Verify if a point is part of quadratic Bezier curve in Java

Question

I want to verify if a Point is part of a quadratic Bezier curve defined by points p0, p1 and p2..

This is my function to obtain a Point in the curve with a certain t:

public static final Point quadratic (Point p0, Point p1, Point p2, double t) {
    double x = Math.pow(1-t, 2) * p0.x + 2 * (1-t) * t * p1.x + Math.pow(t, 2) * p2.x;
    double y = Math.pow(1-t, 2) * p0.y + 2 * (1-t) * t * p1.y + Math.pow(t, 2) * p2.y;

    return new Point((int)x, (int)y);
}

Considering that the point B(t) in a quadratic curve is obtained as follows:

B(t) = (1 - t)^2 * p0 + 2 * t * (1 - t) * p1 + t^2 * p2

I should verify if a point P belongs to a curve by getting the t value for that point and comparing it to the Point obtained using that t param, but in Java I'm having problems with the precision of the variables.

My function to verify a point is the following:

public static final boolean belongsQuadratic (Point p, Point p0, Point p1, Point p2) {
    double[] tx = obtainTs(p.x, p0, p1, p2);
    double[] ty = obtainTs(p.y, p0, p1, p2);

    if (tx[0] >= 0) {
        if ((tx[0] >= ty[0] - ERROR && tx[0] <= ty[0] + ERROR) || (tx[0] >= ty[1] - ERROR && tx[0] <= ty[1] + ERROR)) {
            return true;
        }
    }

    if (tx[1] >= 0) {
        if ((tx[1] >= ty[0] - ERROR && tx[1] <= ty[0] + ERROR) || (tx[1] >= ty[1] - ERROR && tx[1] <= ty[1] + ERROR)) {
            return true;
        }
}


    return false;
}

public static double[] obtainTs (int comp, Point p0, Point p1, Point p2) {
    double a = p0.x - 2*p1.x + p2.x;
    double b = 2*p1.x - 2*p0.x ;
    double c = p0.x - comp;

    double t1 = (-b + Math.sqrt(b*b - 4*a*c)) / (2*a);
    double t2 = (-b - Math.sqrt(b*b - 4*a*c)) / (2*a);

    return new double[] {t1, t2};
}

So if I run the code with this values:

Point p0 = new Point(320, 480);
Point p1 = new Point(320, 240);
Point p2 = new Point(0, 240);
double t = 0.10f;
Point p = Bezier.quadratic(p0, p1, p2, t);
double[] ts = Bezier.obtainTs(p.x, p0, p1, p2);

I obtain the following output:

For t=0.10000000149011612, java.awt.Point[x=316,y=434]
For t1: -0.1118033988749895, java.awt.Point[x=316,y=536]
For t2: 0.1118033988749895, java.awt.Point[x=316,y=429]
java.awt.Point[x=316,y=434] belongs?: false

Should I use BigDecimal to perform the operations? Is there another way to verify this? Thanks

Answer 1

As an alternative answer, to circumvent the problem noted by Martin R, you can simply build a lookup table and resolve coordinates to on-curve-ness that way. When you build the curve, generate an N-point array of coordinates for incremental t values, and then when you need to test whether a coordinate lies on the curve, find the nearest t value to that coordinate by checking whether that coordinate is in the lookup table, or "near enough to" a coordinate in the lookup table. In code:

point[] points = new point[100];
for(i=0; i<100; i++) {
  t = i/100;
  points[i] = new point(computeX(t), computeY(t));
}

and then when you need on-curve-testing:

for(i=0; i<points.length; i++) {
  point = points[i];
  if(abs(point-coordinate)<3) {
    // close enough to the curve to count,
    // so we can use t value map(i,0,100,0,1)
  }
}

building the LUT costs virtually nothing since we're generating those coordinate-for-t values already in order to draw the curve, and you're not going to run an on-curve test without first making sure the coordinate is even in the curve's bounding box.

Answer 2

There is an error here:

double[] ty = obtainTs(p.y, p0, p1, p2);

because obtainTs() uses the x-coordinates of p0, p1, p2 to find the t-parameter for the y-coordinate of p.

If you change the method parameters to int (which can be the x- or y-coordinates of the point):

public static double[] obtainTs (int comp, int p0, int p1, int p2) {
    double a = p0 - 2*p1 + p2;
    double b = 2*p1 - 2*p0 ;
    double c = p0 - comp;

    double t1 = (-b + Math.sqrt(b*b - 4*a*c)) / (2*a);
    double t2 = (-b - Math.sqrt(b*b - 4*a*c)) / (2*a);

    return new double[] {t1, t2};
}

and call it

double[] tx = obtainTs(p.x, p0.x, p1.x, p2.x);
double[] ty = obtainTs(p.y, p0.y, p1.y, p2.y);

then your test code will return "true" (tested with ERROR = 0.02).

---

Note that if you write down the equation

B(t) = (1 - t)^2 * p0 + 2 * t * (1 - t) * p1 + t^2 * p2

for both x- and y-coordinate, then you can eliminate the t^2-Term and get a single linear equation for t. This gives the following method, which might be slightly simpler and does not use square roots:

public static final boolean belongsQuadratic2 (Point p, Point p0, Point p1, Point p2) {
    double ax = p0.x - 2*p1.x + p2.x;
    double bx = 2*p1.x - 2*p0.x ;
    double cx = p0.x - p.x;

    double ay = p0.y - 2*p1.y + p2.y;
    double by = 2*p1.y - 2*p0.y ;
    double cy = p0.y - p.y;

    // "Candidate" for t:
    double t = -(cx*ay - cy*ax)/(bx*ay - by*ax);
    if (t < 0 || t > 1)
        return false;
    // Compute the point corresponding to this candidate value ...
    Point q = Bezier.quadratic(p0, p1, p2, t);
    // ... and check if it is near the given point p:
    return Math.abs(q.x - p.x) <= 1 && Math.abs(q.y - p.y) <= 1;
}

Of course, one would have to check for special cases, such as bx*ay - by*ax == 0.

Note also that it is difficult do decide exactly if a point lies on the curve because the point coordinates are rounded to integers.