Compute the area of a closed curve by counting pixels. http://stackoverflow.com/questions/14220719/ios-how-to-calculate-number-of-pixels-area-enclosed-by-a-curve

## curveArea.m
static CGPathRef createClosedPathWithPoints(const CGPoint *points, size_t count) {
    CGMutablePathRef path = CGPathCreateMutable();
    CGPathAddLines(path, NULL, points, count);
    CGPathCloseSubpath(path);
    return path;
}

static CGRect integralFrameForPath(CGPathRef path) {
    CGRect frame = CGPathGetBoundingBox(path);
    return CGRectIntegral(frame);
}

static size_t bytesPerRowForWidth(CGFloat width) {
    static const size_t kFactor = 64;
    // Round up to a multiple of kFactor, which must be a power of 2.
    return ((size_t)width + (kFactor - 1)) & ~(kFactor - 1);
}

static CGContextRef createBitmapContextWithFrame(CGRect frame, size_t bytesPerRow) {
    CGColorSpaceRef grayscale = CGColorSpaceCreateDeviceGray();
    CGContextRef gc = CGBitmapContextCreate(NULL, frame.size.width, frame.size.height, 8, bytesPerRow, grayscale, kCGImageAlphaNone);
    CGColorSpaceRelease(grayscale);
    CGContextTranslateCTM(gc, -frame.origin.x, -frame.origin.x);
    return gc;
}

static double areaFilledInBitmapContext(gc) {
    size_t width = CGBitmapContextGetWidth(gc);
    size_t height = CGBitmapContextGetHeight(gc);
    size_t stride = CGBitmapContextGetBytesPerRow(gc);
    uint8_t *pixels = CGBitmapContextGetData(gc);
    uint64_t coverage = 0;
    for (size_t y = 0; y < height; ++y) {
        for (size_t x = 0; x < width; ++x) {
            coverage += pixels[y * stride + x];
        }
    }
    return (double)coverage / UINT8_MAX;
}

static double areaOfCurveWithPoints(const CGPoint *points, size_t count) {
    CGPathRef path = createClosedPathWithPoints(points, count);
    CGRect frame = integralFrameForPath(path);
    size_t bytesPerRow = bytesPerRowForWidth(frame.size.width);
    CGContextRef gc = createBitmapContextWithFrame(frame, bytesPerRow);

    CGContextSetFillColorWithColor(gc, [UIColor whiteColor].CGColor);
    CGContextAddPath(gc, path);
    CGContextFillPath(gc);
    CGPathRelease(path);

    double area = areaFilledInBitmapContext(gc);
    CGContextRelease(gc);

    return area;
}