Get outer tangents of two circles.

getOuterTangents.ts

function getOuterTangents(
  x0: number,
  y0: number,
  r0: number,
  x1: number,
  y1: number,
  r1: number
) {
  const dx = x1 - x0,
    dy = y1 - y0,
    dist = Math.hypot(dx, dy)

  // Circles are overlapping, no tangents
  if (dist < Math.abs(r1 - r0)) return

  const a0 = Math.atan2(dy, dx),
    a1 = Math.acos((r0 - r1) / dist)

  return [
    x0 + r0 * Math.cos(a0 + a1),
    y0 + r0 * Math.sin(a0 + a1),
    x1 + r1 * Math.cos(a0 + a1),
    y1 + r1 * Math.sin(a0 + a1),
    x0 + r0 * Math.cos(a0 - a1),
    y0 + r0 * Math.sin(a0 - a1),
    x1 + r1 * Math.cos(a0 - a1),
    y1 + r1 * Math.sin(a0 - a1),
  ]
}

getOuterTangents2.ts

/**
 * Get the outer tangents of two circles. This better preserves the positions of the four tangents.
 *
 * @example
 *   t0______t1    t3______t2
 *   (A) -> (B)    (B) <- (A)
 *   t2 ‾‾‾‾‾t3    t1 ‾‾‾‾‾t0
 *
 * @param A The first circle.
 * @param rA The first radius.
 * @param B The second circle.
 * @param rB The second radius.
 * @returns The outer tangents of the circles.
 */
export function getCircleOuterTangents(
  A: VecLike,
  r0: number,
  B: VecLike,
  r1: number
): Vec2dModel[] {
  const gamma = Math.atan((A.y - B.y) / (B.x - A.x))
  const beta = Math.sin((r1 - r0) / Math.hypot(A.y - B.y, A.x - B.x))

  let result = [
    {
      x: A.x + r0 * Math.cos(TAU - (gamma - beta)),
      y: A.y + r0 * Math.sin(TAU - (gamma - beta)),
    },
    {
      x: B.x + r1 * Math.cos(TAU - (gamma - beta)),
      y: B.y + r1 * Math.sin(TAU - (gamma - beta)),
    },
    {
      x: A.x + r0 * Math.cos(-(TAU + (gamma + beta))),
      y: A.y + r0 * Math.sin(-(TAU + (gamma + beta))),
    },
    {
      x: B.x + r1 * Math.cos(-(TAU + (gamma + beta))),
      y: B.y + r1 * Math.sin(-(TAU + (gamma + beta))),
    },
  ]

  if (A.x <= B.x) {
    result = result.concat(result.splice(0, 2))
  }

  return result
}