I'm working on a problem to eliminate common line segments in a collection of Paths. Many of these paths share the same segment.
It seems that a 2D line would have some way to uniquely identity itself. Like a Key.
So a Line [(A,B), (C,D)] is the same as [(C,D), (A,B)]
Only Solution I could come up with is to sort the points.
This seems like it would be a common problem in Math or Graphics but the solution escapes me.
From a mathematical point of view, this looks like a matter of an undirected graph (as opposed to a directed graph).
Sorting the points is one way to handle this: it's a straightforward way to represent an unordered edge with a single, unambiguously selected value (it shouldn't matter what ordering you choose, as long as it is consistent for all possible segments). You do need to ensure that you maintain this ordering as an invariant: accidentally slipping in a mis-ordered edge could cause problems for anything that depends on the ordering.
However, mathematically speaking, undirected graphs are often defined as directed graphs with a symmetry property: if (A,B) is an edge, then so is (B,A). This suggests another way: ensure that you always store both (A,B) and (B,A). Perhaps both segment orderings could have a link to any common data, and possibly a fast way to access one from the other. (As with the sorted point method, you need to maintain this symmetry as an invariant.)
The best choice depends on your application. If you're using your segments as a key, the sorting method might be best. However, some applications are a better match for the symmetric method. For example, the doubly connected edge list is a data structure which represents each edge as two linked "half-edges", one in each direction.
Since you mention graphics, note that the doubly connected edge list is often used to represent the edges of 3-D polytopes.
Also, note the similarity to oriented triangles: there are good, practical reasons for computer graphics to treat triangles as "one-sided", such that drawing a triangle visible from one side is distinct from drawing the same triangle visible from the other. Like half-edges, this distinction is determined by the order of the vertices: clockwise for one side, counterclockwise for the other.
Related
I am trying to understand how to manually generate objects.
I have a mesh, part of which I delete and create a new geometry in its place. I have information about the normals of deleted vertices. On the basis of which I have to build new faces (in a different size and quantity) looking in the same direction.
But I don’t understand how to choose the correct winding. It sounds easy when the lessons talk about CCW winding in screen space. But what if I have a bunch of almost chaotic points in the model space? How then to determine this CCW, which axis is used for this? I suggest that the nearest old normals might help. But what is the cheapest method to determine the correct order?
It turned out to be easier than I thought. It is necessary to find the cross product of the first two vectors from the vertices of a triangle, then find the dot of the resulting vector and the normal vector, if the result is negative, then during generation it is necessary to change the order of vertices.
I want to write a program with 'geometry automata'. I'd like it to be a companion to a book on artistic designs. There will be different units, like the 'four petal unit' and 'six petal unit' shown below, and users and choose rulesets to draw unique patterns onto the units:
I don't know what the best data structure to use for this project is. I also don't know if similar things have been done and if so, using what packages or languages. I'm willing to learn anything.
All I know right now is 2D arrays to represent a grid of units. I'm also having trouble mathematically partitioning the 'subunits'. I can see myself just overlapping a bunch of unit circle formulas and shrinking the x/y domains (cartesian system). I can also see myself representing the curve from one unit to another (radians).
Any help would be appreciated.
Thanks!!
I can't guarantee that this is the most efficient solution, but it is a solution so should get you started.
It seems that a graph (vertices with edges) is a natural way to encode this grid. Each node has 4 or 6 neighbours (the number of neighbours matches the number of petals). Each node has 8 or 12 edges, two for each neighbour.
Each vertex has an (x,y) co-ordinate, for example the first row in in the left image, starting from the left is at location (1,0), the next node to its right is (3,0). The first node on the second row is (0,1). This can let you make sure they get plotted correctly, but otherwise the co-ordinate doesn't have much to do with it.
The trouble comes from having two different edges to each neighbour, each aligned with a different circle. You could identify them with the centres of their circles, or you could just call one "upper" and the other "lower".
This structure lets you follow edges easily, and can be stored sparsely if necessary in a hash set (keyed by co-ordinate), or linked list.
Data structure:
The vertices can naturally be stored as a 2-dimensional array (row, column), with the special characteristic that every second column has a horizontal offset.
Each vertex has a set of possible connections to those vertices to its right (upper-right, right, or lower right). The set of possible connections depends on the grid. Whether a connection should be displayed as a thin or a thick line can be represented as a single bit, so all possible connections for the vertex could be packed into a single byte (more compact than a boolean array). For your 4-petal variant, only 4 bits need storing; for the 6-petal variant you need to store 6 bits.
That means your data structure should be a 2-dimensional array of bytes.
Package:
Anything you like that allows drawing and mouse/touch interaction. Drawing the connections is pretty straightforward; you could either draw arcs with SVG or you could even use a set of PNG sprites for different connection bit-patterns (the sprites having partial transparency so as not to obscure other connections).
What are some path finding algorithms used in games of all types? (Of all types where characters move, anyway) Is Dijkstra's ever used? I'm not really looking to code anything; just doing some research, though if you paste pseudocode or something, that would be fine (I can understand Java and C++).
I know A* is like THE algorithm to use in 2D games. That's great and all, but what about 2D games that are not grid-based? Things like Age of Empires, or Link's Awakening. There aren't distinct square spaces to navigate to, so what do they do?
What do 3D games do? I've read this thingy http://www.ai-blog.net/archives/000152.html, which I hear is a great authority on the subject, but it doesn't really explain HOW, once the meshes are set, the path finding is done. IF A* is what they use, then how is something like that done in a 3D environment? And how exactly do the splines work for rounding corners?
Dijkstra's algorithm calculates the shortest path to all nodes in a graph that are reachable from the starting position. For your average modern game, that would be both unnecessary and incredibly expensive.
You make a distinction between 2D and 3D, but it's worth noting that for any graph-based algorithm, the number of dimensions of your search space doesn't make a difference. The web page you linked to discusses waypoint graphs and navigation meshes; both are graph-based and could in principle work in any number of dimensions. Although there are no "distinct square spaces to move to", there are discrete "slots" in the space that the AI can move to and which have been carefully layed out by the game designers.
Concluding, A* is actually THE algorithm to use in 3D games just as much as in 2D games. Let's see how A* works:
At the start, you know the coordinates of your current position and
your target position. You make an optimistic estimate of the
distance to your destination, for example the length of the straight
line between the start position and the target.
Consider the adjacent nodes in the graph. If one of them is your
target (or contains it, in case of a navigation mesh), you're done.
For each adjacent node (in the case of a navigation mesh, this could
be the geometric center of the polygon or some other kind of
midpoint), estimate the associated cost of traveling along there as the
sum of two measures: the length of the path you'd have traveled so
far, and another optimistic estimate of the distance that would still
have to be covered.
Sort your options from the previous step by their estimated cost
together with all options that you've considered before, and pick
the option with the lowest estimated cost. Repeat from step 2.
There are some details I haven't discussed here, but this should be enough to see how A* is basically independent of the number of dimensions of your space. You should also be able to see why this works for continous spaces.
There are some closely related algorithms that deal with certain problems in the standard A* search. For example recursive best-first search (RBFS) and simplified memory-bounded A* (SMA*) require less memory, while learning real-time A* (LRTA*) allows the agent to move before a full path has been computed. I don't know whether these algorithms are actually used in current games.
As for the rounding of corners, this can be done either with distance lines (where corners are replaced by circular arcs), or with any kind of spline function for full-path smoothing.
In addition, algorithms are possible that rely on a gradient over the search space (where each point in space is associated with a value), rather than a graph. These are probably not applied in most games because they take more time and memory, but might be interesting to know about anyway. Examples include various hill-climbing algorithms (which are real-time by default) and potential field methods.
Methods to procedurally obtain a graph from a continuous space exist as well, for example cell decomposition, Voronoi skeletonization and probabilistic roadmap skeletonization. The former would produce something compatible with a navigation mesh (though it might be hard to make it equally efficient as a hand-crafted navigation mesh) while the latter two produce results that will be more like waypoint graphs. All of these, as well as potential field methods and A* search, are relevant to robotics.
Sources:
Artificial Intelligence: A Modern Approach, 2nd edition
Introduction to The Design and Analysis of Algorithms, 2nd edition
I have been looking at posts about determining if a point lies within a polygon or not and the answers are either too vague, abstract, or complex for me. So I am going to try to ask my question specific to what I need to do.
I have a set of points that describe a non-straight line (sometimes a closed polygon). I have a rectangular "view" region. I need to determine as efficiently as possible whether any of the line segments (or polygon borders) pass through the view region.
I can't simply test each point to see if it lies within the view region. It is possible for a segment to pass through the region without any point actually inside the region (ie the line is drawn across the region).
Here is an example of what I want to determine (red means the function should return true for the set of points, blue means it should return false, example uses straight lines and rectangles because I am not an artist).
Another condition I want to be able to account for (though the method/function may be a separate one), is to determine not just whether a polygon's border passes through the rectangular region, but whether the region is entirely encompassed by the polygon. The nuance here is that in the situation first described above, if I am only concerned with drawing borders, the method should return false. But in the situation described here, if I need to fill the polygon region then I need the function to return true. I currently do not need to worry about testing "donut" shaped polygons (thank God!).
Here is an example illustrating the nuance (the red rectangle does not have a single vertex or border segment passing through the on-screen region, but it should still be considered on-screen):
For the "does any line segment or polygon border pass through or lie on screen?" problem I know I can come up with a solution (albeit perhaps not an efficient one). Even though it is more verbose, the conditions are clear to me. But the second "is polygon region on screen?" problem is a little harder. I'm hoping someone might have a good suggestion for doing this. And if one solution is easily implemented on top of the other, well, booya.
As always, thank you in advance for any help or suggestions.
PS I have a function for determining line intersection, but it seems like overkill to use it to compare each segment to each side of the on-screen region because the on-screen region is ALWAYS a plain [0, 0, width, height] rectangle. Isn't there some kind of short-cut?
What you are searching for is named a Collision Detection Algorithm A Google search will lead you to plenty of implementations in various language as well as a lot of theory
There are plenty of Geometric theory behind, from the simplest bisector calculus to Constrained Delaunay Triangulations and Voronoi Diagrams (that are just examples). It depends on the shape of Object, the number of dimensions and for sure the ratio between exactness needed and computing time afforded ;-)
Good read
PS I have a function for determining
line intersection, but it seems like
overkill to use it to compare each
segment to each side of the on-screen
region because the on-screen region is
ALWAYS a plain [0, 0, width, height]
rectangle. Isn't there some kind of
short-cut?
It's not an overkill, its neccessary here. The only kind of shortcut I can think of is to hardcode values [0, 0, width, height] into that function and simplify it a bit.
This is just a LARGE generalized question regarding rays (and/or line segments or edges etc) and their place in a software rendered 3d engine that is/not performing raytracing operations. I'm learning the basics and I'm the first to admit that I don't know much about this stuff so please be kind. :)
I wondered why a parameterized line is not used instead of a ray(or are they??). I have looked around at a few cpp files around the internet and seen a couple of resources define a Ray.cpp object, one with a vertex and a vector, another used a point and a vector. I'm pretty sure that you can define an infinate line with only a normal or a vector and then define intersecting points along that line to create a line segment as a subset of that infinate line. Are there any current engines implementing lines in this way, or is there a better way to go about this?
To add further complication (or simplicity?) Wikipedia says that in vector space, the end points of a line segment are often vectors, notably u -> u + v, which makes alot of sence if defining a line by vectors in space rather than intersecting an already defined, infinate line, but I cannot find any implementation of this either which makes me wonder about the validity of my thoughts when applying this in a 3d engine and even further complication is created when looking at the Flash 3D engine, Papervision, I looked at the Ray class and it takes 6 individual number values as it's parameters and then returns them as 2 different Number3D, (the Papervision equivalent of a Vector), data types?!?
I'd be very interested to see an implementation of something which actually uses the CORRECT way of implementing these low level parts as per their true definitions.
I'm pretty sure that you can define an infinate line with only a normal or a vector
No, you can't. A vector would define a direction of the line, but all the parallel lines share the same direction, so to pick one, you need to pin it down using a specific point that the line passes through.
Lines are typically defined in Origin + Direction*K form, where K would take any real value, because that form is easy for other math. You could as well use two points on the line.