How can I use an SVG element’s transformation matrix to calculate destination coordinates?

I suspect there is some way to use an element’s transformation matrix to calculate its coordinates after being transformed, but I don’t know how to do so.

An example diagram explains this best:

I’m afraid I didn’t take the math route into programming. I can sort-of follow the surface-level details of what a transformation matrix is doing, but my understanding is sort of like being able to read music one note at a time, and only very slowly; I don’t really understand the tune at a higher level, so I don’t really understand the sound of a complete musical phrase—let alone the melody.

Similarly, I don’t understand how transformation matrices work. I have tried searching for explanations to grok transformation matrices, but everything I find is loaded with more math jargon I don’t understand. I just know that they work sort of like a function, and that they are incredibly flexible tools, but that’s it.

Of all the methods available to SVGMatrix (supposedly deprecated in favor of DOMMatrix, but Firefox Dev. ed. is still using SVGMatrix), I have no idea whether .inverse() or .multiply() is what I want, and no idea how to coax a simple set of x and y coordinates out of that matrix.

Note:

• I am not interested in translating screen-to-SVG coordinates here.
• I am only concerned with SVG (user-space) coordinates.

You can use simple trigonometric transformation:

const rotatePoint = (point, center, rotateAngle) => {
  const dx = point.x - center.x;
  const dy = point.y - center.y;
  const distance = Math.hypot(dx, dy);
  const currentAngle = Math.atan(dy / dx);
  const nextAngle = currentAngle - rotateAngle;
  const nextDX = distance * Math.cos(nextAngle);
  const nextDY = distance * Math.sin(nextAngle);
  return {x: center.x + nextDX, y: center.y + nextDY};
};

The snippet displays rotation of a blue point around the red one (30 / 90 / 123 degrees counter-clockwise)

const rotatePoint = (point, center, angle) => {
    const dx = point.x - center.x;
  const dy = point.y - center.y;
  const distance = Math.hypot(dx, dy);
  const current = Math.atan(dy / dx);
  const next = current - angle;
  const nextDX = distance * Math.cos(next);
  const nextDY = distance * Math.sin(next);
  return {x: center.x + nextDX, y: center.y + nextDY};
};

const center = {x: 150, y: 150};
const start = {x: 200, y: 30};
const svg = d3.select('svg');

svg.append('circle')
    .attr('cx', center.x)
  .attr('cy', center.y)
  .attr('r', 5)
  .style('fill', 'red');
svg.append('circle')
    .attr('cx', start.x)
  .attr('cy', start.y)
  .attr('r', 5)
  .style('fill', 'blue');  

// Rotate 30 deg
const p30 = rotatePoint(start, center, Math.PI * 30 / 180); 
svg.append('circle')
    .attr('cx', p30.x)
  .attr('cy', p30.y)
  .attr('r', 5)
  .style('fill', 'green');  
// Rotate 90 deg
const p90 = rotatePoint(start, center, Math.PI * 90 / 180); 
svg.append('circle')
    .attr('cx', p90.x)
  .attr('cy', p90.y)
  .attr('r', 5)
  .style('fill', 'orange');
// Rotate 123 deg
const p123 = rotatePoint(start, center, Math.PI * 123 / 180); 
svg.append('circle')
    .attr('cx', p123.x)
  .attr('cy', p123.y)
  .attr('r', 5)
  .style('fill', 'yellow');  
  

<script src="https://cdnjs.cloudflare.com/ajax/libs/d3/5.7.0/d3.min.js"></script>
<svg width="250" height="200"></svg>