Finding coordinates of Koch Curve

问题

Sorry for my language since English is my second language.

I am trying to convert a straight line into a fractal known as Koch curve. The 2 points of the straight line are given and then I need to create the Koch curve where I divide the line to 3 segments and then make the second segment an equilateral triangle. See http://www.tgmdev.be/curvevonkoch.php.

So far we convert the straight line to 4 equally segments, and I need to figure out all the coordinates of the Koch curve.

I have thought of a straight line when the y coordinates of the 2 point are the same which give me horizontal line. if so, I can figure out the 3 points of equilateral triangle by dividing the second segment half and taking the cos(60) of the right triangle. as here: http://www.themathpage.com/atrig/30-60-90-triangle.htm

My problem is how to find all coordinates when the stright line is diagonal, for example a(200,100), b(400,600) or a(400,500), b(100,500).

回答1:

If your base segment is AB, with A(Ax,Ay) and B(Bx,By), then the 4 sub-segments will be AP, PQ, QR, RB as defined below.

First define two orthogonal vectors of same length:

U(Bx-Ax,By-Ay) and
V(Ay-By,Bx-Ax)

Then the points:

P=A+(1/3)*U
Q=A+(1/2)*U+(sqrt(3)/6)*V
R=A+(2/3)*U

The point+vector=point notation is similar to a translation.

Example with A(100,100) and B(400,100):

U(300,0)
V(0,300)
P = (100,100) + (1/3)*(300,0) = (200,100)
Q = (100,100) + (1/2)*(300,0) + (sqrt(3)/6)*(0,300) = (250,186)
R = (100,100) + (2/3)*(300,0) = (300,100)

回答2:

Here is a javascript function based on Eric's algorithm.

export function getChildLinePoints (points, depth = 0) {
  if (depth === 0) {
    const Ax = points[0]
    const Ay = points[1]
    const Bx = points[2]
    const By = points[3]
    const Ux = Bx - Ax
    const Uy = By - Ay
    const Vx = Ay - By
    const Vy = Bx - Ax
    const Px = Ax + ((1 / 3) * Ux)
    const Py = Ay + ((1 / 3) * Uy)
    const Qx = Ax + ((1 / 2) * Ux) + ((Math.sqrt(3) / 6) * Vx)
    const Qy = Ay + ((1 / 2) * Uy) + ((Math.sqrt(3) / 6) * Vy)
    const Rx = Ax + ((2 / 3) * Ux)
    const Ry = Ay + ((2 / 3) * Uy)
    return [[
      Ax, Ay,
      Px, Py
    ], [
      Px, Py,
      Qx, Qy
    ], [
      Qx, Qy,
      Rx, Ry
    ], [
      Rx, Ry,
      Bx, By
    ]]
  } else {
    const xpoints = [...getChildLinePoints(points, depth - 1)]
    return xpoints.reduce((acc, point) => [...acc, ...getChildLinePoints(point)], [])
  }
}

来源：https://stackoverflow.com/questions/15367165/finding-coordinates-of-koch-curve