Mathematical Trickery Mind = Blown
Physics guns are probably more complex than you think.
What Is A Physics Gun?
A Physics Gun is just a device to manipulate physical objects in the world. It simply lets the person wielding the gun grab an object and then move, rotate or scale it. Obviously, this is pretty important for the construct mode which is basically an entire gamemode about moving, rotating and scaling stuff!
I’ve spent the last 4 working days building my physics gun – which was a lot longer than expected. This is mostly because I spent 2 days messing around with dot products and cross products and all sorts of nasty geometric mathematics, which got me nowhere useful, before I finally thought of a very cool trick.
Attempt 1: Physics Guns Move Stuff
Ok, so attempt one was based on the (fairly critical) assumption that a physics gun is all about moving stuff. The scripting interface for tools supplies values indicating where the tool is, and which direction it is facing. So all I had to do was move the grabbed object around by however the direction and position values changed – Simple! This turned out to be something of a train wreck:
local entityToolDirection = GetEntity().GetProperty("tool_direction"); local entityToolPosition = GetEntity().GetProperty("tool_position"); function UpdateTargetPosition() local relativeGrabDirection = (targetPosition - entityToolPosition.Value).Normalize(); --direction from tool to target grab point local requiredDirection = Vector3.New(entityToolDirection.Value).Normalize(); --Vector pointing from tool to where we want target to be --Now rotate target by angle between tool look direction and grab point local axis = relativeGrabDirection.Cross(requiredDirection); local angle = math.acos(relativeGrabDirection.Dot(requiredDirection)); local rotation = Quaternion.CreateFromAxisAngle(axis, angle); --Apply rotation local body = GetBodyFromEntity(target); local transform = Matrix4x4.CreateTranslation(-entityToolPosition.Value) * Matrix4x4.CreateFromQuaternion(rotation) * Matrix4x4.CreateTranslation(entityToolPosition.Value); body.SetWorldTransform(Matrix4x4.New(body.GetWorldTransform()) * transform); end
Well… that was painful. Believe it or not, although this is actually a simplified version of the mess I ended up with, it still doesn’t properly handle grabbing objects anywhere other than their dead center, and it doesn’t really handle translation at all (it tries to but it doesn’t work).
So how is this meant to work? Let’s pretend that all we want to handle is rotation – so that means all we need is to determine an axis (to rotate around) and an angle (the amount to rotate). Taking the cross product of the vector to the grabbed object and the tool look direction gets us an axis, then the dot product of the same two gets us an angle. These things can then be used to create a quaternion (a clever mathematical thing which describes rotations) which we apply to the body to rotate it relative to the physics gun.
This is, to say the least, a little bit ugly. There are several special cases I skipped over in the example above, and it doesn’t even do the entire job! Here’s a video of the gun not working.
Attempt 2: Physics Guns Stop Stuff Moving
Wait, what? That title makes no sense! Physics guns are all about moving stuff, aren’t they? Another way to think of a physics gun is that it makes stuff not move relative to the gun. As the gun moves, the grabbed object will stay in exactly the same position relative to the gun no matter how you wave the gun around. It seems pedantic to restate it like that, but it’s actually very important because there are established ways to make one thing move relative to another which are nice and simple to use. Let’s have a look (this is the complete thing, handling rotation and movement):
local inverse = Matrix4x4.Invert(previousToolTransform); --Invert frame of reference from last time local transform = CalculateToolTransform(); --Calculate the frame of reference defined by the tool previousToolTransform = transform; local body = GetBodyFromEntity(target); body.SetWorldTransform(body.GetWorldTransform() * inverse * transform);
This is conceptually quite simple – all we do is transform the tool into the “frame of reference” defined by the tool. Before we do that though, we have to translate it out of the frame of reference from the last time – this is what the inverse does.
Why Do I Care?
I spent several days working this out, and it was a non obvious trick to get something which seems almost trivially simple to implement. Also, when I inevitably fall behind schedule I can point to this as an excuse ;)
There’s another release out in the usual place. When you create the game, you should select construct mode and then you can create shapes, move them about the world and mess around with constructive solid geometry to create stuff.
I’m moving into mixing construct mode in with the city generation now. The plan is to have this done in a month or two. If everything goes to plan then in March I’ll be releasing a playable construct mode. I’ll also have a much more impressive city (since I will use my own tools to add lots more detail) being prepared, ready for a Greenlight release.
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