如何计算围绕其角旋转的矩形的边界框？

Question

我几天前问过，当矩形围绕其中心旋转时，答案非常适用。

但是，我现在正试图让它在矩形围绕其左上角旋转的情况下工作。

链接答案中的这些行仍然正确并且可用于计算边界框的宽度和高度：

H = w * Abs(Sin(Fi)) + h * Abs(Cos(Fi))
W = w * Abs(Cos(Fi)) + h * Abs(Sin(Fi))

但是，x 和 y 值不再适用于这种左上角的情况。起初我以为它可能只是一样的，只是这个：

x0  = rXCenter - W/2
y0  = rYCenter - H/2

但是好像不行。我认为答案涉及某种 sin 和 cos 函数，但我无法弄清楚哪种组合提供了正确的输出。同样，我在下面包含了问题的完整复制：

let canvas = document.querySelector("canvas");
let ctx = canvas.getContext("2d");  

function drawRectangle(rX, rY, rW, rH) {
  ctx.beginPath();
  ctx.rect(rX, rY, rW, rH);
  ctx.stroke();
}

function degreesToRadians(degrees) { return degrees * (Math.PI / 180); }

function rotateCanvas(radians, centerX, centerY) {
  ctx.translate(centerX, centerY);
  ctx.rotate(radians);
  ctx.translate(-centerX, -centerY);
}

function drawRotatedRectangle(rX, rY, rW, rH, rAngle, rOX, rOY) {
  rotateCanvas(rAngle, rOX, rOY);
  drawRectangle(rX, rY, rW, rH);
  rotateCanvas(-rAngle, rOX, rOY);
}

function computeBB(x, y, w, h, a) {
  let sinA = Math.abs(Math.sin(a));
  let cosA = Math.abs(Math.cos(a));

  let bbH = w * sinA + h * cosA;
  let bbW = w * cosA + h * sinA;

  let bbX = x - bbW / 2;
  let bbY = y - bbH / 2;

  return { x: bbX, y: bbY, w: bbW, h: bbH };
}

let rX = 100;
let rY = 100;
let rW = 100;
let rH = 50;
let rA = degreesToRadians(45);

let rOX = rX;
let rOY = rY;

drawRotatedRectangle(rX, rY, rW, rH, rA, rOX, rOY);

let bb = computeBB(rX, rY, rW, rH, rA);

ctx.strokeStyle = "#ff0000";
drawRectangle(bb.x, bb.y, bb.w, bb.h);

body { margin: 0; overflow: hidden; }

<canvas width="600" height="600"></canvas>

如您所见，当尝试使用相同的代码时，边界框（红色矩形）没有正确包围黑色矩形，我试图让它对 [=20 的所有值都这样做=].

我认为代码片段中只有两行需要更改才能使其正常工作：

let bbX = x - bbW / 2;
let bbY = y - bbH / 2;

但我不太明白我必须将它们更改为什么才能解决问题。

Answer 1

围绕角 x0, y0 旋转角度 Fi 后矩形中心有坐标

cx = x0 + w/2*Cos(Fi) - h/2*Sin(Fi)
cy = y0 + w/2*Sin(Fi) + h/2*Cos(Fi)

边界框中心的坐标相同。所以边界框底角是

bbx = cx - bbW/2
bby = cy - bbH/2

let canvas = document.querySelector("canvas");
let ctx = canvas.getContext("2d");  

function drawRectangle(rX, rY, rW, rH) {
  ctx.beginPath();
  ctx.rect(rX, rY, rW, rH);
  ctx.stroke();
}

function degreesToRadians(degrees) { return degrees * (Math.PI / 180); }

function rotateCanvas(radians, centerX, centerY) {
  ctx.translate(centerX, centerY);
  ctx.rotate(radians);
  ctx.translate(-centerX, -centerY);
}

function drawRotatedRectangle(rX, rY, rW, rH, rAngle, rOX, rOY) {
  rotateCanvas(rAngle, rOX, rOY);
  drawRectangle(rX, rY, rW, rH);
  rotateCanvas(-rAngle, rOX, rOY);
}

function computeBB(x, y, w, h, a) {
  let sinA = Math.abs(Math.sin(a));
  let cosA = Math.abs(Math.cos(a));

  let bbH = w * sinA + h * cosA;
  let bbW = w * cosA + h * sinA;
  
  let cx = x + w/2*Math.cos(a) - h/2*Math.sin(a)
  let cy = y + w/2*Math.sin(a) + h/2*Math.cos(a)

  let bbX = cx - bbW / 2;
  let bbY = cy - bbH / 2;

  return { x: bbX, y: bbY, w: bbW, h: bbH };
}

let rX = 100;
let rY = 100;
let rW = 100;
let rH = 50;
let rA = degreesToRadians(45);

let rOX = rX;
let rOY = rY;

drawRotatedRectangle(rX, rY, rW, rH, rA, rOX, rOY);

let bb = computeBB(rX, rY, rW, rH, rA);

ctx.strokeStyle = "#ff0000";
drawRectangle(bb.x, bb.y, bb.w, bb.h);

<canvas width="600" height="600"></canvas>

如何计算围绕其角旋转的矩形的边界框？

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