Randy Gaul's Game Programming Blog: Collision

Showing posts with label Collision. Show all posts

Tuesday, July 3, 2012

Collision: Basic 2D Collision Detection + Jump Physics

This post will cover some various collision techniques for use in 2D games. Types of collisions to be covered in this article are:

Static point to static circle
Static circle to static circle
Static point to static rectangle
Static rectangle to static rectangle
Static circle to static rectangle
Jumping physics (optimal equation technique)

Static Circle to Static Point

In order to detect whether or not a point is colliding with a circle there are only a few very simple checks required. Luckily a circle is simply a radius and a point. Find the distance from the center of your circle and your colliding point, and check to see if it is larger, greater, or smaller than the radius. Larger means that the point is outside the circle, the same means the point is directly on the edge, and smaller means a collision is occuring.

// Given that CP is the center of the circle, P is the point,

// and R is the radius

if dist( CP, P ) is equal to R: Collision or Non-Collision (you choose)

if dist( CP, P ) is greater than R: No collision

if dist( CP, P ) is less than R: Collision

Static Circle to Static Circle
This collision test is going to be the exact same as Point to Circle, with one difference: you will be comparing the distance from the center of the circles to the radius of the first circle added with the radius of the second circle. This should make sense since the only time two circles will intersect is when their distances are smaller than their radii. Just be sure to avoid this scenario:

Internally tangent intersection.

This internally tangent intersection can be avoided by checking for whether or not the distance between the circles' centers is smaller than the radii combined before checking to see if the distances are equal.

Static Point to Static Rectangle Collision
This algorithm is extremely straightforward. Think of a rectangle as four values: top, left, bottom and right. Here are the checks required to see if a point is within a square or not (it shouldn't require much of any explanation):

// B is bottom of the rectangle

// T is top of the rectangle

// L is the left side of the rectangle

// R is the right side of the rectangle

// P is the point

// Perform the following checks:

if(P.X < L) and if(P.X > R)

and if(P.Y < B) and if(P.Y > T)

then no collision

All of these conditions will fail if there is a collision. To check for a tangent point, check to see if the point's X value equals the left or right side, while being under the top and above the bottom. Vice versa for the top and bottom tangents. This algorithm is going to be very similar to the ActionScript hittest function.

Static Rectange to Static Rectangle
Rectangle to rectangle collision is very simple as well. This should be quite similar to the point to rectangle, except we just use some basic logic to make sure none of the sides of the two rectangles are between each other's sides.

// Given rectangles A and B

if(leftA > rightB) or

if(leftB > rightA) or

if(topA < bottomB) or

if(topB < bottomA)

then no collision

Circle to Rectangle Collision
Circle to rectangle collision is going to be the most complicated algorithm here. First consider a circle to consist of only a point and a radius. The way we detected two circles colliding was by checking the distances from the centers to their radii. Sadly, a rectangle does not have a radius. However, given the closest point on the rectangle to the center of the circle, we can apply the same algorithm as the Static Point to Static Circle collision detection. All we need do now is find out how to find the closest point on the rectangle to the circle's center.

To find this point, clamp a copy of the circle's center coordinate to the edges of the rectangle:

// "Clamping" a point to the side of a rectangle
// Given left right bottom and top are sides of a rectangle
// P is a new variable created and assigned the value of the

// circle's center -- do not modify the circle's coordinate

Point P (new copy of the circles center) = Circle's Center

if(P.X > right) then P.X = right

else(P.X < left) then P.X = left

if(P.Y > top) then P.Y = top

else(P.Y < bottom) then P.Y = bottom then no collision

After the above code is run point P will be somewhere on the edge of the rectangle and will also be the closest point along the rectangle's sides to the center of the circle. Now compare the distance from these two points to the radius with the Static Point to Static Circle algorithm. Results will be the same.

Optimization of Distance Formula
By using some simple algebra one can remove the sqrt function from the distance formula for the purpose of checking two values:

This allows one to compare the distance squared with another distance squared; multiplication is much faster than a sqrt function. This should be applied to all of the above collision algorithms that require a distance function between two points.

Jumping Physics Technique
There are a few different ways to have a character jump in a game. I'm going to share a singular way that I find very efficient and effective. As detailed here in my article on simple physics integration, in order to move something along the screen you are required to use the following equations:

pos.x += vel.x * dt; 

pos.yx += vel.y * dt; 

vel.x += accel.x * dt;

vel.y += accel.y * dt;

This works just fine for moving images around! In order to simulate a jump you suddenly add a large value to the image's y velocity, and then apply a negative acceleration on it each timestep. Like I said, this works just fine. However as a designer this is a headache. How are you going to tweak the jumping height? Just apply random values to the velocity and guess/check to see how high the character goes? It would be best to be able to set how high you'd like the character to jump, and use some form of automation so solve for the necessary velocity.

This equation comes from a conservation of energy. Take a look at the algebraic manipulation:

The equation you start with represents an upward jump. The two sides are equal due to energy being conserved throughout the jump. At the start of a jump (and the end, assuming the start and end positions are the same y value) all of the energy is kinetic, thrusting the object upward. As it slows down energy is converted into potential energy, which is the distance from the ground the object is currently at. Once all kinetic energy is used potential energy is maxed out, which means this is the peak of the jump. We can take advantage of the fact that the kinetic energy is zeroed out and come up with a simple equation to solve for the necessary upward velocity to reach a specific height. Here's what the final equation can look like in use within code to initialize the y velocity for a jump:

// JUMP_HEIGHT will usually be in tiles.

// Example: perhaps to jump exactly 3 tiles, or 4

vel.y = sqrt( 2.f * -GRAVITY * JUMP_HEIGHT );

This technique not only can be used on jumping characters but projectiles as well. I find this equation exceptionally useful in initializing various values! An optimization to this function would be to pre-compute the velocity by hand and assign it to y velocities of objects from a constant. This method would work well in any situation where the velocity of the jumping height is not changing throughout the duration of the game (perhaps turrets shooting bullets in the same path over and over, or an enemy with only one type of jump).

Sunday, July 1, 2012

Hash Tables: Introduction

I've just set up my first hash table after a bit of studying; it's time to write about it! Docendo discimus ~

Wikipedia has a nice definition of what a hash table is: LINK.

In computer science, a hash table or hash map is a data structure that uses a hash function to map identifying values, known as keys (e.g., a person's name), to their associated values (e.g., their telephone number). Thus, a hash table implements an associative array. The hash function is used to transform the key into the index (the hash) of an array element (the slot or bucket) where the corresponding value is to be sought.

The idea of setting up a hash table is that by using a hash function you can index an array without actually knowing what the index is.

For example say you have an array of structures that look like this:

typedef struct _ASSETFILE

{

char *fileName_;
assetFile *asset_;
} ASSETFILE;

This array holds structures that represent asset files for some sort of project (images, sounds, data, etc). Imagine you want anyone to be able to access any asset file in this array by using a function like so: FindAsset( "filename" );. How would you go about creating this FindAsset function? The easiest way would be to use a linear search starting at the beginning of the array. Simply compare the string of the parameter passed to the FindAsset function with the string within the ASSETFILE structure until a match is found.

This can be a slow search. Perhaps the next optimization would be to order the entire list alphabetically from lowest to highest. This would allow a binary search through the array by using strcmp from the standard library. There wouldn't be any faster search method than a binary search in this scenario. Inserting into an ordered list would require a binary search as well.

However by using a hash function you can achieve even faster searches. By using a hash function you can convert the filename provided to the FindAsset function directly into an index to index the array of ASSETFILE structs. The filename is what is called a key. By hashing a key an index to the hash table is generated, and the corresponding ASSETFILE struct can be accessed very quickly. It's important to note that the order in which the entries are stored into slots is random. A good hash function will randomly map keys to indices with an even distribution (avoids clustering).

The relationship between keys and indices is not a one to one ratio however. Keys to indices is of a one-to-many ratio -that is multiple keys can resolve to the same address. Each key must be unique, however, in order for the hash table to function properly.

In short, using a hash table allows for indexing of an array by some other type of data than simply an integer. There are many different hash functions available on the internet, and a simple google search will yield many results.

I've set up a demonstration program to show the usage of a hash function here. I handle collisions with something called linear probing. A collision is whenever a key maps to a location in the table that already contains an entry (two or more keys are resolving to the same slot). A collision can be resolved in many different ways, though the way I chose in my sample program was linear probing. Linear probing is simply moving down one slot in the table until a free slot is found. The entry to insert is then placed. Each time the index is incremented (or decremented, direction doesn't matter as long as your are consistent) to find an empty slot it is called a probe. This means that for each lookup you'll need to compare the key to the entry being looked at and check for a match, as when you get a hashed index it doesn't necessarily mean that the key you have had been placed into that entry; a previous key may have inserted a value there already. When attempting a lookup first check the slot the hash resolves to. If no match is found, check the next slot (using the same direction as before) until you find a key match, or an empty slot. If you find an empty slot then that means your lookup concluded that no entry matches in your table.

Linear probing forms clusters of entries, however, since you resolve collision by placing entries into the next available slot. Each entry placed into the table increases the chances of a collision, and as the table fills up a lot of time ends up being spent on probing looking for an empty slot. There are ways of breaking up these clusters like double hashing, or by having each slot point to a linked list of entries. However with a good hash function clusters can be kept to a minimum as long as the table doesn't get too full.

Load factor is a term that represents the total number of current entries divided by the table size. Once a hash table has a load factor of .7 or so linear probing starts getting dramatically slow.

Image from Wikipedia. As you can see once the load factor of the table hits around .8 the time spent probing increases dramatically.

In order to retrieve an entry from the hash table (with linear probing, as in my sample program) all you'd have to do is take your key and pass it to your hash function. Once this is done you'll have the index to start your search. Check to see if the key matches the key within the index. If it does not match then increment (or decrement, whatever direction you probe in you must do here for consistency) the index by one and compare again. Once a match is found the lookup function has done its job! If no match is found and the search function runs into an empty slot, that means that no match had been found. Here's the lookup function from the sample program:

char *TableLookup( const char *Key, char *table[] )
{
unsigned index = UHashMod( Key, TABLESIZE );
unsigned indexStart = index;

// Only search through clusters, don't hop over empty entries
while(!IsEmpty( table, index ))
{
// Break from loop if found matching entry
if(table[index] != DELETED)
{
if(strcmp(table[index], Key) == 0)
break;
}

++index;

// wraparound to beginning of table
if(index == TABLESIZE)
{
index = 0;
}

// If searched entire table
if(indexStart == index)
{
return NULL;
}
}

return table[index];
}

In the above example the data stored in the table was the same as the key, so the return type is a pointer to the stored data in this instance. In order to support the deletion of entries you must mark a slot as deleted. This is because if a slot is simply marked as empty, then it can mess up the order of searches! Searching relies on the fact that collisions are resolved with linear probing. If you delete an entry that had previous collisions, the entries next to it will not be found in searches. However if you mark slots as "deleted" with a special value, than you can modify searching to not stop on "deleted" slots, and you can modify insertion to insert values into slots that are marked "deleted". You can see in the above code that searches hop over deleted slots, but stop at empty ones.

Now lets talk about this hash function. Creating hash functions seems very difficult, but luckily for around 50 or so years research has been put into them, and as such there lots of well documented hash functions and hash libraries all over the place. Here's the one I chose to use in my demonstration program:

// Hashing function from Program 14.2 in the Sedgewick book
unsigned UHashMod( const char *Key, unsigned TableSize )
{
unsigned hash = 0; // Initial value of hash
unsigned rand1 = 31415; // "Random" 1
unsigned rand2 = 27183; // "Random" 2

while (*Key)
{
hash = hash * rand1; // Multiply hash by random
hash = (hash + *Key); // Add in current char, keep within TableSize
rand1 = (rand1 * rand2); // Update rand1 for next "random" number
Key++;
}

return hash % TableSize;
}

The hash function is simply converting the string into a random (yet consistent) interpretation as an integer. This integer is then modulo'd with the TableSize variable, which is the size of the table to be inserted into to ensure that it is placed randomly within the bounds of the table.

Here's the output of the sample hash table program I wrote. It creates a table with 157 slots (more on why I chose 157 later -hint: it's prime), and then reads a text file line by line and inserts each individual line into the table with a hash function. I then run a demonstration with a hash lookup function. I created support for deletion of entries, and the program gracefully handles the entire table being full (this is easily detected when a search goes through the array and ends up where it started by wrapping from the last element to the first):

The text file I used is provided in the source. Note that duplicate entries were handled by taking no action. I felt that there wasn't a need to hold duplicates within a hash table, and simply just reported that duplicates were found. Here are some final notes I took from one of my CS courses:

Hash tables rely on the fact that the data is uniformly and randomly distributed.
Since we cannot control the data that is provided (from the user), we must ensure that it is randomly distributed by hashing it.
Hash tables whose size is a prime number help to insure good distribution. Also, table sizes that are a power of 2 work well if the stride (second hash in double hashing) is an odd number.
Since the data is not sorted in any meaningful way, operations such as displaying all items in a sorted fashion are expensive. (Use another data structure for that operation.)

There are many different types of collision resolution for hash tables. Using chaining (linked lists) seems to be a great way to store lots of data. The chains can even be ordered to allow for optimized searching (though this is unlikely to be worth the trouble). However for a small amount of keys and a small table, linear probing should work perfectly fine. There are also other probing methods, such as quadratic probing, double hashing, pseudo random probing (seeding with the key). If probing is the collision method chosen, then support for growing the hash table like a vector may need to be implemented, depending on the requirements of the table. Chaining avoids the growing issue altogether.

I will be using the code constructed here for my next large project. It will actually be solving the hypothetical issue described at the beginning of the post! This will be very handy, as nobody will have to manually keep track of any files anymore; I'll have code traverse a single directory full of assets and then automatically add them as entries to a hash table. Lookups will be highly optimized (which is important, lookups are highly common in a game project).

Additional resources:

Thursday, April 5, 2012

Tile Maps: Binary Collision

Back in the day binary collision maps (also known as a type of tile map) were used as a simple and efficient way to check for collision between an object with coordinates of floating point value, and square blocks. What sorts of games used binary collision arrays? Well, pretty much any game that had square tile based levels likely used some form of binary collision, here's a few NES screenshots:

Zelda

Mario

Metroid

Zelda

A binary collision array is an array of booleans representing collision or non-collision. In our above examples any square tile that the player could not walk on, or was not empty space would have been represented by a value meaning collision. There would then be a separate array of the same dimensions as the binary collision array, which would represent the different images to draw for each tile.

Example Collision and Data Arrays

Above is an example of what the collision and data arrays might look like for the Zelda screenshot with a bridge. Each 1 would be a block that is unwalkable by the character, and each 0 would be a space the character could walk on. The data array holds different numbers, and each number would represent a different image. These image values could perhaps have been stored in some sort of enumeration, or a CBlock in assembly.

Though, how is the character handled? The character in these games seems to have a position of floating point value. I'm not actually sure if they did or didn't for optimization reasons, but what I'll show you uses a floating point coordinate for different characters/enemies. Here's an image showing an example of a floating point coordinate perhaps representing the player's character:

This point would represent the center of the position of the player. The image drawn for this character at this point would be centered directly onto the point. So now how does one go about detecting when the sides of an image on this point are touching a box that represents collision? In order to do that I'll show a method I've learned called "hotspots".

Hotspots are simply points that are offsets of a coordinate, like the coordinate above. The smallest amount of offset hotspots you can have that I'd recommend is four: one for each side of your square image. When implementing a binary collision array for a simple platformer, I found that eight hotspots worked very well. Here's an example image showing eight hotspots on the edges of a square:

Each of those dots are offset by 1/4 of the square's side length, and 1/2 of the square's side length. For example the bottom two hotspots are offset from the center by -1/2 * side length (assuming origin bottom left of the screen). By using these hotspots you can then detect when any of these hotspots points are within the bounds of a binary tile. There are multiple ways to do this, but the way I chose to was to just floor the floating point value of each hotspot and look at only the integer value left. You can do these with an integer typecast, or a floor function. This works since any floating point value in the range of 1.0 through less then 2.0 will still be within the coordinate tile of 1. If this float truncation is confusing, read the next paragraph, if you understand go ahead and skip the next paragraph.

Say you have a coordinate of 3.7 and 4.3 representing one of your hotspots, and you need to check to see if this coordinate is within a tile that represents collision. If you take a look you'll realize that all values from 3.0 to just before 4.0 are all within the x axis on of the value 3 (as in on column 3 of your grid). This goes the same for 4.3; all values from 4.0 through just before 5.0 are in row 4. This means that any coordinate from 3.0 - 3.99999, 4.0 - 4.999999 are within the tile of 3, 4. So then to check to see where any floating point coordinate is on your integer coordinate system, just throw away your decimal value and treat your float as an integer. 3.7 would be 3, and 4.3 would be 4. Thus your resulting coordinate is 3, 4.

Once this is accomplished you can then execute your code to move the character back to the tile they are currently on. This can be done by also flooring the value of the character's coordinate (not the hotspots). However, you want to snap your character back to the closest tile, not just floor the entire floating point value. To do this you use a simple trick:

coordinate = (int)(coordinate + 0.5f);

This will round your floating point number to the nearest tile by offset the decimal portion of your value by 0.5.

Lastly, if you ever implement a coordinate system like here I suggest normalizing your coordinate system. That is have one tile be represented by the value of 1. This allows for simpler mathematics, and allows you to easily transform (scale, rotate, translate) your entire grid/game to whatever exact size you desire. If you normalize your entire grid in your data, you can separate the drawing logic from your data thus decoupling your code overall.

Here's a screenshot for a platformer I worked on for a CS assignment! I had a normalized grid system with the origin represented by the bottom left of the map. I was scaling this map by a factor of 3. I also implemented some simple AI with a statemachine for the red guys! They walk left and right up the edge of their platform, and turn around at the edge, or turn around when they hit a wall.

Randy Gaul's Game Programming Blog

Tuesday, July 3, 2012

Collision: Basic 2D Collision Detection + Jump Physics

Sunday, July 1, 2012

Hash Tables: Introduction

Thursday, April 5, 2012

Tile Maps: Binary Collision

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Tuesday, July 3, 2012

Collision: Basic 2D Collision Detection + Jump Physics

Sunday, July 1, 2012

Hash Tables: Introduction

Thursday, April 5, 2012

Tile Maps: Binary Collision

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