/// <summary>
/// This function calculates the intersection point between a plane and a line. Replace the x,y,z components of the line equation into the plane equation and solve for t, then replace t in the line equation to get the x,y,z
/// </summary>
/// <param name="line"></param>
/// <param name="plane"></param>
/// <returns></returns>
public static Point IntersectionLinePlane(Line line, Plane plane)
{
// could write in one line but it'd be too hard to debug, or read!
// replace the x,y,z components of the line equation into the plane equation and solve for t
double numerator = -plane.d - plane.a * line.P0.x - plane.b * line.P0.y - plane.c * line.P0.z;
double denominator = plane.a * line.v.x + plane.b * line.v.y + plane.c * line.v.z;
double t = numerator / denominator;
// replace t in the line equation to get the x,y,z
double x = t * line.v.x + line.P0.x;
double y = t * line.v.y + line.P0.y;
double z = t * line.v.z + line.P0.z;
return new Point(x, y, z);
}
private void buttonTest_Click(object sender, EventArgs e)
{
// test the function
// setup test with one triangle and one plane
// input a set of triangles
List<Triangle> triangles = new List<Triangle>();
triangles.Add(new Triangle(new Point(-5, 10, 0), new Point(10, 10, 0), new Point(5, -10, 0)));
triangles.Add(new Triangle(new Point(-5, 10, 5), new Point(10, 10, 5), new Point(5, -10, 5)));
triangles.Add(new Triangle(new Point(-3, 10, 4), new Point(1, 10, -3), new Point(10, -13, 8)));
triangles.Add(new Triangle(new Point(-5, 100, 0), new Point(10, 100, 0), new Point(5, 100, 0)));
triangles.Add(new Triangle(new Point(-50, 13, 2), new Point(0, -4, 0), new Point(2, -10, 0)));
// define the plane
Plane plane = new Plane(-0.5, 1, 0.5, 0);
// slice using a plane defined by normal equation, the plane does not have bounds
List<Segment> segments = new List<Segment>();
foreach (Triangle triangle in triangles)
{
try
{
Segment segment = Triangle.TriangleInPlane(triangle, plane);
if (segment != null)
{
segments.Add(segment);
}
}
catch (Exception ex)
{
textBoxDebug.Text = ex.ToString();
}
}
// segments contain polylines from slicing a set of triangles
textBoxDebug.Text = "done";
}
/// <summary>
/// this function calculates th distance between a point and a plane. Using the normal equation, we plug the coordinates of the point and get the distance
/// </summary>
/// <param name="point"></param>
/// <param name="plane"></param>
/// <returns></returns>
public double DistancePointPlane(Point point, Plane plane)
{
// the equation is distance = (a*x0 + b*y0 + c*z0 + d)/sqrt(a*a + b*b + c*c)
// split the equation in two for ease of reading
double numerator = (plane.a * point.x + plane.b * point.y + plane.c * point.z + plane.d);
double denominator = Math.Sqrt(plane.a * plane.a + plane.b * plane.b + plane.c * plane.c);
return numerator / denominator;
}
/// <summary>
/// this function finds the intersection between the triangle and the plane
/// --> checks if the triangle intersects the plane
/// --> if it intersects, it will look like a segment
/// --> the ends of the segment are the intersections of the two edges of the triangle going through the plane and the plane
/// </summary>
/// <param name="triangle">the input triangle</param>
/// <param name="plane">the input plane</param>
/// <returns>segment if there is intersection, else return null</returns>
public static Segment TriangleInPlane(Triangle triangle, Plane plane)
{
//
// find side +/- of each of 3 points of triangle
bool SidePoint1 = SidePointPlane(triangle.point1, plane);
bool SidePoint2 = SidePointPlane(triangle.point2, plane);
bool SidePoint3 = SidePointPlane(triangle.point3, plane);
Point p1, p2;
// point1 is lonely
if (SidePoint1 != SidePoint2
&& SidePoint2 == SidePoint3)
{
// get lines made by Point1-2 and Point1-3
Line l12 = new Line(triangle.point1, triangle.point2);
Line l13 = new Line(triangle.point1, triangle.point3);
// get intersection of lines and plane
p1 = Line.IntersectionLinePlane(l12,plane);
p2 = Line.IntersectionLinePlane(l13, plane);
// return the segment made of those two points
return new Segment(p1,p2);
}
else if (SidePoint2 != SidePoint1
&& SidePoint1 == SidePoint3)// point2 is lonely
{
// get lines made by Point1-2 and Point1-3
Line l21 = new Line(triangle.point2, triangle.point1);
Line l23 = new Line(triangle.point2, triangle.point3);
// get intersection of lines and plane
p1 = Line.IntersectionLinePlane(l21, plane);
p2 = Line.IntersectionLinePlane(l23, plane);
// return the segment made of those two points
return new Segment(p1, p2);
}
else if (SidePoint3 != SidePoint2
&& SidePoint1 == SidePoint2)// point3 is lonely
{
// get lines made by Point1-2 and Point1-3
Line l31 = new Line(triangle.point3, triangle.point1);
Line l32 = new Line(triangle.point3, triangle.point2);
// get intersection of lines and plane
p1 = Line.IntersectionLinePlane(l31, plane);
p2 = Line.IntersectionLinePlane(l32, plane);
// return the segment made of those two points
return new Segment(p1, p2);
}
else
{// both points are on same side of plane, meaning that the triangle is not intersecting the plane
return null;
}
}
/// <summary>
/// this is the same as the distance but we only want the sign so we skip the division part which is time consuming
/// </summary>
/// <param name="point"></param>
/// <param name="plane"></param>
/// <returns></returns>
public static bool SidePointPlane(Point point, Plane plane)
{
// side is a bit arbitrary because it depends on definition of the plane's normal vector
// the equation is distance = (a*x0 + b*y0 + c*z0 + d)/sqrt(a*a + b*b + c*c)
// since we only want the sign, we can ignore the division part
// sign(a*x0 + b*y0 + c*z0 + d)
return (plane.a * point.x + plane.b * point.y + plane.c * point.z + plane.d) >= 0; // return true if positive, false if negative
}