Detecting wether a point is inside a SVG Path

I'm trying to detecting wether a given point is inside a closed SVG path in Objective-C. I'm having trouble figuring out how to do the math.

I have a path's coordinates and I'd like to determine wether a random point is inside or outside the path.

Here's an example of a path's coordinates:

"M673 460 c2 0 4 -1 5 -2 1 -1 2 -2 2 -4 0 -2 0 -3 0 -3 0 0 -3 1 -5 1 -3 1 -5 2 -5 3 0 1 0 3 0 4 1 0 2 1 3 1z:"

I'm aware of the CoreGraphics's containsPoint: method, but I'd like to avoid using this method.

How can I do this by writing my own method?

EDIT:

I'm trying to avoid containsPoint: because the function seems to crash on some coordinates / paths when using it. It looks quite random when it crashes and when it doesn't.

Here are some examples of paths where containsPoint: makes the app crash:

"M661 446 c1 -1 3 -1 4 -1 1 -1 2 -2 2 -4 0 -2 0 -2 -2 -2 0 1 -2 1 -3 1 -2 0 -3 1 -3 2 0 1 0 2 0 3 0 0 1 1 2 1z" 
"M535 460 c0 0 1 -1 1 -2 1 -2 0 -3 -1 -3 0 0 -1 0 -2 1 0 1 0 2 0 3 1 0 1 1 2 1z"

Xcode breaks with and EXC_BAD_ACCESS in the assembly by the following two functions: get_y_inflections and get_cubic_coefficients.

EDIT 2:

I posted a new question about the containsPoint: problem.

So, as i needed a Polygon/Interpolation from a Curved Path myself, i might have a solution for your problem as well:

The two 'Public' Functions:

CGPathRef CGPathCreatePolygonPath(CGPathRef path, int quality);

Creates an Path with only Line-Elements, quality is the number of segments a curve is split into

BOOL CGPathContainsPointInterpolated(CGPathRef path, const CGAffineTransform *m, CGPoint point, bool eoFill, int quality);

Does CGPathContainsPoint and Interpolation in one step, quality is again the number of segments a curve is split into.

Here is the implementation.

typedef struct {
    CGMutablePathRef path;
    int quality;
} InterpolationInfo;

static inline float cubeInterp(float t, float p0, float p1, float p2, float p3) {
    return powf(1-t, 3)*p0 + 3*powf(1-t, 2)*t*p1 + 3*(1-t)*powf(t, 2)*p2 + powf(t, 3)*p3;
}

static void pointsForCubeCurve(CGPoint cp0, CGPoint cp1, CGPoint cp2, CGPoint ep, CGPoint *buffer, int numberPoints) {
    for (int i = 0; i<numberPoints; i++) {
        float t = (i+1)/(float)numberPoints;
        buffer[i] = CGPointMake(cubeInterp(t, cp0.x, cp1.x, cp2.x, ep.x), cubeInterp(t, cp0.y, cp1.y, cp2.y, ep.y));
    }
}

static inline float quadInterp(float t, float p0, float p1, float p2) {
    return powf(1-t, 2)*p0 + 2*(1-t)*t*p1 + powf(t, 2)*p2;
}

static void pointsForQuadCurve(CGPoint cp0, CGPoint cp, CGPoint ep, CGPoint *buffer, int numberPoints) {
    for (int i = 0; i<numberPoints; i++) {
        float t = (i+1)/(float)numberPoints;
        buffer[i] = CGPointMake(quadInterp(t, cp0.x, cp.x, ep.x), quadInterp(t, cp0.x, cp.x, ep.x));
    }
}

static void CGPathElementConvertToPolygon(void *info, const CGPathElement *element) {
    InterpolationInfo *interpInfo = info;
    switch (element->type) {
        case kCGPathElementMoveToPoint:
            CGPathMoveToPoint(interpInfo->path, NULL, element->points[0].x, element->points[0].y);
            break;
        case kCGPathElementAddLineToPoint:
            CGPathAddLineToPoint(interpInfo->path, NULL, element->points[0].x, element->points[0].y);
            break;
        case kCGPathElementAddQuadCurveToPoint: {
            int nr = interpInfo->quality;
            CGPoint buffer[nr];
            pointsForQuadCurve(CGPathGetCurrentPoint(interpInfo->path), element->points[0], element->points[1], buffer, nr);
            for (int i = 0; i<nr; i++) {
                CGPathAddLineToPoint(interpInfo->path, NULL, buffer[i].x, buffer[i].y);
            }
            break;
        }
        case kCGPathElementAddCurveToPoint: {
            int nr = interpInfo->quality;
            CGPoint buffer[nr];
            pointsForCubeCurve(CGPathGetCurrentPoint(interpInfo->path), element->points[0], element->points[1], element->points[2], buffer, nr);
            for (int i = 0; i<nr; i++) {
                CGPathAddLineToPoint(interpInfo->path, NULL, buffer[i].x, buffer[i].y);
            }
            break;
        }
        case kCGPathElementCloseSubpath:
            CGPathCloseSubpath(interpInfo->path);
            break;
        default:
            break;
    }
}

static CGPathRef CGPathCreatePolygonPath(CGPathRef path, int quality) {
    CGMutablePathRef newPath = CGPathCreateMutable();
    InterpolationInfo info;
    info.path = newPath;
    info.quality = quality;
    CGPathApply(path, &info, CGPathElementConvertToPolygon);
    return newPath;
}

static BOOL CGPathContainsPointInterpolated(CGPathRef path, const CGAffineTransform *m, CGPoint point, bool eoFill, int quality) {
    CGPathRef polygon = CGPathCreatePolygonPath(path, quality);
    BOOL returnValue = CGPathContainsPoint(polygon, m, point, eoFill);
    CGPathRelease(polygon);
    return returnValue;
}