A quick experiment with box2d and p5.js to cook up a game based on the angle at which polygons collide. The idea seems promising

polywar.js

 
//aliases for box2d stuff. Makes life easier.
var b2Vec2 = Box2D.Common.Math.b2Vec2
, b2AABB = Box2D.Collision.b2AABB
, b2BodyDef = Box2D.Dynamics.b2BodyDef
, b2Body = Box2D.Dynamics.b2Body
, b2FixtureDef = Box2D.Dynamics.b2FixtureDef
, b2Fixture = Box2D.Dynamics.b2Fixture
, b2World = Box2D.Dynamics.b2World
, b2MassData = Box2D.Collision.Shapes.b2MassData
, b2PolygonShape = Box2D.Collision.Shapes.b2PolygonShape
, b2CircleShape = Box2D.Collision.Shapes.b2CircleShape
, b2DebugDraw = Box2D.Dynamics.b2DebugDraw
,  b2MouseJointDef =  Box2D.Dynamics.Joints.b2MouseJointDef
;

//inorder to convert everything to box2d measurements
var SCALE = 30.0;

var Globals = {
	world: null,
	renderStack: [],
	player: null
};

//A generic unit
function Unit(world, x_, y_, w_, h_, mass_){

	this.x = x_;
	this.y = y_;
	this.mass = mass_;
	this.width = w_;
	this.height = h_;
	this.body = null;

	this.hitPoints = 100;

	this.attack = 10;

	var fixDef = new b2FixtureDef;
	fixDef.density = 1.0;
	fixDef.restitution = 0.2;
	fixDef.friction = 0.3;

	var bodyDef = new b2BodyDef;
	bodyDef.position.x = this.x/SCALE;
	bodyDef.position.y = this.y/SCALE;
	bodyDef.type = b2Body.b2_dynamicBody;

	fixDef.shape = new b2PolygonShape;
	fixDef.shape.SetAsBox(this.width/(SCALE*2), this.height/(SCALE*2));

	this.body = world.CreateBody(bodyDef);
	this.body.CreateFixture(fixDef);

	//so that we can access element from box2d
	this.body.SetUserData(this);
}

Unit.prototype.render = function(){
	var x = this.body.GetPosition().x;
	var y = this.body.GetPosition().y;

	push();
	translate(x * SCALE, y * SCALE);
	rotate((this.body.GetAngle()));
	rect(0, 0, this.width, this.height);
	text(this.hitPoints, 0, 0);
	pop();
}

function init(world){

	//set boundaries

	var fixDef = new b2FixtureDef;
	fixDef.density = 1.0;
	fixDef.restitution = 0.2;
	fixDef.friction = 0.3;

	var bodyDef = new b2BodyDef;
	bodyDef.position.x = 0;
	bodyDef.position.y = 0;
	bodyDef.type = b2Body.b2_staticBody;

	fixDef.shape = new b2PolygonShape;
	fixDef.shape.SetAsBox(10/SCALE, height/(SCALE));
	world.CreateBody(bodyDef).CreateFixture(fixDef);
	
	bodyDef.position.x = width/SCALE;
	bodyDef.position.y = 0;
	fixDef.shape.SetAsBox(10/SCALE, height/SCALE);
	world.CreateBody(bodyDef).CreateFixture(fixDef);

	//add player and npc
	var npc = new Unit(world, width/2, height/2, 50, 50, 10);
	var player = new Unit(world, 100, 200, 25, 25, 20);
	player.isPlayer = true;
	Globals.player = player;
	Globals.renderStack.push(npc);
	Globals.renderStack.push(player);

	var listener = new Box2D.Dynamics.b2ContactListener;       
    
    listener.BeginContact = function(contact){
   
    	var bodyA = contact.GetFixtureA().GetBody();
    	var bodyB = contact.GetFixtureB().GetBody();

    	a = bodyA.GetUserData();
    	b = bodyB.GetUserData();

    	if(bodyA.GetAngle()%1.57 > bodyB.GetAngle()%1.57){

    		b.hitPoints -= int(a.attack * (bodyA.GetAngle()%1.57 - bodyB.GetAngle()%1.57)*2);
    	}
    	else
    		a.hitPoints -= int(b.attack * (bodyB.GetAngle()%1.57 - bodyA.GetAngle()%1.57)*2);
    }
    
    world.SetContactListener(listener);
}

function setup(){

	createCanvas(800, 700);
	frameRate(30);
	rectMode(CENTER);

	var world = new b2World(new b2Vec2(0, 0), true);
	Globals.world = world;

	var debugDraw = new b2DebugDraw();
    debugDraw.SetSprite(document.getElementById("defaultCanvas").getContext("2d"));
    debugDraw.SetDrawScale(SCALE);
    debugDraw.SetFillAlpha(0.3);
    debugDraw.SetLineThickness(1.0);
    debugDraw.SetFlags(b2DebugDraw.e_shapeBit | b2DebugDraw.e_jointBit);
    world.SetDebugDraw(debugDraw); 

    init(world);
}

function draw(){

	background(220);
	Globals.world.DrawDebugData();
	for(var i=0; i<Globals.renderStack.length; i++){
		Globals.renderStack[i].render();
	}

	Globals.world.Step(1/60, 10, 10);
}

function mousePressed(){

	Globals.player.body.SetAwake(true);
	Globals.player.body.SetLinearVelocity(new b2Vec2(mouseX/SCALE - Globals.player.body.GetPosition().x, mouseY/SCALE - Globals.player.body.GetPosition().y));
}