Graphic algorithm Unions, intersect, subtract - graphics

Question

I need a good source for reading up on how to create a algorithm to take two polylines (a path comprised of many lines) and performing a union, subtraction, or intersection between them. This is tied to a custom API so I need to understand the underlying algorithm.
Plus any sources in a VB dialect would be doubly helpful.

Answer 1

This catalogue of implementations of intersection algorithms from the Stony Brook Algorithm Repository might be useful. The repository is managed by Steven Skiena,
author of a very well respected book on algorithms: The Algorithm Design Manual.
That's his own Amazon exec link by the way :)

Answer 2

Several routines for you here. Hope you find them useful :-)
// routine to calculate the square of either the shortest distance or largest distance
// from the CPoint to the intersection point of a ray fired at an angle flAngle
// radians at an array of line segments
// this routine returns TRUE if an intersection has been found in which case flD
// is valid and holds the square of the distance.
// and returns FALSE if no valid intersection was found
// If an intersection was found, then intersectionPoint is set to the point found
bool CalcIntersection(const CPoint &cPoint,
const float flAngle,
const int nVertexTotal,
const CPoint *pVertexList,
const BOOL bMin,
float &flD,
CPoint &intersectionPoint)
{
float d, dsx, dsy, dx, dy, lambda, mu, px, py;
int p0x, p0y, p1x, p1y;
// get source position
const float flSx = (float)cPoint.x;
const float flSy = -(float)cPoint.y;
// calc trig functions
const float flTan = tanf(flAngle);
const float flSin = sinf(flAngle);
const float flCos = cosf(flAngle);
const bool bUseSin = fabsf(flSin) > fabsf(flCos);
// initialise distance
flD = (bMin ? FLT_MAX : 0.0f);
// for each line segment in protective feature
for(int i = 0; i < nVertexTotal; i++)
{
// get coordinates of line (negate the y value so the y-axis is upwards)
p0x = pVertexList[i].x;
p0y = -pVertexList[i].y;
p1x = pVertexList[i + 1].x;
p1y = -pVertexList[i + 1].y;
// calc. deltas
dsx = (float)(cPoint.x - p0x);
dsy = (float)(-cPoint.y - p0y);
dx = (float)(p1x - p0x);
dy = (float)(p1y - p0y);
// calc. denominator
d = dy * flTan - dx;
// if line & ray are parallel
if(fabsf(d) < 1.0e-7f)
continue;
// calc. intersection point parameter
lambda = (dsy * flTan - dsx) / d;
// if intersection is not valid
if((lambda <= 0.0f) || (lambda > 1.0f))
continue;
// if sine is bigger than cosine
if(bUseSin){
mu = ((float)p0x + lambda * dx - flSx) / flSin;
} else {
mu = ((float)p0y + lambda * dy - flSy) / flCos;
}
// if intersection is valid
if(mu >= 0.0f){
// calc. intersection point
px = (float)p0x + lambda * dx;
py = (float)p0y + lambda * dy;
// calc. distance between intersection point & source point
dx = px - flSx;
dy = py - flSy;
d = dx * dx + dy * dy;
// compare with relevant value
if(bMin){
if(d < flD)
{
flD = d;
intersectionPoint.x = RoundValue(px);
intersectionPoint.y = -RoundValue(py);
}
} else {
if(d > flD)
{
flD = d;
intersectionPoint.x = RoundValue(px);
intersectionPoint.y = -RoundValue(py);
}
}
}
}
// return
return(bMin ? (flD != FLT_MAX) : (flD != 0.0f));
}
// Routine to calculate the square of the distance from the CPoint to the
// intersection point of a ray fired at an angle flAngle radians at a line.
// This routine returns TRUE if an intersection has been found in which case flD
// is valid and holds the square of the distance.
// Returns FALSE if no valid intersection was found.
// If an intersection was found, then intersectionPoint is set to the point found.
bool CalcIntersection(const CPoint &cPoint,
const float flAngle,
const CPoint &PointA,
const CPoint &PointB,
const bool bExtendLine,
float &flD,
CPoint &intersectionPoint)
{
// get source position
const float flSx = (float)cPoint.x;
const float flSy = -(float)cPoint.y;
// calc trig functions
float flTan = tanf(flAngle);
float flSin = sinf(flAngle);
float flCos = cosf(flAngle);
const bool bUseSin = fabsf(flSin) > fabsf(flCos);
// get coordinates of line (negate the y value so the y-axis is upwards)
const int p0x = PointA.x;
const int p0y = -PointA.y;
const int p1x = PointB.x;
const int p1y = -PointB.y;
// calc. deltas
const float dsx = (float)(cPoint.x - p0x);
const float dsy = (float)(-cPoint.y - p0y);
float dx = (float)(p1x - p0x);
float dy = (float)(p1y - p0y);
// Calc. denominator
const float d = dy * flTan - dx;
// If line & ray are parallel
if(fabsf(d) < 1.0e-7f)
return false;
// calc. intersection point parameter
const float lambda = (dsy * flTan - dsx) / d;
// If extending line to meet point, don't check for ray missing line
if(!bExtendLine)
{
// If intersection is not valid
if((lambda <= 0.0f) || (lambda > 1.0f))
return false; // Ray missed line
}
// If sine is bigger than cosine
float mu;
if(bUseSin){
mu = ((float)p0x + lambda * dx - flSx) / flSin;
} else {
mu = ((float)p0y + lambda * dy - flSy) / flCos;
}
// if intersection is valid
if(mu >= 0.0f)
{
// calc. intersection point
const float px = (float)p0x + lambda * dx;
const float py = (float)p0y + lambda * dy;
// calc. distance between intersection point & source point
dx = px - flSx;
dy = py - flSy;
flD = (dx * dx) + (dy * dy);
intersectionPoint.x = RoundValue(px);
intersectionPoint.y = -RoundValue(py);
return true;
}
return false;
}
// Fillet (with a radius of 0) two lines. From point source fired at angle (radians) to line Line1A, Line1B.
// Modifies line end point Line1B. If the ray does not intersect line, then it is rotates every 90 degrees
// and tried again until fillet is complete.
void Fillet(const CPoint &source, const float fThetaRadians, const CPoint &Line1A, CPoint &Line1B)
{
if(Line1A == Line1B)
return; // No line
float dist;
if(CalcIntersection(source, fThetaRadians, Line1A, Line1B, true, dist, Line1B))
return;
if(CalcIntersection(source, CalcBaseFloat(TWO_PI, fThetaRadians + PI * 0.5f), Line1A, Line1B, true, dist, Line1B))
return;
if(CalcIntersection(source, CalcBaseFloat(TWO_PI, fThetaRadians + PI), Line1A, Line1B, true, dist, Line1B))
return;
if(!CalcIntersection(source, CalcBaseFloat(TWO_PI, fThetaRadians + PI * 1.5f), Line1A, Line1B, true, dist, Line1B))
ASSERT(FALSE); // Could not find intersection?
}
// routine to determine if an array of line segments cross gridSquare
// x and y give the float coordinates of the corners
BOOL CrossGridSquare(int nV, const CPoint *pV,
const CRect &extent, const CRect &gridSquare)
{
// test extents
if( (extent.right < gridSquare.left) ||
(extent.left > gridSquare.right) ||
(extent.top > gridSquare.bottom) ||
(extent.bottom < gridSquare.top))
{
return FALSE;
}
float a, b, c, dx, dy, s, x[4], y[4];
int max_x, max_y, min_x, min_y, p0x, p0y, p1x, p1y, sign, sign_old;
// construct array of vertices for grid square
x[0] = (float)gridSquare.left;
y[0] = (float)gridSquare.top;
x[1] = (float)(gridSquare.right);
y[1] = y[0];
x[2] = x[1];
y[2] = (float)(gridSquare.bottom);
x[3] = x[0];
y[3] = y[2];
// for each line segment
for(int i = 0; i < nV; i++)
{
// get end-points
p0x = pV[i].x;
p0y = pV[i].y;
p1x = pV[i + 1].x;
p1y = pV[i + 1].y;
// determine line extent
if(p0x > p1x){
min_x = p1x;
max_x = p0x;
} else {
min_x = p0x;
max_x = p1x;
}
if(p0y > p1y){
min_y = p1y;
max_y = p0y;
} else {
min_y = p0y;
max_y = p1y;
}
// test to see if grid square is outside of line segment extent
if( (max_x < gridSquare.left) ||
(min_x > gridSquare.right) ||
(max_y < gridSquare.top) ||
(min_y > gridSquare.bottom))
{
continue;
}
// calc. line equation
dx = (float)(p1x - p0x);
dy = (float)(p1y - p0y);
a = dy;
b = -dx;
c = -dy * (float)p0x + dx * (float)p0y;
// evaluate line eqn. at first grid square vertex
s = a * x[0] + b * y[0] + c;
if(s < 0.0f){
sign_old = -1;
} else if(s > 1.0f){
sign_old = 1;
} else {
sign_old = 0;
}
// evaluate line eqn. at other grid square vertices
for (int j = 1; j < 4; j++)
{
s = a * x[j] + b * y[j] + c;
if(s < 0.0f){
sign = -1;
} else if(s > 1.0f){
sign = 1;
} else {
sign = 0;
}
// if there has been a chnage in sign
if(sign != sign_old)
return TRUE;
}
}
return FALSE;
}
// calculate the square of the shortest distance from point s
// and the line segment between p0 and p1
// t is the point on the line from which the minimum distance
// is measured
float CalcShortestDistanceSqr(const CPoint &s,
const CPoint &p0,
const CPoint &p1,
CPoint &t)
{
// if point is at a vertex
if((s == p0) || (s == p1))
return(0.0F);
// calc. deltas
int dx = p1.x - p0.x;
int dy = p1.y - p0.y;
int dsx = s.x - p0.x;
int dsy = s.y - p0.y;
// if both deltas are zero
if((dx == 0) && (dy == 0))
{
// shortest distance is distance is to either vertex
float l = (float)(dsx * dsx + dsy * dsy);
t = p0;
return(l);
}
// calc. point, p, on line that is closest to sourcePosition
// p = p0 + l * (p1 - p0)
float l = (float)(dsx * dx + dsy * dy) / (float)(dx * dx + dy * dy);
// if intersection is beyond p0
if(l <= 0.0F){
// shortest distance is to p0
l = (float)(dsx * dsx + dsy * dsy);
t = p0;
// else if intersection is beyond p1
} else if(l >= 1.0F){
// shortest distance is to p1
dsx = s.x - p1.x;
dsy = s.y - p1.y;
l = (float)(dsx * dsx + dsy * dsy);
t = p1;
// if intersection is between line end points
} else {
// calc. perpendicular distance
float ldx = (float)dsx - l * (float)dx;
float ldy = (float)dsy - l * (float)dy;
t.x = p0.x + RoundValue(l * (float)dx);
t.y = p0.y + RoundValue(l * (float)dy);
l = ldx * ldx + ldy * ldy;
}
return(l);
}
// Calculates the bounding rectangle around a set of points
// Returns TRUE if the rectangle is not empty (has area), FALSE otherwise
// Opposite of CreateRectPoints()
BOOL CalcBoundingRectangle(const CPoint *pVertexList, const int nVertexTotal, CRect &rect)
{
rect.SetRectEmpty();
if(nVertexTotal < 2)
{
ASSERT(FALSE); // Must have at least 2 points
return FALSE;
}
// First point, set rectangle (no area at this point)
rect.left = rect.right = pVertexList[0].x;
rect.top = rect.bottom = pVertexList[0].y;
// Increst rectangle by looking at other points
for(int n = 1; n < nVertexTotal; n++)
{
if(rect.left > pVertexList[n].x) // Take minimum
rect.left = pVertexList[n].x;
if(rect.right < pVertexList[n].x) // Take maximum
rect.right = pVertexList[n].x;
if(rect.top > pVertexList[n].y) // Take minimum
rect.top = pVertexList[n].y;
if(rect.bottom < pVertexList[n].y) // Take maximum
rect.bottom = pVertexList[n].y;
}
rect.NormalizeRect(); // Normalise rectangle
return !(rect.IsRectEmpty());
}