我想使用节点在搅拌机中制作 Cornu 螺旋

Question

And I have no clue where to start. In case you don't know what a cornu spiral is: https://mathworld.wolfram.com/CornuSpiral.html

All that I see are intrincated maths whose terms I'm not familiar with and doesn't translate as far as I know directly to blender interface which also I don't know in depth.

I had some success setting the handle absolute positions given point position on the bezier but I have no idea how to relate them in order to achieve my goal. Can you lend me any formula to calculate the correct brezier handle positions relative to point position and also the positions of the points? I'm about a week on this and I'm still starting.

Answer 1

We can draw Cornu spiral using small line segments. Example in Delphi:

  cx := 10;
  cy := 700;
  scale := 700;
  step := 0.02;
  Canvas.MoveTo(cx, cy);

  for i := 1 to 250 do begin
    t := i * step;
    fresnelintegral(t, c, s);
    x := cx + Round(scale * s);
    y := cy - Round(scale * c);
    Canvas.LineTo(x, y);
  end;

here t is curve parameter (proportional to arc length), and we calculate coordinate of current point using Fresnel integrals and scaling (to make a picture of viewable size). This code generates black curve:

If we want to approximate Cornu spiral by Bezier curves, we have to get endpoints with the same equations, then get direction angles in previous point and in the current point (slope equation (2) in your link) to define intermediate control points. But we also need to calculate distances to control points - seems rather hard problem. I made quick code to demonstrate an approach.

Curvature of Cornu spiral rises along the curve. To approximate spiral better, I had to diminish steps, so Bezier segments become smaller and smaller. Small circles are Bezier endpoints, they become closer.

I tried to make coeff depending on t value, but have not found good equation. The code below generates blue curve. It is very close to black one (we can align them) in chosen t range.

  Canvas.Pen.Color := clBlue;
  cx := 500;
  step := 0.2;
  prevt := 0;
  t := 0;
  Canvas.MoveTo(cx, cy);
  Pts[0] := Point(cx, cy);

  for i := 1 to 70 do begin
    t := t + step;

    fresnelintegral(prevt, c, s);
    prevx := cx + Round(scale * s);
    prevy := cy - Round(scale * c);
    prevfi := 0.5 * Pi * prevt * prevt;

    fresnelintegral(t, c, s);
    x := cx + Round(scale * s);
    y := cy - Round(scale * c);
    pts[i*3] := Point(x, y);
    fi := 0.5 * Pi * t * t;
    Canvas.Ellipse(x-2, y-2, x+3, y+3);

    coeff := 1/3;
    dist := Hypot(x - prevx, y - prevy) * coeff;

    px := prevx + Round(dist * Sin(prevfi));
    py := prevy - Round(dist * Cos(prevfi));
    pts[i * 3 - 2] := Point(px, py);

    px := x - Round(dist * Sin(fi));
    py := y + Round(dist * Cos(fi));
    pts[i * 3 - 1] := Point(px, py);

    step := step * 0.96;
    prevt := t;
  end;
  Canvas.PolyBezier(pts);