I have a 3D shape (primitive shapes like cuboid, sphere, pyramid etc.). The shape is described by a set of necessary geometric parameters. For example, for a cube, I have information about the vertices, dimensions and orientation of the cube. Now, I have a voxel (kind of 3D grid of pixels). How can I determine which cells of the voxel needs to be filled and which will remain empty for "rendering" the cube in 3D?
Are there existing libraries for the same? Or, can someone point out the approach/algorithm that would help in this task? Also, what is this process exactly called, so that I can search on the Internet in the right direction?
It is called "Mesh to Voxel conversion" or "Mesh voxelization" or to be a bit more general "Volume Visualization".
If you are interested in math here is some good articles about it:
cs.swansea.ac.uk/~csmark/PDFS/cgfvoxel.pdf
Complete Polygonal Scene Voxelization
Ray Cast Methods
Related
What is the best way to project an arbitrary 2D polygon onto a 3D triangle mesh?
To make thing clearer, here is a visualization of the problem:
The triangle mesh is representing terrain and thus can be considered 2.5D. I want to be able to treat the projected polygon as a separate object.
This particular implementation is done in WebGL and three.js but any solution that fits an interactive 3D application is of interest.
If your question is not how to texture map the surface, then you really have to generate new 3D polygons.
You will be using some projection mechanism (such as a parallel one) that turns your 3D problem to 2D.
First backproject the surface onto the polygon plane. The polygon will be overlaid on a corresponding 2D mesh. Now for every facet, find the intersection (in the Boolean sense) of the facet and the polygon.
You will need a polygon intersection machinery for that purpose, such as the Weiler-Atherton or Sutherland-Hodgman clipping algorithms (the latter is much simpler, but works on convex windows only). (Also check http://www.angusj.com/delphi/clipper.php)
After clipping, you project to the original facet plane.
I understand that to create a shape (let's say a 3D sphere for an example) that I have to first find the vertex locations of the shape and second, use the parametric equation in order to create the x, y, z points of the triangle meshes. I am currently looking at a sample code to create shapes and it appears that after using the parametric equation in order to find the vectors of the triangle meshes, unit normals to the sphere at the vertices are found.
I understand why regular vectors in the first step are used to create the 3D shape and that a normal vector is perpendicular to the shape object, but I don't understand why the unit normal vectors at the vertices are used to create the shapes? What's the purpose of finding the normal of the vectors?
I am not sure I totally understand your question, but one very important use for normals in computer graphics is calculating reflections. For instance, if you're writing a simple raytracer, Lambertian reflectance is quite easy to compute if you know the normal vector where your camera ray intersects a surface. Normals are similarly required for (off the top of my head) the majority of calculations involved in more complex rendering techniques.
I have two objects: A sphere and an object. Its an object that I created using surface reconstruction - so we do not know the equation of the object. I want to know the intersecting points on the sphere when the object and the sphere intersect. If we had a sphere and a cylinder, we could solve for the equation and figure out the area and all that but the problem here is that the object is not uniform.
Is there a way to find out the intersecting points or area on the sphere?
I'd start by finding the intersection of triangles with the sphere. First find the intersection of each triangle's plane and the sphere, which gives a circle. Then find the circle's intersection/s with the triangle edges in 2D using line/circle tests. The result will be many arcs which I guess you could approximate with lines. I'm not really sure where to go from here without knowing the end goal.
If it's surface area you're after, maybe a numerical approach would be better. I'd cover the sphere in points and count the number inside the non-uniform object. To find if a point is inside, maybe trace outwards and count the intersections with the surface (if it's odd, the point is inside). You could use the stencil buffer for this if you wanted (similar to stencil shadows).
If you want the volume of intersection a quick google search gives "carve", a mesh based CSG library.
Starting with triangles versus the sphere will give you the points of intersection.
You can take the arcs of intersection with each surface and combine them to make fences around the sphere. Ideally your reconstructed object will be in winged-edge format so you could just step from one fence segment to the next, but with reconstructed surfaces I guess you might need to apply some slightly fuzzy logic.
You can determine which side of each fence is inside the reconstructed object and which side is out by factoring in the surface normals along the fence.
You can then cut the sphere along the fences and add the internal bits to the display.
For the other side of things you could remove any triangle completely inside the sphere and cut those that intersect.
I'm using DirectX10 to simulate a water surface, and I'm now with a height map,which is a 2D array of the heights(y) at the points (x,z). But to draw it on the screen, I must turn it into a mesh or have a index to draw triangle topology.
But the data is too large to do it manually. Are there any methods for me to draw it on the screen. I hope it's easy to implement. If there is function included in DirectX10 which can make it, the it's the best one for me.
Create a mesh that format a grid of squares (each made of two triangles) and set all vertices y = 0. In the vertex shader sample the heightmap and add the value stored in the heightmap to the y of the vertice.
This might help you.
P.S: If the area you want it to cover is too big you should take a look at terrain LOD techniques (should work the same for water).
I'm sure you can make a mesh out of it. I doubt you can generate the heightmap for a water surface that is too large to "meshify".
Why are you looking at Diamond square. For a 512x512 heightmap all you need to do is define a set of point and then generate the triangles for it. Its really very simple.
I have a class that holds a 4x4 matrix for scaling and translations. How would I implement rotation methods to this class? And should I implement the rotation as a separate matrix?
You can multiply Your current matrix with a rotation matrix. Take a look at http://en.wikipedia.org/wiki/Rotation_matrix
There's a site which I use every time when I need to look up the details of a 3D transformation, called http://www.euclideanspace.com. The particular page on matrix rotations can be found here.
Edit: Rotation around a given axis, look at the axis & angle representation. This page also links to a description on how to translate one representation to another.
If you need to rotate around mutiple axes, simply multiply the corresponding matrices.
Answering the second half of the question, a single 4x4 matrix is perfectly capable of holding a scaling, a translation, and a rotation. So unless you've put special limitations on what sort of 4x4 matrices you can handle, a single 4x4 is a fine for what you want.
As for rotation about an arbitrary vector (as you are asking in comments), look at the "Rotation about an arbitrary vector" section in the Wikipedia article yabcok links to. You will want to extend that to a 4x4 matrix by padding it out with zeros except for the 4,4 (scaling) position, which should be one. Then use matrix multiplication with your scaling/translation 4x4 to generate a new 4x4 matrix.
You want to make sure you find a reference which talks about the right kind of matrix that's used for computer graphics (namely 3D homogeneous coordinates using a 4x4 transformation matrix for rotation/translation/skewing).
See a computer graphics "bible" such as Foley and Van Dam (pg. 213), or one of these:
The Mathematics of the 3D Rotation Matrix
Mathematics of 3D Graphics
MSDN 3D graphics tutorial
SIGGRAPH article about 3D rotation
other page from CProgramming.com
This page has quite a bit of useful information:
http://knol.google.com/k/matrices-for-3d-applications-translation-rotation