BezierInfo-2/docs/chapters/catmullconv/notes.txt

THE FOLLOWING CODE IS FOR COMPUTING THE FAR MORE COMPLEX CENTRIPETAL CATMULL ROM CURVE AND IS PRETTY MUCH OUT OF SCOPE (for now?)


let points, knots;

setup() {
    points = [
        {x:38,y:136},
        {x:65,y:89},
        {x:99,y:178},
        {x:149,y:93},
        {x:191,y:163},
        {x:227,y:122},
        {x:251,y:132}
    ];
    setMovable(points);
    knots = [0, 1/3, 2/3, 1];
    setSlider(`.slide-control.tension`, `tension`, 0.5);
}

onTension(v) {
    if (v < 0.5) {
        v = map(v,0.5,0,1,4);
        v = 1/v;
    } else {
        v = map(v,0.5,1,1,4);
    }
    this.tension = v;
}

draw() {
    clear();

    const [first, last] = this.generateVirtualPoints();
    const full = [first, ...points, last];

    for (let i=0, e=full.length-3; i<e; i++) {
        this.dragSegment(
            full[i],
            full[i+1],
            full[i+2],
            full[i+3]
        );
    }

    points.forEach(p => {
        setColor( randomColor() );
        circle(p.x, p.y, 3);
    });
}

generateVirtualPoints() {
    // see http://www.sdmath.com/math/geometry/reflection_across_line.html#formulasmb
    const n = points.length,
          p1 = points[0],
          p2 = points[1],
          p3 = points[n-2],
          p4 = points[n-1],
          m = (p4.y-p1.y)/(p4.x-p1.x),
          b = (p4.x*p1.y-p1.x*p4.y)/(p4.x-p1.x),
          ratio = 0.5;

    return [[p2,p1], [p3,p4]].map(pair => {
        const p = pair[0],
              M = pair[1],
              reflected = {
                x: M.x - (p.x - M.x),
                y: M.y - (p.y - M.y),
              },
              projected = {
                x: ((1 - m**2)*p.x + 2*m*p.y - 2*m*b) / (m**2 + 1),
                y: ((m**2 - 1)*p.y + 2*m*p.x + 2*b) / (m**2 + 1)
              };
        return {
            x: (1-ratio) * reflected.x + ratio * projected.x,
            y: (1-ratio) * reflected.y + ratio * projected.y
        };
    });
}

dragSegment(p0, p1, p2, p3) {
    const alpha = 0.5,
          t0 = 0,
          // See https://en.wikipedia.org/wiki/Centripetal_Catmull%E2%80%93Rom_spline#Definition
          t1 = t0 + ((p1.x-p0.x)**2 + (p1.y-p0.y)**2)**alpha,
          t2 = t1 + ((p2.x-p1.x)**2 + (p2.y-p1.y)**2)**alpha,
          t3 = t2 + ((p3.x-p2.x)**2 + (p3.y-p2.y)**2)**alpha,
          s = (t2 - t1) / this.tension,
          // See https://stackoverflow.com/a/23980479/740553
          tangent1 = {
            x: s * ((p1.x-p0.x)/(t1-t0) - (p2.x-p0.x)/(t2-t0) + (p2.x-p1.x)/(t2-t1)),
            y: s * ((p1.y-p0.y)/(t1-t0) - (p2.y-p0.y)/(t2-t0) + (p2.y-p1.y)/(t2-t1))
          },
          tangent2 = {
            x: s * ((p2.x-p1.x)/(t2-t1) - (p3.x-p1.x)/(t3-t1) + (p3.x-p2.x)/(t3-t2)),
            y: s * ((p2.y-p1.y)/(t2-t1) - (p3.y-p1.y)/(t3-t1) + (p3.y-p2.y)/(t3-t2))
          };

    noFill();
    setStroke(randomColor() );

    start();
    this.markCoordinate(0, p1,p2,tangent1,tangent2);
    for(let s=0.01, t=s; t<1; t+=0.01) this.markCoordinate(t, p1,p2,tangent1,tangent2);
    this.markCoordinate(1, p1,p2,tangent1,tangent2);
    end();
}

markCoordinate(t, p0, p1, m0, m1) {
    let c = 2*t**3 - 3*t**2,
        c0 = c + 1,
        c1 = t**3 - 2*t**2 + t,
        c2 = -c,
        c3 = t**3 - t**2,
        point = {
            x: c0 * p0.x + c1 * m0.x + c2 * p1.x + c3 * m1.x,
            y: c0 * p0.y + c1 * m0.y + c2 * p1.y + c3 * m1.y
        };
    vertex(point.x, point.y);
}

onMouseDown() {
    if (!this.currentPoint) {
        let {x, y} = this.cursor;
        points.push({ x, y });
        resetMovable(points);
        redraw();
    }
}