我的碰撞分辨率的线性分量相对较好，但我无法弄清楚如何对角度分量做同样的事情。从我读过的内容来看，它就像... torque = point of collision x linear velocity。 （跨产品）我试图将example I found合并到我的代码中，但实际上当对象碰撞时我看不到任何旋转。另一个小提琴完美地适用于分离轴定理和角速度计算的基本实现。这就是我想出的......
属性定义（方向，角速度和角加速度）：
rotation: 0,
angularVelocity: 0,
angularAcceleration: 0
计算碰撞响应中的角速度：
var pivotA = this.vector(bodyA.x, bodyA.y);
bodyA.angularVelocity = 1 * 0.2 * (bodyA.angularVelocity / Math.abs(bodyA.angularVelocity)) * pivotA.subtract(isCircle ? pivotA.add(bodyA.radius) : {
x: pivotA.x + boundsA.width,
y: pivotA.y + boundsA.height
}).vCross(bodyA.velocity);
var pivotB = this.vector(bodyB.x, bodyB.y);
bodyB.angularVelocity = 1 * 0.2 * (bodyB.angularVelocity / Math.abs(bodyB.angularVelocity)) * pivotB.subtract(isCircle ? pivotB.add(bodyB.radius) : {
x: pivotB.x + boundsB.width,
y: pivotB.y + boundsB.height
}).vCross(bodyB.velocity);
更新更新循环中的方向：
var torque = 0;
torque += core.objects[o].angularVelocity * -1;
core.objects[o].angularAcceleration = torque / core.objects[o].momentOfInertia();
core.objects[o].angularVelocity += core.objects[o].angularAcceleration;
core.objects[o].rotation += core.objects[o].angularVelocity;
我会发布我用于计算惯性矩的代码，但每个对象都有一个单独的代码，因此有点......冗长。尽管如此，这是一个圆圈的例子：
return this.mass * this.radius * this.radius / 2;
只是为了显示结果，这是我的fiddle。如图所示，物体在碰撞时不会旋转。 （圆圈不完全可见，但它应该适用于零和七）
我做错了什么？
编辑：他们根本没有轮换的原因是因为响应函数中的组出错 - 它现在旋转，只是不正确。但是，我现在已经评论过，因为它搞砸了。
另外，我尝试了另一种旋转方法。这是响应中的代码：
_bodyA.angularVelocity = direction.vCross(_bodyA.velocity) / (isCircle ? _bodyA.radius : boundsA.width);
_bodyB.angularVelocity = direction.vCross(_bodyB.velocity) / (isCircle ? _bodyB.radius : boundsB.width);
请注意direction指的是“碰撞正常”。
答案 0 :(得分：9)
由于施加的力引起的角度和方向加速度是同一物体的两个组成部分，不能分开。要获得一个，你需要为两者解决。
定义计算
从简单的物理学和站在肩膀上我们知道以下内容。
F is force (equivalent to inertia)
Fv is linear force
Fa is angular force
a is acceleration could be linear or rotational depending on where it is used
v is velocity. For angular situations it is the tangential component only
m is mass
r is radius
对于线性力
F = m * v
我们从中得出
m = F / v
v = F / m
对于旋转力（v是切向速度）
F = r * r * m * (v / r) and simplify F = r * m * v
我们从中得出
m = F / ( r * v )
v = F / ( r * m )
r = F / ( v * m )
因为我们应用的力是瞬时的，我们可以交换a加速度和v速度来给出以下所有公式
线性
F = m * a
m = F / a
a = F / m
旋转
F = r * m * a
m = F / ( r * a )
a = F / ( r * m )
r = F / ( a * m )
因为我们只对线性和旋转解决方案的速度变化感兴趣
a1 = F / m
a2 = F / ( r * m )
其中a1是每帧像素的加速度 2 和a2是每帧弧度的加速度 2 （帧平方仅表示它是加速度）
从1D到2D
因为这是一个2D解决方案而且以上都是1D我们需要使用向量。我对于这个问题使用两种形式的2D矢量。具有幅度（长度，距离等）和方向的极地。具有x和y的笛卡儿。矢量代表什么取决于它的使用方式。
以下功能在解决方案中用作帮助程序。它们是用ES6编写的，因此对于不兼容的浏览器，你必须对它们进行调整，尽管我不会建议你使用它们，因为它们是为了方便而编写的，它们的效率很低，并且进行了大量的冗余计算。
将矢量从极地转换为笛卡儿，返回一个新的
function polarToCart(pVec, retV = {x : 0, y : 0}) {
retV.x = Math.cos(pVec.dir) * pVec.mag;
retV.y = Math.sin(pVec.dir) * pVec.mag;
return retV;
}
将矢量从笛卡儿转换为极性，返回一个新的
function cartToPolar(vec, retV = {dir : 0, mag : 0}) {
retV.dir = Math.atan2(vec.y, vec.x);
retV.mag = Math.hypot(vec.x, vec.y);
return retV;
}
创建极向量
function polar(mag = 1, dir = 0) {
return validatePolar({dir : dir,mag : mag});
}
将矢量创建为笛卡儿
function vector(x = 1, y = 0) {
return {x : x, y : y};
}
确实是arg vec是极性形式的向量
function isPolar(vec) {
if (vec.mag !== undefined && vec.dir !== undefined) {return true;}
return false;
}
如果arg vec是笛卡尔形式的向量
，则返回truefunction isCart(vec) {
if (vec.x !== undefined && vec.y !== undefined) {return true;}
return false;
}
以极坐标形式返回一个新向量，同时确保vec.mag为正
function asPolar(vec){
if(isCart(vec)){ return cartToPolar(vec); }
if(vec.mag < 0){
vec.mag = - vec.mag;
vec.dir += PI;
}
return { dir : vec.dir, mag : vec.mag };
}
将未知的vec复制并转换为购物车（如果尚未
）function asCart(vec){
if(isPolar(vec)){ return polarToCart(vec); }
return { x : vec.x, y : vec.y};
}
计算可能导致负值，但这对于某些计算是有效的，这导致不正确的向量（反向）这简单地验证了极向量具有正幅度它不会仅仅改变矢量符号和方向
function validatePolar(vec) {
if (isPolar(vec)) {
if (vec.mag < 0) {
vec.mag = - vec.mag;
vec.dir += PI;
}
}
return vec;
}
The Box
现在我们可以定义一个可以用来玩的对象。一个简单的盒子，有位置，大小，质量，方向，速度和旋转
function createBox(x,y,w,h){
var box = {
x : x, // pos
y : y,
r : 0.1, // its rotation AKA orientation or direction in radians
h : h, // its height
w : w, // its width
dx : 0, // delta x in pixels per frame 1/60th second
dy : 0, // delta y
dr : 0.0, // deltat rotation in radians per frame 1/60th second
mass : w * h, // mass in things
update :function(){
this.x += this.dx;
this.y += this.dy;
this.r += this.dr;
},
}
return box;
}
对对象施加力
所以现在我们可以重新定义一些术语
F（力）是矢量力，幅度是力，它有方向
var force = polar(100,0); // create a force 100 units to the right (0 radians)
如果没有应用它的位置，这种力就毫无意义。
位置是一个只保持x和y位置的矢量
var location = vector(canvas.width/2, canvas.height/2); // defines a point in the middle of the canvas
方向向量保持位置向量之间的方向和距离
var l1 = vector(canvas.width/2, canvas.height/2); // defines a point in the middle of the canvas
var l2 = vector(100,100);
var direction = asPolar(vector(l2.x - l1.x, l2.y - l1.y)); // get the direction as polar vector
direction现在有从画布中心到点（100,100）的方向和距离。
我们需要做的最后一件事是沿着方向向量从力矢量中提取组件。对物体施加力时，力被分成两个，一个是沿着线到物体中心的力并加到物体加速度，另一个力是90度到物体中心的线（切线）这就是改变轮换的力量。
为了获得这两个分量，你可以得到力矢量和从力被施加到物体中心的方向矢量之间的方向差异。
var force = polar(100,0); // the force
var forceLoc = vector(50,50); // the location the force is applied
var direction2Center = asPolar(vector(box.x - forceLoc.x, box.y - forceLoc.y)); // get the direction as polar vector
var pheta = direction2Center - force.dir; // get the angle between the force and object center
现在你有了这个角度pheta，力可以被分成带有trig的旋转和线性分量。
var F = force.mag; // get the force magnitude
var Fv = Math.cos(pheta) * F; // get the linear force
var Fa = Math.sin(pheta) * F; // get the angular force
现在可以将力转换回线性加速度a = F / m和角度a = F /（m * r）
accelV = Fv / box.mass; // linear acceleration in pixels
accelA = Fa / (box.mass * direction2Center.mag); // angular acceleration in radians
然后将线性力转换回具有指向对象中心方向的矢量
var forceV = polar(Fv, direction2Center);
转换回笛卡尔坐标，因此我们可以将它添加到对象deltaX和deltaY
forceV = asCart(forceV);
并将加速度添加到方框
box.dx += forceV.x;
box.dy += forceV.y;
旋转加速度只是一维，所以只需将其添加到框的增量旋转
box.dr += accelA;
就是这样。
对Box施加强制的功能
如果附加到框中的功能将在框的位置应用力矢量。
像这样附在盒子上
box.applyForce = applyForce; // bind function to the box;
然后您可以通过方框
调用该功能box.applyForce(force, locationOfForce);
function applyForce(force, loc){ // force is a vector, loc is a coordinate
var toCenter = asPolar(vector(this.x - loc.x, this.y - loc.y)); // get the vector to the center
var pheta = toCenter.dir - force.dir; // get the angle between the force and the line to center
var Fv = Math.cos(pheta) * force.mag; // Split the force into the velocity force along the line to the center
var Fa = Math.sin(pheta) * force.mag; // and the angular force at the tangent to the line to the center
var accel = asPolar(toCenter); // copy the direction to center
accel.mag = Fv / this.mass; // now use F = m * a in the form a = F/m to get acceleration
var deltaV = asCart(accel); // convert acceleration to cartesian
this.dx += deltaV.x // update the box delta V
this.dy += deltaV.y //
var accelA = Fa / (toCenter.mag * this.mass); // for the angular component get the rotation
// acceleration from F=m*a*r in the
// form a = F/(m*r)
this.dr += accelA;// now add that to the box delta r
}
演示
演示只是关于函数applyForce与重力和弹跳有关的事情只是非常糟糕的近似值，不应该用于任何物理类型的东西，因为它们不能节省能量。
单击并拖动以沿着移动鼠标的方向向对象施加力。
const PI90 = Math.PI / 2;
const PI = Math.PI;
const PI2 = Math.PI * 2;
const INSET = 10; // playfeild inset
const ARROW_SIZE = 6
const SCALE_VEC = 10;
const SCALE_FORCE = 0.15;
const LINE_W = 2;
const LIFE = 12;
const FONT_SIZE = 20;
const FONT = "Arial Black";
const WALL_NORMS = [PI90,PI,-PI90,0]; // dirction of the wall normals
var box = createBox(200, 200, 50, 100);
box.applyForce = applyForce; // Add this function to the box
// render / update function
var mouse = (function(){
function preventDefault(e) { e.preventDefault(); }
var i;
var mouse = {
x : 0, y : 0,buttonRaw : 0,
bm : [1, 2, 4, 6, 5, 3], // masks for setting and clearing button raw bits;
mouseEvents : "mousemove,mousedown,mouseup".split(",")
};
function mouseMove(e) {
var t = e.type, m = mouse;
m.x = e.offsetX; m.y = e.offsetY;
if (m.x === undefined) { m.x = e.clientX; m.y = e.clientY; }
if (t === "mousedown") { m.buttonRaw |= m.bm[e.which-1];
} else if (t === "mouseup") { m.buttonRaw &= m.bm[e.which + 2];}
e.preventDefault();
}
mouse.start = function(element = document){
if(mouse.element !== undefined){ mouse.removeMouse();}
mouse.element = element;
mouse.mouseEvents.forEach(n => { element.addEventListener(n, mouseMove); } );
}
mouse.remove = function(){
if(mouse.element !== undefined){
mouse.mouseEvents.forEach(n => { mouse.element.removeEventListener(n, mouseMove); } );
mouse.element = undefined;
}
}
return mouse;
})();
var canvas,ctx;
function createCanvas(){
canvas = document.createElement("canvas");
canvas.style.position = "absolute";
canvas.style.left = "0px";
canvas.style.top = "0px";
canvas.style.zIndex = 1000;
document.body.appendChild(canvas);
}
function resizeCanvas(){
if(canvas === undefined){
createCanvas();
}
canvas.width = window.innerWidth;
canvas.height = window.innerHeight;
ctx = canvas.getContext("2d");
if(box){
box.w = canvas.width * 0.10;
box.h = box.w * 2;
box.mass = box.w * box.h;
}
}
window.addEventListener("resize",resizeCanvas);
resizeCanvas();
mouse.start(canvas)
var tempVecs = [];
function addTempVec(v,vec,col,life = LIFE,scale = SCALE_VEC){tempVecs.push({v:v,vec:vec,col:col,scale:scale,life:life,sLife:life});}
function drawTempVecs(){
for(var i = 0; i < tempVecs.length; i ++ ){
var t = tempVecs[i]; t.life -= 1;
if(t.life <= 0){tempVecs.splice(i, 1); i--; continue}
ctx.globalAlpha = (t.life / t.sLife)*0.25;
drawVec(t.v, t.vec ,t.col, t.scale)
}
}
function drawVec(v,vec,col,scale = SCALE_VEC){
vec = asPolar(vec)
ctx.setTransform(1,0,0,1,v.x,v.y);
var d = vec.dir;
var m = vec.mag;
ctx.rotate(d);
ctx.beginPath();
ctx.lineWidth = LINE_W;
ctx.strokeStyle = col;
ctx.moveTo(0,0);
ctx.lineTo(m * scale,0);
ctx.moveTo(m * scale-ARROW_SIZE,-ARROW_SIZE);
ctx.lineTo(m * scale,0);
ctx.lineTo(m * scale-ARROW_SIZE,ARROW_SIZE);
ctx.stroke();
}
function drawText(text,x,y,font,size,col){
ctx.font = size + "px "+font;
ctx.textAlign = "center";
ctx.textBaseline = "middle";
ctx.setTransform(1,0,0,1,x,y);
ctx.globalAlpha = 1;
ctx.fillStyle = col;
ctx.fillText(text,0,0);
}
function createBox(x,y,w,h){
var box = {
x : x, // pos
y : y,
r : 0.1, // its rotation AKA orientation or direction in radians
h : h, // its height, and I will assume that its depth is always equal to its height
w : w, // its width
dx : 0, // delta x in pixels per frame 1/60th second
dy : 0, // delta y
dr : 0.0, // deltat rotation in radians per frame 1/60th second
getDesc : function(){
var vel = Math.hypot(this.dx ,this.dy);
var radius = Math.hypot(this.w,this.h)/2
var rVel = Math.abs(this.dr * radius);
var str = "V " + (vel*60).toFixed(0) + "pps ";
str += Math.abs(this.dr * 60 * 60).toFixed(0) + "rpm ";
str += "Va " + (rVel*60).toFixed(0) + "pps ";
return str;
},
mass : function(){ return (this.w * this.h * this.h)/1000; }, // mass in K things
draw : function(){
ctx.globalAlpha = 1;
ctx.setTransform(1,0,0,1,this.x,this.y);
ctx.rotate(this.r);
ctx.fillStyle = "#444";
ctx.fillRect(-this.w/2, -this.h/2, this.w, this.h)
ctx.strokeRect(-this.w/2, -this.h/2, this.w, this.h)
},
update :function(){
this.x += this.dx;
this.y += this.dy;
this.dy += 0.061; // alittle gravity
this.r += this.dr;
},
getPoint : function(which){
var dx,dy,x,y,xx,yy,velocityA,velocityT,velocity;
dx = Math.cos(this.r);
dy = Math.sin(this.r);
switch(which){
case 0:
x = -this.w /2;
y = -this.h /2;
break;
case 1:
x = this.w /2;
y = -this.h /2;
break;
case 2:
x = this.w /2;
y = this.h /2;
break;
case 3:
x = -this.w /2;
y = this.h /2;
break;
case 4:
x = this.x;
y = this.y;
}
var xx,yy;
xx = x * dx + y * -dy;
yy = x * dy + y * dx;
var details = asPolar(vector(xx, yy))
xx += this.x;
yy += this.y;
velocityA = polar(details.mag * this.dr, details.dir + PI90);
velocityT = vectorAdd(velocity = vector(this.dx, this.dy), velocityA);
return {
velocity : velocity, // only directional
velocityT : velocityT, // total
velocityA : velocityA, // angular only
pos : vector(xx, yy),
radius : details.mag,
}
},
}
box.mass = box.mass(); // Mass remains the same so just set it with its function
return box;
}
// calculations can result in a negative magnitude though this is valide for some
// calculations this results in the incorrect vector (reversed)
// this simply validates that the polat vector has a positive magnitude
// it does not change the vector just the sign and direction
function validatePolar(vec){
if(isPolar(vec)){
if(vec.mag < 0){
vec.mag = - vec.mag;
vec.dir += PI;
}
}
return vec;
}
// converts a vector from polar to cartesian returning a new one
function polarToCart(pVec, retV = {x : 0, y : 0}){
retV.x = Math.cos(pVec.dir) * pVec.mag;
retV.y = Math.sin(pVec.dir) * pVec.mag;
return retV;
}
// converts a vector from cartesian to polar returning a new one
function cartToPolar(vec, retV = {dir : 0, mag : 0}){
retV.dir = Math.atan2(vec.y,vec.x);
retV.mag = Math.hypot(vec.x,vec.y);
return retV;
}
function polar (mag = 1, dir = 0) { return validatePolar({dir : dir, mag : mag}); } // create a polar vector
function vector (x= 1, y= 0) { return {x: x, y: y}; } // create a cartesian vector
function isPolar (vec) { if(vec.mag !== undefined && vec.dir !== undefined) { return true; } return false; }// returns true if polar
function isCart (vec) { if(vec.x !== undefined && vec.y !== undefined) { return true; } return false; }// returns true if cartesian
// copy and converts an unknown vec to polar if not already
function asPolar(vec){
if(isCart(vec)){ return cartToPolar(vec); }
if(vec.mag < 0){
vec.mag = - vec.mag;
vec.dir += PI;
}
return { dir : vec.dir, mag : vec.mag };
}
// copy and converts an unknown vec to cart if not already
function asCart(vec){
if(isPolar(vec)){ return polarToCart(vec); }
return { x : vec.x, y : vec.y};
}
// normalise makes a vector a unit length and returns it as a cartesian
function normalise(vec){
var vp = asPolar(vec);
vap.mag = 1;
return asCart(vp);
}
function vectorAdd(vec1, vec2){
var v1 = asCart(vec1);
var v2 = asCart(vec2);
return vector(v1.x + v2.x, v1.y + v2.y);
}
// This splits the vector (polar or cartesian) into the components along dir and the tangent to that dir
function vectorComponentsForDir(vec,dir){
var v = asPolar(vec); // as polar
var pheta = v.dir - dir;
var Fv = Math.cos(pheta) * v.mag;
var Fa = Math.sin(pheta) * v.mag;
var d1 = dir;
var d2 = dir + PI90;
if(Fv < 0){
d1 += PI;
Fv = -Fv;
}
if(Fa < 0){
d2 += PI;
Fa = -Fa;
}
return {
along : polar(Fv,d1),
tangent : polar(Fa,d2)
};
}
function doCollision(pointDetails, wallIndex){
var vv = asPolar(pointDetails.velocity); // Cartesian V make sure the velocity is in cartesian form
var va = asPolar(pointDetails.velocityA); // Angular V make sure the velocity is in cartesian form
var vvc = vectorComponentsForDir(vv, WALL_NORMS[wallIndex])
var vac = vectorComponentsForDir(va, WALL_NORMS[wallIndex])
vvc.along.mag *= 1.18; // Elastic collision requiers that the two equal forces from the wall
vac.along.mag *= 1.18; // against the box and the box against the wall be summed.
// As the wall can not move the result is that the force is twice
// the force the box applies to the wall (Yes and currently force is in
// velocity form untill the next line)
vvc.along.mag *= box.mass; // convert to force
//vac.along.mag/= pointDetails.radius
vac.along.mag *= box.mass
vvc.along.dir += PI; // force is in the oppisite direction so turn it 180
vac.along.dir += PI; // force is in the oppisite direction so turn it 180
// split the force into components based on the wall normal. One along the norm the
// other along the wall
vvc.tangent.mag *= 0.18; // add friction along the wall
vac.tangent.mag *= 0.18;
vvc.tangent.mag *= box.mass //
vac.tangent.mag *= box.mass
vvc.tangent.dir += PI; // force is in the oppisite direction so turn it 180
vac.tangent.dir += PI; // force is in the oppisite direction so turn it 180
// apply the force out from the wall
box.applyForce(vvc.along, pointDetails.pos)
// apply the force along the wall
box.applyForce(vvc.tangent, pointDetails.pos)
// apply the force out from the wall
box.applyForce(vac.along, pointDetails.pos)
// apply the force along the wall
box.applyForce(vac.tangent, pointDetails.pos)
//addTempVec(pointDetails.pos, vvc.tangent, "red", LIFE, 10)
//addTempVec(pointDetails.pos, vac.tangent, "red", LIFE, 10)
}
function applyForce(force, loc){ // force is a vector, loc is a coordinate
validatePolar(force); // make sure the force is a valid polar
// addTempVec(loc, force,"White", LIFE, SCALE_FORCE) // show the force
var l = asCart(loc); // make sure the location is in cartesian form
var toCenter = asPolar(vector(this.x - l.x, this.y - l.y));
var pheta = toCenter.dir - force.dir;
var Fv = Math.cos(pheta) * force.mag;
var Fa = Math.sin(pheta) * force.mag;
var accel = asPolar(toCenter); // copy the direction to center
accel.mag = Fv / this.mass; // now use F = m * a in the form a = F/m
var deltaV = asCart(accel); // convert it to cartesian
this.dx += deltaV.x // update the box delta V
this.dy += deltaV.y
var accelA = Fa / (toCenter.mag * this.mass); // for the angular component get the rotation
// acceleration
this.dr += accelA;// now add that to the box delta r
}
// make a box
ctx.globalAlpha = 1;
var lx,ly;
function update(){
// clearLog();
ctx.setTransform(1, 0, 0, 1, 0, 0);
ctx.clearRect(0, 0, canvas.width, canvas.height);
ctx.setTransform(1, 0, 0, 1, 0, 0);
ctx.lineWidth = 1;
ctx.strokeStyle = "black";
ctx.fillStyle = "#888";
ctx.fillRect(INSET, INSET, canvas.width - INSET * 2, canvas.height - INSET * 2);
ctx.strokeRect(INSET, INSET, canvas.width - INSET * 2, canvas.height - INSET * 2);
ctx.lineWidth = 2;
ctx.strokeStyle = "black";
box.update();
box.draw();
if(mouse.buttonRaw & 1){
var force = asPolar(vector(mouse.x - lx, mouse.y - ly));
force.mag *= box.mass * 0.1;
box.applyForce(force,vector(mouse.x, mouse.y))
addTempVec(vector(mouse.x, mouse.y), asPolar(vector(mouse.x - lx, mouse.y - ly)), "Cyan", LIFE, 5);
}
lx = mouse.x;
ly = mouse.y;
for(i = 0; i < 4; i++){
var p = box.getPoint(i);
// only do one collision per frame or we will end up adding energy
if(p.pos.x < INSET){
box.x += (INSET) - p.pos.x;
doCollision(p,3)
}else
if( p.pos.x > canvas.width-INSET){
box.x += (canvas.width - INSET) - p.pos.x;
doCollision(p,1)
}else
if(p.pos.y < INSET){
box.y += (INSET) -p.pos.y;
doCollision(p,0)
}else
if( p.pos.y > canvas.height-INSET){
box.y += (canvas.height - INSET) -p.pos.y;
doCollision(p,2)
}
drawVec(p.pos,p.velocity,"blue")
}
drawTempVecs();
ctx.globalAlpha = 1;
drawText(box.getDesc(),canvas.width/2,FONT_SIZE,FONT,FONT_SIZE,"black");
drawText("Click drag to apply force to box",canvas.width/2,FONT_SIZE +17,FONT,14,"black");
requestAnimationFrame(update)
}
update();