While using Cohen-Sutherland line clipping algorithm, the clipping window is a rectangle. Is it possible to clip a line using a triangular or circular windows using a similar technique?
The idea can be reused for clipping against a triangle, with seven regions instead of nine. You can easily see what pair of origin/destination regions result in no visibility or full visibility. For the remaining cases deeper analysis is required.
For circles the coding is of less use because two "outside" codes are not enough to decide. But clipping a segment against a circle is simple: write the parametric equation of the segment and find the values of the parameter giving a point inside the circle (this amounts to the resolution of a quadratic equation).
Related
Using only a box function, what is the proper way to draw an annulus (wide circle) using Bresenham's algorithm? I assume that consecutive parallel lines could be drawn, but that using an angled line instead of a point would be more feasible, but also involve trigonometry.
I am using Python, but examples in any language appreciated.
You cannot fill all ring points with radial lines, because for R2=2*R1 outer circumference contains twice as much points in it's raster representations, and there will be empty places near outer circle.
Graphics engines (DirectX, OpenGL and so on) often use triangle fans to fill the circles, ellipses, rings.
I am working on a 3d application and am currently looking for a way to project a line segment defined by two points in screen-space onto a three-dimensional polygonal mesh (in my case a triangle mesh). The goal is to find the intersection points in world-space of the line segment with the edges of the mesh.
I can only think of two ways to do this, but neither is ideal. The first is to sample the line segment (in screen-space) at small intervals and ray trace at those intervals to find the world-space coordinates where the ray hits the mesh, but this does not easily give me the intersection points of the line segment with the mesh edges.
The other way I can think of is to somehow back-project the mesh into screen-space, find the intersections there (in 2d) and then project those intersection points back to 3d. The problem with this is that the screen-space coordinate system may change between the selection of the first and second endpoints of the line segment (due to moving the camera).
If any of that was confusing, then here is an image that approximately shows what I'm trying to do (the white dots indicate the points that I want to find). However, in my case the yellow curve is simply a line segment.
[Yunjin Lee, et al. "Mesh scissoring with minima rule and part salience." 2005]
Any help is very much appreciated.
Here's my suggestion:
Project the screen line into world space (getting a plane in world space).
Intersect the plane with the triangles in the mesh, getting a set of edges.
Add the edges to a data structure that keeps only the parts of the edges that are closest to the camera plane (see the diagram below, in which the red line segments and their endpoints are the ones we want to keep). This is like building up an image via a Z-buffer, except that because we know that this set is piecewise linear, we don't have to rasterize it, we can just maintain a sorted list of endpoints.
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 know the Bresenham and related algorithms, and I found a good algorithm to draw a circle with a 1-pixel wide border. Is there any 'standard' algorithm to draw a circle with an n-pixel wide border, without restoring to drawing n circles?
Drawing the pixel and n2 surrounding pixels might be a solution, but it draws many more pixels than needed.
I am writing a graphics library for an embedded system, so I am not looking for a way to do this using an existing library, although a library that does this function and is open source might be a lead.
Compute the points for a single octant for both radii at the same time and simultaneously replicate it eight ways, which is how Bresenham circles are usually drawn anyway. To avoid overdrawing (e.g., for XOR drawing), the second octant should be constrained to draw outside the first octant's x-extents.
Note that this approach breaks down if the line is very thick compared to the radius.
Treat it as a rasterization problem:
Take the bounding box of your annulus.
Consider the image rows falling in the bounding box.
For each row, compute the intersection with the 2 circles (ie solve x^2+y^2=r^2, so x=sqrt(r^2-y^2) for each, for x,y relative to the circle centres.
Fill in the spans. Repeat for next row.
This approach generalizes to all sorts of shapes, can produce sub-pixel coordinates useful for anti-aliasing and scales better with increasing resolution than hacky solutions involving multiple shifted draws.
If the sqrt looks scary for an embedded system, bear in mind there are fast approximate algorithms which would probably be good enough, especially if you're rounding off to the nearest pixel.
I need to create a (large) set of spatial polygons for test purposes. Is there an algorithm that will create a randomly shaped polygon staying within a bounding envelope? I'm using OGC Simple stuff so a routine to create the well known text is the most useful, Language of choice is C# but it's not that important.
Here you can find two examples of how to generate random convex polygons. They both are in Java, but should be easy to rewrite them to C#:
Generate Polygon example from Sun
from JTS mailing list, post Minimum Area bounding box by Michael Bedward
Another possible approach based on generating set of random points and employ Delaunay tessellation.
Generally, problem of generating proper random polygons is not trivial.
Do they really need to be random, or would some real WKT do? Because if it will, just go to http://koordinates.com/ and download a few layers.
What shape is your bounding envelope ? If it's a rectangle, then generate your random polygon as a list of points within [0,1]x[0,1] and scale to the size of your rectangle.
If the envelope is not a rectangle things get a little more tricky. In this case you might get best performance simply by generating points inside the unit square and rejecting any which lie in the part of the unit square which does not scale to the bounding envelope of your choice.
HTH
Mark
Supplement
If you wanted only convex polygons you'd use one of the convex hull algorithms. Since you don't seem to want only convex polygons your suggestion of a circular sweep would work.
But you might find it simpler to sweep along a line parallel to either the x- or y-axis. Assume the x-axis.
Sort the points into x-order.
Select the leftmost (ie first) point. At the y-coordinate of this point draw an imaginary horizontal line across the unit square. Prepare to create a list of points along the boundary of the polygon above the imaginary line, and another list along the boundary below it.
Select the next point. Add it to the upper or lower boundary list as determined by it's y-coordinate.
Continue until you're out of points.
This will generate convex and non-convex polygons, but the non-convexity will be of a fairly limited form. No inlets or twists and turns.
Another Thought
To avoid edge crossings and to avoid a circular sweep after generating your random points inside the unit square you could:
Generate random points inside the unit circle in polar coordinates, ie (r, theta).
Sort the points in theta order.
Transform to cartesian coordinates.
Scale the unit circle to a bounding ellipse of your choice.
Off the top of my head, that seems to work OK