even
The distance from one point to its closest neighbour is either 1, sqrt(2) or sqrt(3)
You probably want the marching cubes algorithm.
The Marching Cubes algorithm will do exactly what you want. For a working implementation (using Three.js for rendering the graphics), check out:
http://stemkoski.github.com/Three.js/Marching-Cubes.html
For more details on the theory, I think the best article is the website:
http://paulbourke.net/geometry/polygonise/
Related
My problem involves matching a set of 2d points to a set of 3d points, with known correspondence between the two. Basically I have points on an image, and I need the optimal translation and rotation to fit the points to a known 3d point cloud. Kabsch algorithm is originally meant for finding the best fit of 3d points to another point cloud, and there are implementations out there for 2d to 2d, but not something I can use. I do know it's possible, but just don't know how to go about it. I searched for code out there and came up empty. I'm programming in matlab at the moment, but any language would do.
Thank you.
Edit: The goal is getting a rotation and translation of the 3d point cloud to best match the 2d points when it is projected onto an image plane.
I should also mention that the 3d to 2d projection is done using a weak perspective.
So basically, you have a "plane" or a "line" of points, like the third dimension was 0. You could threat them like this, and use the tipicall kabsh algorithm of squared distance minimisation, don't you?
EDIT: maybe it's a nonsense, but what about projecting the 3d body to 2d coordinates, and do a 2d comparison? Computationally is expensive, so it includes exploring all the angles of the 3d object + projection, but it's easier losing one dimension by applying a projection, that adding a new dimenssion to a 2d point.
I've been scouring the internet for days, but have been unable to find a good answer (or at least one that made sense to me) to what seems like it should be a common question. How does one scale an arbitrary polygon? In particular, concave polygons. I need an algorithm which can handle concave (definitely) and self-intersecting (if possible) polygons. The obvious and simple algorithm I've been using to handle simple convex polygons is calculating the centroid of the polygon, translating that centroid to the origin, scaling all the vertices, and translating the polygon back to its original location.
This approach does not work for many (or maybe all) concave polygons as the centroid often falls outside the polygon, so the scaling operation also results in a translation and I need to be able to scale the polygon "in place" without the final result being translated.
Is anybody aware of a method for scaling concave polygons? Or maybe a way of finding the "visual center" which can be used as a frame of reference for the scaling operation?
Just to clarify, I'm working in 2D space and I would like to scale my polygons using the "visual center" as the frame of reference. So maybe another way to ask the question would be, how do I find the visual center of a concave and/or self-intersecting polygon?
Thanks!
I'm not sure what your problem is.
You're working in an affine space, and you're looking for an affine transformation to scale your polygon ?
If i'm right, just write the transformation matrix:
scaling matrix
homotethy
And transform your polygon with matrix
You can look up for affine transformation matrix.
hope it helps
EDIT
if you want to keep the same "center", you can just do an homotethy of parameter lambda with center G = barycenter of the polygon:
it verifies :
G won't move since it's the center of the homotethy.
It will still verify the relation below, so it will still be the barycenter. (you just multiply the relation by lambda)
in your case G is easy to determinate: G(x,y) : (average of x values of points, average of y values of points)
and it should do what you need
Perhaps Craig is looking for a "polygon offset" algorithm - where each edge in the polygon is offset by a given value. For example, given a clockwise oriented polygon, offsetting edges towards the left will increase the size of the polygon. If this is what Craig is looking for then this has been asked and answered before here - An algorithm for inflating/deflating (offsetting, buffering) polygons.
If you're looking for a ready made (opensource freeware) solution, I've also created a clipping library (Clipper) written in Delphi, C++ and C# which includes a rather simple polygon offsetting function.
The reason why you can't find a good answer is because you are being imprecise with your requirements. First explicitly define what you mean by "in-place". What is being kept constant?
Once you have figured that out, then translate the constant point to the origin, scale the polygon as usual, and translate back.
We have a grid with red squares on it. Meaning we have an array of 3 squares (with angles == 90 deg) which as we know have same size, lying on the same plane and with same rotation relative to the plane they are lying on, and are not situated on same line on plane.
We have a projection of the space which contains the plane with squares.
We want to turn our plane projection with squares so that we would see it like it's facing us, in general we need a formula for turning each point of that original plane projection so that it would be facing us like on the image below.
What formulas can be used for solving such problem, how to solve it, has any one faced something like this before?
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 square, 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 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.
Starting with a 3D mesh, how would you give a rounded appearance to the edges and corners between the polygons of that mesh?
Without wishing to discourage other approaches, here's how I'm currently approaching the problem:
Given the mesh for a regular polyhedron, I can give the mesh's edges a rounded appearance by scaling each polygon along its plane and connecting the edges using cylinder segments such that each cylinder is tangent to each polygon where it meets that polygon.
Here's an example involving a cube:
Here's the cube after scaling its polygons:
Here's the cube after connecting the polygons' edges using cylinders:
What I'm having trouble with is figuring out how to deal with the corners between polygons, especially in cases where more than three edges meet at each corner. I'd also like an algorithm that works for all closed polyhedra instead of just those that are regular.
I post this as an answer because I can't put images into comments.
Sattle point
Here's an image of two brothers camping:
They placed their simple tents right beside each other in the middle of a steep walley (that's one bad place for tents, but thats not the point), so one end of each tent points upwards. At the point where the four squares meet you have a sattle point. The two edges on top of each tent can be rounded normally as well as the two downward edges. But at the sattle point you have different curvature in both directions and therefore its not possible to use a sphere. This rules out Svante's solution.
Selfintersection
The following image shows some 3D polygons if viewed from the side. Its some sharp thing with a hole drilled into it from the other side. The left image shows it before, the right after rounding.
.
The mass thats get removed from the sharp edge containts the end of the drill hole.
There is someething else to see here. The drill holes sides might be very large polygons (lets say it's not a hole but a slit). Still you only get small radii at the top. you can't just scale your polygons, you have to take into account the neighboring polygon.
Convexity
You say you're only removing mass, this is only true if your geometry is convex. Look at the image you posted. But now assume that the viewer is inside the volume. The radii turn away from you and therefore add mass.
NURBS
I'm not a nurbs specialist my self. But the constraints would look something like this:
The corners of the nurbs patch must be at the same position as the corners of the scaled-down polygons. The normal vectors of the nurb surface at the corners must be equal to the normal of the polygon. This should be sufficent to gurarantee that the nurb edge will be a straight line following the polygon edge. The normals also ensure that no visible edges will result at the border between polygon and nurbs patch.
I'd just do the math myself. nurbs are just polygons. You'll have some unknown coefficients and your constraints. This gives you a system of equations (often linear) that you can solve.
Is there any upper bound on the number of faces, that meet at that corner?
You might you might employ concepts from CAGD, especially Non-Uniform Rational B-Splines (NURBS) might be of interest for you.
Your current approach - glueing some fixed geometrical primitives might be too inflexible to solve the problem. NURBS require some mathematical work to get used to, but might be more suitable for your needs.
Extrapolating your cylinder-edge approach, the corners should be spheres, resp. sphere segments, that have the same radius as the cylinders meeting there and the centre at the intersection of the cylinders' axes.
Here we have a single C++ header for generating triangulated rounded 3D boxes. The code is in C++ but also easy to transplant to other coding languages. Also it's easy to be modified for other primitives like quads.
https://github.com/nepluno/RoundCornerBox
As #Raymond suggests, I also think that the nepluno repo provides a very good implementation to solve this issue; efficient and simple.
To complete his answer, I just wrote a solution to this issue in JS, based on the BabylonJS 3D engine. This solution can be found here, and can be quite easily replaced by another 3D engine:
https://playground.babylonjs.com/#AY7B23
I have a point (Lat/Lon) and a heading in degrees (true north) for which this point is traveling along. I have numerous stationary polygons (Points defined in Lat/Lon) which may or may not be convex.
My question is, how do I calculate the closest intersection point, if any, with a polygon. I have seen several confusing posts about Ray Tracing but they seem to all relate to 3D when the Ray and Polygon are not on the same Plane and also the Polygons must be convex.
sounds like you should be able to do a simple 2d line intersection...
However I have worked with Lat/Long before and know that they aren't exactly true to any 2d coordinate system.
I would start with a general "IsPointInPolygon" function, you can find a million of them by googling, and then test it on your poly's to see how well it works. If they are accurate enough, just use that. But it is possible that due to the non-square nature of lat/long coordinates, you may have to do some modifications using Spherical geometry.
In 2D, the calculations are fairly simple...
You could always start by checking to make sure the ray's endpoint is not inside the polygon (since that's the intersection point in that case).
If the endpoint is out of the line, you could do a ray/line segment intersection with each of the boundary features of the polygon, and use the closest found location. That handles convex/concave features, etc.
Compute whether the ray intersects each line segment in the polygon using this technique.
The resulting scaling factor in (my accepted) answer (which I called h) is "How far along the ray is the intersection." You're looking for a value between 0 and 1.
If there are multiple intersection points, that's fine! If you want the "first," use the one with the smallest value of h.
The answer on this page seems to be the most accurate.
Question 1.E GodeGuru