# Slides: Intro to Engine Dev with Component Based Design

I just found some slides I made a few months ago. They cover the basics of component based engine design and how to take advantage of simple serialization to data drive the construction of objects. Hopefully these slides can be of use to someone:

# Powerful C++ Messaging

A prerequisite to this information is most of the previous C++ type introspection stuff I have been writing about for a while now. Assuming the previous information has been covered, lets move on:

There exists a design of messaging, specifically for C++, of which has minimal downsides and many positive advantages. Ideally messaging should not involve any polling or implicitly required searching (as in searching through game space to see who to message, which requires expensive collision queries). It should also have a very intuitive usage, and not be very complex to work with.

If such a messaging system can be achieved then inter-object communication can be setup, to create game logic, within a scripting language.

Here are some slides I wrote on this topic for my university, but are available for public viewing:

# C++ Reflection Part 6: Lua Binding

Binding C/C++ functions to Lua is a tedious, error prone and time consuming task when done by hand. A custom C++ introspection system can aide in the automation of binding any callable C or C++ function or method a breeze. Once such a functor-like object exists the act of binding a function to Lua can look like this, as seen in a CPP file:

The advantage of such a scheme is that only a single CPP file would need to be modified in order to expose new functionality to Lua, allowing for efficient pipe-lining of development cycles.

Another advantage of this powerful functor is that communication and game logic can be quickly be created in a script, loaded from a text file, or even setup through a visual editor. Here is a quick example of what might be possible with a good scripting language:

In the above example a simple enemy is supposed to follow some target object. If the target is close enough then the enemy damages it. If the target dies, the enemy flashes a bright color and then acquires a new target.

The key here is the message subscription within the initialization routine. During run-time objects can subscribe to know about messages emitted by any other object!

So by now hopefully one would have seen enough explanation of function binding to understand how powerful it is. I’ve written some slides on the topic available in PDF format here (do note that these slides were originally made for a lecture at my university):

# Templates and Metaprogramming

Recently I’ve begun to take over a club called Game Engine Architecture Club at DigiPen. The founder of the club graduated and asked me to try running it. I’ve found it be to a lot of fun! The first lecture I did was on template metaprogramming in C++ (TMP).

The lecture was recorded live and is currently up on DigiPen’s youtube channel for public viewing. Though the slides are visible during the video, I don’t think I can share the slides themselves publicly.

Hope you enjoy the video!

# Component Based Engine Design

## What is Component Based Design?

Component based engine design was originally pioneered in order to avoid annoying class hierarchies that inheritance introduces. The idea is to package all functionality of game objects into separate objects. A single game object is just a collection of the components, as so the components of a game object define it’s behavior, appearance and functionality. This is perfectly fine, though there are plenty of resources out that talk about this topic. However I’d like to take a step back and start from the top.

It should be noted that the implementation presented here is just one way of going about things, and comes directly from my highly subjective opinion. Perhaps you as a reader can come up with or know of better solutions or designs.

Here is an example game object, note that the game object is generic and simply contains some components:

## The Actual Engine

The engine of a game can be thought of as a manager of systems. As to what a system is, we’ll get to that later, for now think of a system as either Physics, Graphics or AI. The engine ought to have a main loop function, as well as an update function. The update function calls update on all contained systems in a specific order. The main loop function is just a small infinite loop that calls update.

Often times the main loop will deal with timestepping itself. Have a look at the linked article to learn more about proper timestepping.

It is important to have your engine expose the update function, as sometimes your engine will need to be compiled as a static library and linked to externally. In this case the main loop of your simulation may reside outside of the engine library altogether. A common usage I’ve seen for this sort of design choice is when creating an editor of some sort, perhaps a level or content editor. Often times these editors will have a veiwport to preview the game, and in order to do so access to some sort of engine update function is necessary.

Beyond containing and calling update, the Engine also forwards global messages to all the systems. More on messaging later.

## Singletons?

Creating more than one instance of an engine should never happen. The question of “should I make this a singleton” will sometimes arise. In my experience the answer is often no. Unless you’re working on a large team of programmers where the chances of some moron making an instance of some Engine or System class, making things a singleton is just a waste of time. Especially if retrieving data from that singleton incurs a little bit of overhead.

## Systems

Each system in the engine corresponds to one type of functionality. This idea is best shown by example, so here are the various systems I usually have in engines I work on:

• Graphics
• Physics
• GameLogic
• Windowing/Input
• UI
• Audio
• ObjectFactory – Creates objects and components from string or integral ID

Each system’s primary functionality is to operate upon game objects. You can think of a system as a transform function: data is input, modified somehow, and then data is output. The data passed to each system should be a list of game objects (or of components). However a system only updates components on game objects, and the components to be updated are the ones related to that system. For example the graphics system would only update sprite or graphics related components.

Here’s an example header file for a system:

The naive approach to a system update would be to pass a list of game objects like so:

The above code is assuming the ObjectFactory contains all game objects, and can be accessed somehow (perhaps by pointer). This code will work, and it’s exactly what I started with when I wrote my first engine. However you’ll soon realize the folly involved here.

## Cache is King

That’s right, those who have the gold makes the rules; the golden rule. In a more serious sense, cache is king due to processing speed related to memory access speed. The bottleneck in all engines I have ever seen or touched in the past couple years has been due to poor memory access patterns. Not a single serious bottleneck was due computation.

Currently modern hardware performs very very fast. Reaching for data in memory (RAM, not just hard disk) is orders of magnitude slower. So caches come the rescue. A cache can be thought of, in a simplified sense, as a small chunk of memory right next to the CPU (or GPU). Accessing the cache memory is way faster than going all the way out to main RAM. Whenever memory is fetched from RAM memory around that RAM location is also placed into the CPU cache. The idea here is that when you retrieve something from RAM the likelyhood of requiring to fetch something very nearby is high, so all the data in that area is grabbed all at once.

Long story short, if you place things that need to be accessed at around the same time next to each other in memory, huge performance benefits will be reaped. The best performance comes from traversing memory linearly, as if iterating over an array. This means that if we can stick things into arrays and traverse these arrays linearly, there will be no faster form of memory access.

Fetching memory that does not exist in the cache is called a cache miss.

## Cache and Components

Lets revisit the naive approach to updating systems. Assuming a system is handed a generic game object, that system must then retrieve its corresponding component(s) to update, like so:

As this loop is run the memory of every game object and every component type that corresponds to the system is touched. A cache miss will likely be incurred over and over as the loop runs bouncing around in memory. This is even worse if the ObjectFactory is just creating random objects with new, as every memory access to every object and every component will likely incur cache misses.

What do? The solution to all these memory access problems is to simplify the data into arrays.

## Arrays GameObjects + Components

I suggest having every game object exist within a single giant array. This array should probably be contained within the ObjectFactory. Usage of std::vector for such a task is recommended. This keeps game objects together in memory, and even though deletion of a game object is of O(n) complexity, that O(n) operation traverses an array, and usually will turn out to be unnoticeable. A custom vector or array class can be created that avoids the O(n) operation entirely by taking the element at the end, and placing it into the deleted slot. This can only be done if references into the array are translated handles (more on handles momentarily).

Every component type should be in a giant array too. Each component array should be stored within their respective systems (but can be “created” from the Factory). Again, an array like data structure would be ideal.

This simple setup allows for linear traversal of most memory in the entire engine, so long as the update function of each system is redesigned slightly. Instead of handing a list of game objects to each system, the system can just iterate over its related components directly, since the components are stored within the systems already.

## So, how are Game Objects “handled” now?

Since components have been moved into large arrays, and the game objects themselves are in a big array, what exactly should the relationship between a game object and a component be? In the naive implementation some sort of map would have worked perfectly fine, as the memory location of each component could be anywhere due to the use of new calls. However the relation isn’t so carefree.

Since things are stored in arrays its time to switch from pointer-centric relationships to handle based relationships. A handle can be thought of in its simplest form an index into an array. Since game objects and components are stored in arrays, it is only natural that to access a game object you do so by index. This allows for these giant arrays to grow and shrink as necessary without obliterating dangling pointers in the rest of the program.

Here’s a code example:

As you can see, an array of handles is stored to represent the containment of components. There is one slot in the array for each type of component. By design this limits each component to be of unique type within a game object. Each handle is an index into a large array of components. This index is used to retrieve components that correspond to a game object. A special value (perhaps -1, or by some other mechanism) can be used to denote “no component”.

Handles can get quite a bit more versatile than just a plain ol’ integer. I myself created a HandleManager for translating an integer into a pointer. Here’s a great resource for creating your own handle manager.

The idea of translating a handle into a pointer is such that once the pointer is used it is not kept around. Just let it be reclaimed back into the stack. This makes it so that every time a pointer is required there is a translation of handle to a single pointer somewhere in memory. This constant translation allows for the actual pointer value to be translated to, to be swapped for another pointer at any time without fear of leaving dangling pointers.

## Where does the Code go?

Code for update routines can be put into either systems or components. The choice is yours entirely. A more data oriented approach would put as much code into systems as possible, and just use components as buckets of data. I myself prefer this approach. However once you hit game logic components virtual functionality is likely to be desired, and so code will likely be attached directly to such components.

The last engine I built used the naive approach to component based design, and it worked wonderfully. I used a block allocator so cache misses weren’t as high as with raw new calls.

The point is, do what makes most sense and keep things simple. If you want to store routines within your components and use virtual function calls, then you’ll probably have trouble storing things in an array unless you place all memory in the base class. If you can externalize as much code from your components as possible, it may be simpler to keep all your components in an array.

There is a tradeoff between flexibility and efficiency. My personal preference is to store performance sensitive components in huge arrays, and keep AI and game logic related things together in memory as much as possible, but not really stress too much about it. Game logic and AI should probably just be rather flexible, and so details about memory locations aren’t so important. One might just allocate these types of components with a block allocator and call it good.

## Messaging

The last major devil in an engine is messaging. Messaging is transferring data from one location to another. In this sense the most basic form of messaging is a simple function call.

Taking this a step further, we’d like to be able to send messages over a connection in a generic manner. It should not matter what type of message we send; all messages should be sent the same way to reduce code duplication and complexity. The most basic form of this is dynamic dispatch, or virtual function calls. An ID is introduced to the message so the internal data can be typecasted to the correct type.

Still, we can do better. Lets imagine we have a sort of GameLogic component or system. We need a way to send a message to this object that contains some data. Lets not focus much on memory access patterns, as simplicity and flexibility are key here. Take a look at this code:

This code highlights the usage of messaging quite well. Say the player emits some messages as it walks around, perhaps something like “I’m here” to all nearby things in a level. The player can blindly send these messages across the SendMessage function without caring about whether or not it will respond or do anything. As you can see, the implementation of the SendMessage function ignores most message types and responds to a few.

In this example when the player nears the treasure chest it will glimmer a bit. Perhaps when the player gets closer it glimmers brighter and brighter. In order to do so, the MSG object sent to the treasure chest ought to contain the player coordinates, and so it can be typecasted to the appropriate message type.

The eActivate message may be emitted by the player whenever they hit the “e” button. Anything that could possibly respond to an eActive message will do so, and the rest of the objects receiving the message will safely ignore it.

This type of messaging is simple and easy to implement, quite efficient (if a block allocator or stack memory is used for the messages), and rather powerful.

A more advanced version of messaging makes heavy use of C++ introspection. A future article will likely be devoted to this topic, as it’s a hefty topic altogether.

Resources:

# Introduction

During the previous article in this Custom Physics Engine series we talked about impulse resolution. Though understanding the mathematics and physics presented there are important, not much could be put into practice without both collision detection and manifold generation.

Collision detection is a pretty widely documented area, and so I won’t go into too much detail on how to achieve collision detection in this article series. Instead I’ll focus on manifold generation, which is in my opinion much more difficult and less-documented compared to collision detection.

In general collision detection is useful for retrieving a boolean result of “are these two things colliding”. The usefulness of such a result ends when this collision needs to be resolved. This is where manifold generation comes in.

# Manifold Generation – Summary

A manifold, in context of physics engines, is a small structure that contains data about the details of a collision between two objects. The two bodies are commonly referred to as A and B. Whenever referring to a “collision” as a system, A is usually the reference object, as in the problem is viewed from A’s orthonormal basis.

I am not actually sure why this structure is called “the manifold”, and I do not know anyone that actually knows. So don’t ask! Either way this structure should be passed around by reference or pointer to avoid unnecessary copies. I also pool all my manifolds, and intrusively link them in order to keep a list of active manifolds during a scene’s step (the term scene is defined in the previous article).

Manifold generation involves gathering three pieces of information:

• Points of contact
• Penetration depth
• Vector of resolution, or collision normal

The points of contact are 2D points (or 3D for a 3D engine) that mark where one shape overlaps another. Usually these contact points are placed onto the vertex of one shape that resides within another.

Two points of contact for two boxes found intersecting.

The penetration depth is the depth of which the two shapes are intersecting. This is found using the Separating Axis Test (SAT). There are lots of resources around that talk about SAT, so I suggest googling for them. The penetration depth is defined as the axis of least penetration. In this case (assuming blue’s frame of reference and vertical as y axis) the y axis is the axis of least penetration.

The collision normal is used to describe in which direction to press both objects away from one another. The collision normal will be a face normal, and in this case it would be a normal pointing towards the brown box, where the normal corresponds to the blue box’s top face.

Generating these three pieces of information can be quite a challenge. Now lets view what a possible setup of the manifold structure might look like:

Note that there can be a variable amount of contact points. I suggest having a contactCount of zero signify “no collision”. I also suggest having only two possible points of contact for a 2D simulation, and to start just use a single possible point of contact. More than one point of contact isn’t necessary until advanced constraint resolution is used (a future article).

It is important to just keep an array of contacts within this data structure as to keep strong cache coherency. There’s no reason to dynamically allocate the array of contacts.

# Circle to Circle

I’ll be covering how to gather manifold information for specialized cases of couple different types of shapes. Lets first go over circle to circle first. Here is what the definition of a circle would look like:

The first thing to do is to see if they are colliding or not. Again, throughout this article I’m going to mostly blaze through the collision detection and focus just on gathering important manifold information. Feel free to google or ask specific questions in the comments about collision detection.

The above code is really quite simple. The important thing to note is that our contact normal will always be a vector from A to B. In order to create a vector from one point to another you take your endpoint minus your starting pointer. In this case B’s position subtracted by A’s position. This results in a vector from A to B. This vector normalized will be the collision normal, or in other words the direction in which to resolve the collision.

It is important to note that no square root functions are called before the early out condition is checked. Most of the time your shapes are probably not colliding, and so there’s no reason to use a square rooted value.

The last tricky thing is to check if the two shapes are right on top of each other. Though this is unlikely in a dynamic environment, sometimes shapes can be placed directly upon one another through an editor. It is important to, in all collision detection functions, to handle all special cases, even if the handling is bad. Whatever you do just make sure you are consistent. I just chose a random vector to resolve in the direction of.

The nice thing about cirlce to circle collision is that only one collision point is really needed. Just  be sure to be consistent in how you choose your collision point in order to reduce simulation jitter.

# AABB to AABB

Collision detection between two AABBs is a bit more complicated than two circles but still quite simple. The idea is to make use of min and max. Lets assume we’re storing our AABBs in a structure like so:

This allows a very simple algorithm to find points of contact. For convex polygons that are not axis aligned often times Sutherland-Hodgman clipping will need to be performed. In our case we can implicitly deduce our contact area due to the nature of AABBs.

First determine if the two AABBs are overlaping at all. When an axis of least penetration is found the collision area can then be deduced.

AABB intersection with intersection points and area of intersection. Min the maxes and max the mins.

The idea is to perform the SAT while storing each overlap value. The least overlap is your axis of separation. To get the contact area and two points of intersection you can min your maxes and max your mins (I’m talking about the extents of each AABB).

This sounds silly, but that’s how you do it. I suggest drawing it out. Here’s how to find your collision area given by two points (intersection points of the AABBs):

The last bit of info required would be to record the penetration and contact normal. Penetration is your axis of least overlap, so after you’ve found your axis of least overlap you can just assign a vector value as your contact normal. If you have found the axis of least penetration to be on the x axis, you want to point towards object B along the x axis. If the y axis is the axis of least penetration, you want to point towards object B along the y axis.

That’s all there is to the AABB to AABB intersection. Be sure to properly record the number of contacts found (if any), and if neither axis x or y are actually overlapping, then that means there is no intersection.

# AABB to Circle

I will leave AABB to Circle collision an exercise for the reader, though I will quickly provide an explanation paragraph behind the idea. What needs to be done is to first determine if the shapes are overlapping at all. I have a previous post on my blog that explains the Circle to AABB intersection, and more information about such an intersection check can be found around the internet.

Lets assume A is the AABB and Circle is B and we have a collision. The collision normal will again be the vector from A to B, except slightly modified. The early out condition involves finding the closest point on the AABB to the Circle. The collision normal is the translation vector from A to B subtracted by a vector to the closest point on the AABB. This will represent a vector from the circle’s center to the closest point.

The contact point will be residing on the circle’s radius in the direction of the contact normal. This should be easy to perform if you understood the Circle vs Circle collision detailed above. The penetration depth will be the length of the collision normal before it is normalized.

There is one special case that must be properly handled: if the center of the circle is within the AABB. This is quite simple to handle; clamp the center of the circle to the edge of the AABB along the edge closest to the circle’s center. Then flip the collision normal (so it points away from the AABB instead of to the center) and normalize it.

# OBBs and Orientation

Now lets start talking about adding in some manifold generation for some more complex oriented shapes! The first thing that must be learned is how to properly change from one orthonomormal basis to another (that is shift from one frame of reference to another). This will vastly simplify collision detection involving OBB shapes.

Changing a basis involves taking the orientation and translation of an OBB and applying the inverse of these two it to another shape. In this way you can then treat the OBB as an AABB as long as you are still referring to your transformed object. Lets go over this in some more detail with some pictures.

Here is what an OBB is like in the OBB’s frame of reference (left), and the OBB in model space (right).

Note: origin is (0, 0) in model and reference space.

The important thing to realize is that in order to place an object into an OBB’s frame of reference it must have inverse translation and rotation of the OBB’s translation and rotation applied to it. This takes the OBB’s position to the origin, and the OBB can then be treated as an AABB.

If the inverse rotation of the OBB, in this case -45 degrees, is applied to both the OBB and an object near it, this is what happens:

Change of basis going from left to right.

As you can visually see, once the circle has been inversely transformed into the OBB’s frame of reference the OBB can be viewed as a simple AABB centered at the origin. The extents of the OBB can be used to mimic an AABB, and the OBB to Circle intersection and manifold generation can be treated identically to the AABB to Circle intersection, if a proper inverse transformation is performed. Again, this inverse transformation is called a “change of basis”. It means you transform the Circle into the OBB’s frame of reference.

# Mat2 Rotation Matrices

Lets go over rotations in 2D extremely quickly. I won’t go over derivation here for brevity’s sake (as you will see, brevity is a close friend of mine in these Physics Engine articles haha). Instead I will just show you how to create your own 2 by 2 matrix and use it as apart of whatever math library you currently have (which you should hand-code yourself!). Really the only useful thing about having a 2 by 2 matrix is to do rotation operations.

For those using C++ you’re in luck for I know how to use unions.

The above is a proper usage of the unnamed union trick. The elements of the 2 by 2 array can be accessed as if they are a two dimensional array, single dimensional array, or separate floating point values. Additionally you can stick two vectors into your union for column or row access, if you so wish.

I want to briefly hit all the important methods without writing an entire book, so get ready for code snippets to be thrown at you.

The first thing you should realize is that the default constructor does nothing. This is important. Often times you will create a matrix only to briefly thereafter assign some value to it. Do not default construct your matrix to zero values as an optimization. Force the user to use the Set function like so: mat.Set( 0, 0, 0, 0 ).

The interesting functions here are the rotation constructor and SetRotation functions. Each one computes cosine and sine from a given radian value and caches the result. Caching the results prevents unneeded additional calls to cosine and sine. Note the format in which sine and cosine are stored. It is also important to realize that m00 and m10 represent a transformation of the x axis, and m01 and m11 represent a transformation of the y axis. Each of these two are columns, both columns can be viewed as unit vectors.

Multiplying a Mat2 with a vector will rotate the vector’s x and y components around the origin. It is important to realize where your origin is before you apply a rotation. If you want to jump into an OBB’s frame of reference you must do an inverse translation to set the OBB as the origin. This allows you to then apply the OBB’s inverse rotation (perhaps with the inverse operator of your Mat2, see Box2D if you don’t know how to inverse a Mat2) and rotate about the origin (which is about the OBB).

# OBB Representation

Every oriented shape will need to store its orientation somehow. I suggest the following:

The OBB should store its current orientation in both a single floating point value along with a matrix to represent that radian value as a rotation matrix. When you need to rotate the OBB during integration, you can just add or subtract a little bit to the radians value, and then call u.SetRotate( radians ) to update the rotation matrix. This makes use of a simple and organized way to cache results from sine and cosine calls, and minimizes the amount of calls to these functions that you require.

# OBB to OBB

Now lets talk about the big one. How in the world can you see if two OBBs intersect? Both boxes are oriented, so the problem would involve a lot of complex calculations involving trigonometric computations.

Lets make things easier: transform one OBB into the other OBB’s frame of reference, and treat the transformed object as an AABB. Now the problem becomes much simpler.

First perform a separating axis check and find the axis of least penetration. In order to perform the SAT you must find a projected radius onto the axis you are currently testing.

If the sums of the projected radii from both OBBs are larger than the distance between the center of each OBB (along your respective axis), then they are intersecting on that axis. This method works for all convex polygons in 2D.

The way I perform this check is by taking the translation vector from A to B, lets call it T. Then I rotate T into A’s frame of reference and subtract A’s extent vector, and subtract that entire result by   B’s extent vector rotated into A’s frame of reference. This results in a vector holding the overlap along the x and y axis for object A. Due to symmetry only two axes need to be tested. The same operation can be done for object B to find B’s separating axis. If no separating axis is found the shapes are intersecting.

This is where things start to get difficult. Since we just performed an early out test, now we need to find the axis of least separation. However you cannot just blindly perform floating point comparisons due to floating point error during rotation of the translation vector from A to B. You must bias your comparisons to favor one axis over another in a consistent manner. This is important for stability!

In the above picture, which set of contact points/normal is the “correct” one? Each axis of separation is very close to the same distance, so floating point error could account for which axis is chosen. If your simulation flops between the two you’ll end up with strange jitter and your simulation will be less believable. The solution is to just favor one axis over another using an error EPSILON value.

Here’s a function (you’re lucky I just gave it to you!) that will check which value is greater than the other. Each value is modified slightly, and a small bias is fed into the comparison based off of how large each value passed in is. This can be used to favor one axis of separation over another, until a threshold larger than floating point error is breached.

Carefully record which direction your normal goes (from A to B) depending on what axis is separating. This is a similar operation to the one found in AABB to AABB as seen above.

Once an axis is found two line segments must be identified: the reference face and incident face. The reference face corresponds to your normal you recorded. So the reference face is the face that corresponds to your axis of least penetration. If your axis of least penetration is on A, then your  reference face is on A. The incident face is the one with the information we need to generate our manifold.

The incident face it the face on the other object that the reference face has hit. We must compute it. All that needs be done is find out which face is most facing the normal (has the most negative dot product). Looping through all faces performing dot products is the simplest way to achieve this. A more optimized algorithm is the follow the sign of the flipped normal. Your normal will have an x and y component (and z in 3D), and each component will be positive or negative. This gives you a total of four combinations of positive or negative.

First check to see if the normal is point more towards the x axis or y axis (after you transform it into the incident face’s frame of reference). Then check the sign of the y axis. You know know to which face your normal is most pointing.

Think of it this way: if the normal is pointing more towards the x axis (absolute value of n.x is greater than absolute value of n.y), then you are going to be pointing more towards either the left or right face on your OBB (in the OBB’s frame of reference). All that you need to know from there, is if you’re pointing left or right on the x axis, which is denoted by the sign of the normal.

Your incident face segment endpoints are the extents of the x axis of the OBB, which are aligned with the x axis in the OBB’s frame of reference. You can then take the OBB x half-extent and use the positive and negative version of it to form two points: (-x, 0) and (x, 0) where x is the half-extent of the OBB on its x axis. Rotate these points with the OBB’s rotation matrix, and then translate them into world space with the OBB’s position vector, and you now have your endpoints for your incident face in world space.

All that is left is the clip the incident face to the reference face side planes using Sutherland-Hodgman clipping. Here’s a diagram showing this:

This is a fairly difficult thing to do unless you know your math fairly well. Each side plane can be simply computed once you know your reference normal. Here’s the process for getting two side planes:

The above code was hand-derived by myself, but you’ll find something very similar within Box2D Lite (where I originally learned the math from). If this is confusing to you I suggest reading up on the various types of representations of lines in 2D.

Here’s another diagram I just found in my own source files:

You might have noticed I’m storing the c value. This is important as the c value stored within the Line structure can be used to find the distance of a point to the line like so:

I’ll be nice and provide you my clipping notes I created for Sutherland-Hodgman clipping :)

However since we are clipping a line to a single plane the algorithm will need to be slightly modified. In some cases you need to push extra points, since Sutherland-Hodgman assumes to be clipping two polygons in a loop. See Box2D Lite for a good implementation of the incident to line clipping. I however use my own hand-derived algorithm that works in a very similar way. I’ll share some pseudo code for clipping a segment to a Line, assuming the Line is in the format of offset c and a normal n:

After clipping to the side planes floating point error must be accounted for. If our clipping process went as expected we must have two resulting points. If we end up with less than two output points that means floating point error has screwed us over, and we must treat the entire process as if the two OBBs are non-intersecting.

Assuming we have two points output from our clipping we then need to only consider points that are behind the reference face. Use the DistanceToLine function I provided above to check this. Record each point behind the reference face as a contact point!

# OBB to AABB

If you’ve been reading along diligently and deriving your own understanding, you should be able to figure out how to perform such a check. This test is the exact same as OBB to OBB, except with less rotating from one basis to another. You can recall that the OBB to OBB test rotated one OBB into the frame of the other completely, turning the test into an AABB to OBB test. The same thing can be done here without the preliminary change of basis, and perhaps some other small optimizations. I will leave the section out as a challenge for the reader.

# Conclusion

I hope you have a solid understanding of various types of hand-crafted intersection tests! Feel free to email me or comment here with questions or comments. I refer you to Box2D Lite’s source code as a reference to the material in this article, along with all of Erin Catto’s GDC slides, especially the one from 2007. The next article in this series is likely to talk about creating a simple O(N^2) broadphase, and how to cull out duplicate contact pairs. Until then this article along with the previous one is more than enough to create a simple physics engine. Be aware that with impulse resolution only one point of contact should be necessary. This article goes over grabbing multiple contact points, and this is because more advanced collision resolution techniques can make use of more contact points.

# EDIT:

This series ended up on Tuts+, and these are sort of deprecated. Please visit Tuts+ to see the finalized articles.

# Personal Update

Beyond C++ reflection I’ve taken a huge liking to physics programming for games. Reflection in C++ is a largely “solved” area of study. Though resources on the internet may be few and far between for creating custom introspection, people that know how to create such systems have explored much of what is to be accomplished.

However physics is another story entirely. Up until just a few years ago physics for games was in a highly primitive state; techniques just hadn’t been created due to the inability of hardware to perform necessary computation. There are a lot of new and exciting problems to solve.

So this next year while attending DigiPen I’ll be focusing my studies around: engine architecture, data oriented design and optimization, C++ introspection and game physics. This last topic being the topic I’m going to start an article series on.

# Game Physics High Level Detail

A game physics engine performs some functionality for ensuring that shapes behave in a particular way. To state this one could say: a physics engine removes degrees of freedom from moving bodies, and enforces these imposed rules.

From here on out, until specified otherwise I’ll be talking specifically about convex rigid bodies within a physics engine. A rigid body is just a shape that is implicitly rigid; the engine does not naturally support deformation of a set of points that make up a shape. Using this implicit definition the mathematics behind manipulating such shapes can be known intuitively and implemented efficiently.

Here’s a short feature list of some features a custom physics system may provide, assuming the engine uses rigid bodies:

• Collision detection
• Collision resolution
• Linear movement, angular rotation and integration
• Raycasting
• Islanding/sleeping/portals
• Friction simulation
• Spatial partitioning/bounding volumes
• Fracturing/splitting
• Multi-part bodies
• Various types of shapes
• Advanced mathematical constraints (joints, motors, contact constraints, etc.)
Raycasting from Box2D.

In this article series I will attempt to talk about all of the above topics in my own time. I’ve learned a lot about physics and physics engines in a short amount of time and by documenting what I have learned I hope to solidify my understanding, as well as help out others to achieve what I have. I would not have learned nearly as much as I currently know without help from others so it is only natural to want to do the same.

The best example of an open source physics engine that employs much of the feature list shown above would be Box2D by Erin Catto. The rest of this article series will be detailing the physics engine that I myself have written. There are of course other methods I choose not to talk about, or just don’t know about.

# Architecture

There are two main objects that make up a physics engine: shapes and bodies. A body is a small package of data that defines intrinsic properties of a physics object. Properties such as density, restitution (elasticity), friction coefficients, inertia tensors, along with any other flags or settings should be stored within the body. These bits of data are properties that can be isolated away from the shape definition. The shape itself is contained with the body through a pointer.

The shape definition defines what sort of shape a physics object represents. Some physics engines can attach multiple shapes to a body, which are referred to as fixtures. A shape stores any data required to define the shape itself, and provides various functions to operate on the shape, such as testing for line or point intersection, or generating a bounding volume.

Together the body and shape represent a physics object, which by the end of this article series will be able to perform a lot of interesting interactions with other physics objects.

A tower of stacked oriented boxes within a custom physics engine called iiD.

All bodies should be contained within what is known as a scene or world. I refer to this object as a scene. The scene should contain a list of all live objects, as well as functionality inserting or removing bodies from the scene. The scene should also have callbacks for processing shape or ray queries. A query just checks to see if any bodies intersect with something like a point or ray.

The scene has one particular function called step, which steps the scene forward in time by a single delta time (dt). This step function steps all objects forward in time by integration. The integration step just moves the objects forward by using their velocity, position and acceleration to determine their next position.

During the step collisions are detected and then resolved. Often times a broadphase of some sort is used to detect possible collisions, and expensive collisions operations are only used when really needed.

All of this organization allows the user of your physics engine to worry about three main operations: creating and removing bodies, and stepping the scene. The rest is automated and taken care of within the physic system’s internals.

The last major isolated system would be the broadphase. There are two major phases in collision detection: the broad and narrow phases. The narrow phase is the one which determines if two shapes intersect or not. The broad phase determines if two shapes can possibly be intersecting or not. The broadphase should be constructed such that intersection queries are very very fast and simple. An axis-aligned bounding box (AABB) will suffice perfectly.

Once the broadphase chooses which objects could perhaps be colliding, it sends them off to the narrow phase. The narrow phase performs much more intensive operations to determine if two objects are colliding or not. The whole point of this is to reduce the amount of times the narrow phase has to be used, as it is expensive to operate with.

Once the narrow phase finds a pair of bodies to be colliding information about the collision is gathered and placed into a small container called the manifold. Do not ask why it is called a manifold, for I have no idea and neither does anyone else seem to! The manifold contains three important pieces of information:

• Collision penetration depth
• Direction to resolve in
• Points of contact

These pieces of information are used to fill in formulas that are used to resolve the penetration and solve for a single unknown: the magnitude of the resolution vector. Here’s a small diagram:

c – Collision contact
n – Resolution normal
d – Penetration distance

It is also useful to to store pointers or handles to the two objects that formed this collision info. This allows some useful functions to placed into the manifold object directly: solve and resolve. Solve would be the function to collect the three pieces of collision information. If no contact points are found, then there is no collision. If contact points are found, then resolving performs a resolution operation to force both objects to not be intersecting after integration.

# Velocity

Complex physics manipulation is performed on the velocities of objects. Trying to manipulate the positions of objects directly is very difficult, as it poses a problem that isn’t linear. By using derived position equations for velocity, the problem is thus simplified. Most of the time we will be only dealing with velocity manipulation.

# Impulse Resolution in 2D (No Rotation or Friction)

The act of resolving collisions is something that isn’t covered very well on the internet. There are some scattered documents and details, though the information isn’t always easy to find. Most of what I know I learned by talking with other people, but I know most people will not have such an opportunity. Hopefully this article series can provide a strong resource for understanding and constructing a simple physics engine.

The toughest place to code is in my opinion resolution of collision. There exists tons of information on collision detection and broadphase, and thus creating these portions of a physics engine is in my opinion not too difficult. Some resources for collision detection are: Real-Time Collision Detection by Christer Ericson, and Game Physics Engine Developement by Ian Millington. I have both of these books right next to me as I write this :)

Generating contact manifolds and resolving such manifolds are what most programmers get caught up in. So lets hit the ground running and tackle a portion of code that will bring your entire physics system to life: contact resolution.

The best type of contact resolution to start with is impulse resolution. The idea behind impulse resolution is to take your contact manifold and solve for a single velocity vector that can be used to add into an object’s current velocity, in order to make it not penetrating the other object during the next frame. The reason for starting with impulse resolution is that it’s quite simple and easy to get up and running, and more complicated and advanced techniques require you to understand impulse resolution anyway.

2D impulse simulation.

Now the contact manifold is assumed to contain the direction of our velocity vector we are solving for. I will cover how to generate the contact manifold in another article in this series. The only unknown left to solve for is the magnitude of this vector. It so happens that it’s possible to solve for this magnitude in relation to both objects. If this magnitude is known, you add velocity along the normal scaled by the magnitude for one object, and you do the same operation to the other object in the direction opposite to the manifold normal. Lets start from the top.

We have two objects moving along in our scene, and there exists a relative velocity between the two, with object A as the reference object at the origin:

The relative velocity can be expressed in terms of the collision normal (from the collision manifold) with a dot product:

This can be thought of as the relative normal velocity between the two objects. The next thing to include in this derivation is the coefficient of restitution (elasticity factor). Newton’s Law of Restitution states that:

Often times the restitution identifier is specified by an e, or epsilon symbol. Knowing this it’s fairly simple to include it within our current equation:

Now we need to go to another topic and model an impulse. I have said the term “impulse” quite a few times without defining it, so here is the definition I use: an impulse is an instantaneous change in velocity. We will use an impulse to change the velocity of an object. Here’s how you could use an impulse to modify the velocity of a single object:

Here Impulse would be a scalar floating point value. This isn’t too useful however, as it only scales an object’s current velocity, and thusly makes the object move slower or faster along the positive or negative direction of the vector.

What is needed is a way to do component-wise modification of the vector, so we can make it point slightly in one direction or another, allowing objects to make turns and change directions.

In the above equation we can take a direction vector with a magnitude of 1, and scale it by our impulse. By adding this new vector to our velocity we can then modify the velocity in any way we wish. Our end goal is to solve for our Impulse scalar that will separate two objects from collision, so in the end we’ll need to distribute this scalar across two equations in terms of velocity.

An impulse is defined to be a change in momentum. Thus we get:

To isolate our velocity after we can rearrange into:

Now lets change equation 4 into one that contains velocities under the influence of impulses. However we’ll want to express our VelocityNew as one that is acted upon by impulse, and substitute in equation 9:

Remember that the impulse is a scalar. Also note that all values on the right hand side of the equation are all known, including the Direction which was solved for by the collision detection.

All that is left here is to distribute this scalar impulse over each object’s velocity vector proportional to the total mass of the system (system being collision between both objects).

The total mass is MassA + MassB, so to get an even distribution you do: impulse * Mass / TotalMass. To simplify this one could use the following:

This can be done twice, once per object. The total impulse will be applied, except only a portion of the impulse will be applied to each object. This ensures smaller objects move more than larger ones during impact.

One thing you must ensure is that if the velocities are separating (objects moving away from one another) that you do nothing. Here’s a sample version of finalized code for impulse resolution in 2D without rotation or friction:

# Rotation

Now that we have covered resolution without these two factors adding them in to our final equation (equation 10) will be quite a bit simpler. We will need to understand the concept of “mass” in terms of rotations. The inertia tensor of an object describes how “hard it is to turn” along some axis. In 2D there’s only a single axis of rotation (along the z) so a single tensor is all that’s needed. Here’s a new version of equation 11.

For the sake of brevity here’s the final equation, where r is the vector from center of mass to a point on the body (contact point). The velocity of a point on the body relative to it’s center is given by:

Final equation (Direction substituted for n):

And there we have it! This will solve for a separating impulse given a specific contact point. Now you might have noticed there are a couple cross products that are kinda strange in 2D. I won’t cover what they are here, but they still hold true. I’ll just give them to you:

# Friction

Friction is the simplest thing to do in this entire resolution article, assuming we’re dealing with 2D! I actually haven’t derived this portion myself, so again I’m going to just throw the equations at you. Assuming we have our resolution direction in the manifold (the collision normal), we can calculate a tangent vector, which is a vector perpendicular to the collision normal. Just replace all instances of n in equation 14 with this tangent vector.

To solve for the tangent vector you do:

Again, just take the above TangentVector and replace it in equation 14 for n.

There is something I have missed. When solving for the force of friction there’s a max, which is the manifold normal * the coefficient of friction (from the body definition). This is due to the Coloumb friction law. Treat the coefficient of friction * normal as your “friction cap” when solving for your tangential impulse. This is just a simple capping of the final impulse vector.

# Penetration Resolution

Due to floating point error energy will always be lost from the system over time. As such, when objects start coming to a rest they will sink into each other due to gravity. How to solve? Well the solution is to actually just handle the leftover penetration manually by just moving the objects apart directly. This will prevent objects from sinking into one another, though doesn’t add any energy into the system.

To resolve penetration, simply move each object along the manifold normal in opposite directions. However there’s an art to doing so; you need to be gentle. If you just resolve 100% of the penetration each frame objects underneath other objects will jitter violently due to the naive penetration resolution scheme I am presenting. To avoid this, try only resolving a percentage of the leftover penetration, perhaps 20 to 80%.

Additionally you only want to resolve penetration if the penetration depth after the impulse is applied is above some constant arbitrary (but small) threshold, (try 0.01 to 0.1). If the penetration is below this, then don’t move either object.

This method of penetration resolution is called linear projection. Here’s a snippet of C++ code demonstrating this:

# Iterative Solving

There is one one additional tweak you can do to increase the believability of your physics simulation: iterate over all contacts and solve + resolve the impulses many times (perhaps 5 to 20 iterations). Since the large equation 14 has relative velocity within it, each iteration will feed in the previous result to come up with a new one.

This will allow the step to propagate energy throughout multiple objects contacting one another within a single timestep. This is essential for allowing the “billiards balls” effect to ensue.

This simple iteration is a very easy way to vastly improve the results of resolution.

• http://chrishecker.com/images/e/e7/Gdmphys3.pdf
• http://www.cs.cmu.edu/~baraff/sigcourse/notesd2.pdf

# Game Object Factory: Distributed Factory

When creating game objects within your game, you will likely want to start using new or malloc all over the place. This is bad! Every object should have a clear owner, and there should be a single interface for creating any object. Take a look at this code:

The above code does work, although it creates a giant mess of file dependencies. Suddenly one would require a header file for all the various object definitions in order to use the new operator everywhere. Furthermore, handling of attaching the object to some sort of owner will have to be done everywhere that the new keyword is used.

A factory should be used to create game objects in a unified way. Here’s an example of the simplest form of an object factory:

However there are some problems with this type of factory. One is that the factory will still have annoying file dependencies requiring the user to make another include, and modify the switch statement each time a new game object is designed. Examine the following:

The above example shows the usage of the object factory when strings are mapped to types of objects. This setup is the ideal interface one would desire, as a single include to the ObjectFactory header is all that is required to create any type of object. The previous examples were mapping integers of an enumeration type to object types, which would require modification to the switch statement run in the enum, as well as manual updating to the enumeration itself.

So now that we’ve identified the ideal interface for creating game objects, we still need to make sure that the game objects created have a clear owner. Mapping of strings to game object types will need to be implemented, and cutting down on file dependencies of the the GameObjectFactory file itself is still something to be solved as well. By using a map (perhaps std::map) or hash table we can use a string identifier to fetch some value. Here’s what we want our factory to look like:

By mapping a string to a creator function of the type of object desired, all file dependencies except that of those required to utilize the table (map in the above example) are eliminated. Furthermore, whenever a new game object type is designed, all that is needed to be done to ensure compatibility with our factory is a single registration call.

There are some details about the Create function that should be discussed. The Create function is specific to each type of game object, and the interface for each Create function needs to be identical. To ensure this, each type of game object should have a single class called the creator. The creator object for each class derives from a base Creator class to ensure a clear and consistent interface.

What we’ve now created is called a distributed factory. All file dependencies are hidden by registration in using a hash table data structure to provide the mapping of typeless strings to actual types of game objects. The creation of objects has a unified interface, and there’s a single access point (the object factory) that allows the creation of all types of game objects that are registered within its data table of types. There’s one last small problem to solve: who owns the created objects? This answer can of course vary, but luckily there can be a single point of object management, like so:

And there we have it! A properly structured object factory. To further the awesome might of the factory, one could easily add in deserialization to the factory, since the factory is already working with abstract strings as type identifiers. Serialization, however,  is for a future post!

Factories like this also interface very well with scripting languages :)