//
#include "geometry.h"
-
-
-Point Geometry::IntersectionOfLineAndLine(Point p1, Point p2, Point p3, Point p4)
-{
- // Find the intersection of the lines by formula:
- // px = (x1y2 - y1x2)(x3 - x4) - (x1 - x2)(x3y4 - y3x4)
- // py = (x1y2 - y1x2)(y3 - y4) - (y1 - y2)(x3y4 - y3x4)
- // d = (x1 - x2)(y3 - y4) - (y1 - y2)(x3 - x4) = 0 if lines are parallel
- // Intersection is (px / d, py / d)
-
- double d = ((p1.x - p2.x) * (p3.y - p4.y)) - ((p1.y - p2.y) * (p3.x - p4.x));
-
- // Check for parallel lines, and return sentinel if so
- if (d == 0)
- return Point(0, 0, -1);
-
- double px = (((p1.x * p2.y) - (p1.y * p2.x)) * (p3.x - p4.x))
- - ((p1.x - p2.x) * ((p3.x * p4.y) - (p3.y * p4.x)));
- double py = (((p1.x * p2.y) - (p1.y * p2.x)) * (p3.y - p4.y))
- - ((p1.y - p2.y) * ((p3.x * p4.y) - (p3.y * p4.x)));
-
- return Point(px / d, py / d, 0);
-}
+#include <math.h>
+#include <stdio.h>
+#include "global.h"
+#include "mathconstants.h"
// Returns the parameter of a point in space to this vector. If the parameter
// is between 0 and 1, the normal of the vector to the point is on the vector.
-double Geometry::ParameterOfLineAndPoint(Point lp1, Point lp2, Point point)
+// Note: lp1 is the tail, lp2 is the head of the line (vector).
+double Geometry::ParameterOfLineAndPoint(Point tail, Point head, Point point)
{
// Geometric interpretation:
// The parameterized point on the vector lineSegment is where the normal of
// the perpendicular lies beyond the 1st endpoint. If pp > 1, then the
// perpendicular lies beyond the 2nd endpoint.
- Vector lineSegment = lp1 - lp2;
+ Vector lineSegment = head - tail;
double magnitude = lineSegment.Magnitude();
- Vector pointSegment = point - lp2;
+ Vector pointSegment = point - tail;
double t = lineSegment.Dot(pointSegment) / (magnitude * magnitude);
return t;
}
-Point Geometry::MirrorPointAroundLine(Point point, Point p1, Point p2)
+Point Geometry::MirrorPointAroundLine(Point point, Point tail, Point head)
{
// Get the vector of the intersection of the line and the normal on the
// line to the point in question.
- double t = ParameterOfLineAndPoint(p1, p2, point);
- Vector v = Vector(p1, p2) * t;
+ double t = ParameterOfLineAndPoint(tail, head, point);
+ Vector v = Vector(tail, head) * t;
- // Get the point normal to point to the line passed in (p2 is the tail)
- Point normalOnLine = p2 + v;
+ // Get the point normal to point to the line passed in
+ Point normalOnLine = tail + v;
// Make our mirrored vector (head - tail)
Vector mirror = -(point - normalOnLine);
return mirroredPoint;
}
+
+//
+// point: The point we're rotating
+// rotationPoint: The point we're rotating around
+//
+Point Geometry::RotatePointAroundPoint(Point point, Point rotationPoint, double angle)
+{
+ Vector v = Vector(rotationPoint, point);
+ double px = (v.x * cos(angle)) - (v.y * sin(angle));
+ double py = (v.x * sin(angle)) + (v.y * cos(angle));
+
+ return Vector(rotationPoint.x + px, rotationPoint.y + py, 0);
+}
+
+
+double Geometry::Determinant(Point p1, Point p2)
+{
+ return (p1.x * p2.y) - (p2.x * p1.y);
+}
+
+
+void Geometry::Intersects(Object * obj1, Object * obj2)
+{
+ Global::numIntersectPoints = Global::numIntersectParams = 0;
+
+ if ((obj1->type == OTLine) && (obj2->type == OTLine))
+ CheckLineToLineIntersection(obj1, obj2);
+ else if ((obj1->type == OTCircle) && (obj2->type == OTCircle))
+ CheckCircleToCircleIntersection(obj1, obj2);
+ else if ((obj1->type == OTLine) && (obj2->type == OTCircle))
+ CheckLineToCircleIntersection(obj1, obj2);
+ else if ((obj1->type == OTCircle) && (obj2->type == OTLine))
+ CheckLineToCircleIntersection(obj2, obj1);
+}
+
+
+/*
+Intersecting line segments:
+An easier way:
+Segment L1 has edges A=(a1,a2), A'=(a1',a2').
+Segment L2 has edges B=(b1,b2), B'=(b1',b2').
+Segment L1 is the set of points tA'+(1-t)A, where 0<=t<=1.
+Segment L2 is the set of points sB'+(1-s)B, where 0<=s<=1.
+Segment L1 meet segment L2 if and only if for some t and s we have
+tA'+(1-t)A=sB'+(1-s)B
+The solution of this with respect to t and s is
+
+t=((-b?'a?+b?'b?+b?a?+a?b?'-a?b?-b?b?')/(b?'a?'-b?'a?-b?a?'+b?a?-a?'b?'+a?'b?+a?b?'-a?b?))
+
+s=((-a?b?+a?'b?-a?a?'+b?a?+a?'a?-b?a?')/(b?'a?'-b?'a?-b?a?'+b?a?-a?'b??+a?'b?+a?b?'-a?b?))
+
+So check if the above two numbers are both >=0 and <=1.
+*/
+
+
+// Finds the intersection between two lines (if any)
+void Geometry::CheckLineToLineIntersection(Object * l1, Object * l2)
+{
+ Global::numIntersectPoints = Global::numIntersectParams = 0;
+
+ Vector r(l1->p[0], l1->p[1]);
+ Vector s(l2->p[0], l2->p[1]);
+ Vector v1 = l2->p[0] - l1->p[0]; // q - p
+
+ double rxs = (r.x * s.y) - (s.x * r.y);
+ double t, u;
+
+ if (rxs == 0)
+ {
+ double qpxr = (v1.x * r.y) - (r.x * v1.y);
+
+ // Lines are parallel, so no intersection...
+ if (qpxr != 0)
+ return;
+
+ // Check to see which endpoints are connected... Four possibilities:
+ if (l1->p[0] == l2->p[0])
+ t = 0, u = 0;
+ else if (l1->p[0] == l2->p[1])
+ t = 0, u = 1.0;
+ else if (l1->p[1] == l2->p[0])
+ t = 1.0, u = 0;
+ else if (l1->p[1] == l2->p[1])
+ t = 1.0, u = 1.0;
+ else
+ return;
+ }
+ else
+ {
+ t = ((v1.x * s.y) - (s.x * v1.y)) / rxs;
+ u = ((v1.x * r.y) - (r.x * v1.y)) / rxs;
+ }
+
+ Global::intersectParam[0] = t;
+ Global::intersectParam[1] = u;
+
+ // If the parameters are in range, we have overlap!
+ if ((t >= 0) && (t <= 1.0) && (u >= 0) && (u <= 1.0))
+ Global::numIntersectParams = 1;
+}
+
+
+void Geometry::CheckCircleToCircleIntersection(Object * c1, Object * c2)
+{
+ // Set up global vars
+ Global::numIntersectPoints = Global::numIntersectParams = 0;
+
+ // Get the distance between the centers of the circles
+ Vector centerLine(c1->p[0], c2->p[0]);
+ double d = centerLine.Magnitude();
+ double clAngle = centerLine.Angle();
+
+ // If the distance between centers is greater than the sum of the radii or
+ // less than the difference between the radii, there is NO intersection
+ if ((d > (c1->radius[0] + c2->radius[0]))
+ || (d < fabs(c1->radius[0] - c2->radius[0])))
+ return;
+
+ // If the distance between centers is equal to the sum of the radii or
+ // equal to the difference between the radii, the intersection is tangent
+ // to both circles.
+ if (d == (c1->radius[0] + c2->radius[0]))
+ {
+ Global::intersectPoint[0].x = c1->p[0].x + (cos(clAngle) * c1->radius[0]);
+ Global::intersectPoint[0].y = c1->p[0].y + (sin(clAngle) * c1->radius[0]);
+ Global::numIntersectPoints = 1;
+ return;
+ }
+ else if (d == fabs(c1->radius[0] - c2->radius[0]))
+ {
+ double sign = (c1->radius[0] > c2->radius[0] ? +1 : -1);
+ Global::intersectPoint[0].x = c1->p[0].x + (cos(clAngle) * c1->radius[0] * sign);
+ Global::intersectPoint[0].y = c1->p[0].y + (sin(clAngle) * c1->radius[0] * sign);
+ Global::numIntersectPoints = 1;
+ return;
+ }
+
+/*
+ c² = a² + b² - 2ab·cos µ
+2ab·cos µ = a² + b² - c²
+ cos µ = (a² + b² - c²) / 2ab
+*/
+ // Use the Law of Cosines to find the angle between the centerline and the
+ // radial line on Circle #1
+ double a = acos(((c1->radius[0] * c1->radius[0]) + (d * d) - (c2->radius[0] * c2->radius[0])) / (2.0 * c1->radius[0] * d));
+
+ // Finally, find the points of intersection by using +/- the angle found
+ // from the centerline's angle
+ Global::intersectPoint[0].x = c1->p[0].x + (cos(clAngle + a) * c1->radius[0]);
+ Global::intersectPoint[0].y = c1->p[0].y + (sin(clAngle + a) * c1->radius[0]);
+ Global::intersectPoint[1].x = c1->p[0].x + (cos(clAngle - a) * c1->radius[0]);
+ Global::intersectPoint[1].y = c1->p[0].y + (sin(clAngle - a) * c1->radius[0]);
+ Global::numIntersectPoints = 2;
+}
+
+
+//
+// N.B.: l is the line, c is the circle
+//
+void Geometry::CheckLineToCircleIntersection(Object * l, Object * c)
+{
+ // Set up global vars
+ Global::numIntersectPoints = Global::numIntersectParams = 0;
+
+ // Step 1: Find shortest distance from center of circle to the infinite line
+ double t = ParameterOfLineAndPoint(l->p[0], l->p[1], c->p[0]);
+ Point p = l->p[0] + (Vector(l->p[0], l->p[1]) * t);
+ Vector radial = Vector(c->p[0], p);
+ double distance = radial.Magnitude();
+
+ // Step 2: See if we have 0, 1, or 2 intersection points
+
+ // Case #1: No intersection points
+ if (distance > c->radius[0])
+ return;
+ // Case #2: One intersection point (possibly--tangent)
+ else if (distance == c->radius[0])
+ {
+ // Only intersects if the parameter is on the line segment!
+ if ((t >= 0.0) && (t <= 1.0))
+ {
+ Global::intersectPoint[0] = c->p[0] + radial;
+ Global::numIntersectPoints = 1;
+ }
+
+ return;
+ }
+
+ // Case #3: Two intersection points (possibly--secant)
+
+ // So, we have the line, and the perpendicular from the center of the
+ // circle to the line. Now figure out where the intersection points are.
+ // This is a right triangle, though do we really know all the sides?
+ // Don't need to, 2 is enough for Pythagoras :-)
+ // Radius is the hypotenuse, so we have to use c² = a² + b² => a² = c² - b²
+ double perpendicularLength = sqrt((c->radius[0] * c->radius[0]) - (distance * distance));
+
+ // Now, find the intersection points using the length...
+ Vector lineUnit = Vector(l->p[0], l->p[1]).Unit();
+ Point i1 = p + (lineUnit * perpendicularLength);
+ Point i2 = p - (lineUnit * perpendicularLength);
+
+ // Next we need to see if they are on the line segment...
+ double u = ParameterOfLineAndPoint(l->p[0], l->p[1], i1);
+ double v = ParameterOfLineAndPoint(l->p[0], l->p[1], i2);
+
+ if ((u >= 0.0) && (u <= 1.0))
+ {
+ Global::intersectPoint[Global::numIntersectPoints] = i1;
+ Global::numIntersectPoints++;
+ }
+
+ if ((v >= 0.0) && (v <= 1.0))
+ {
+ Global::intersectPoint[Global::numIntersectPoints] = i2;
+ Global::numIntersectPoints++;
+ }
+}
+
+
+// should we just do common trig solves, like AAS, ASA, SAS, SSA?
+// Law of Cosines:
+// c² = a² + b² - 2ab * cos(C)
+// Solving for C:
+// cos(C) = (c² - a² - b²) / -2ab = (a² + b² - c²) / 2ab
+// Law of Sines:
+// a / sin A = b / sin B = c / sin C
+
+// Solve the angles of the triangle given the sides. Angles returned are
+// opposite of the given sides (so a1 consists of sides s2 & s3, and so on).
+void Geometry::FindAnglesForSides(double s1, double s2, double s3, double * a1, double * a2, double * a3)
+{
+ // Use law of cosines to find 1st angle
+ double cosine1 = ((s2 * s2) + (s3 * s3) - (s1 * s1)) / (2.0 * s2 * s3);
+
+ // Check for a valid triangle
+ if ((cosine1 < -1.0) || (cosine1 > 1.0))
+ return;
+
+ double angle1 = acos(cosine1);
+
+ // Use law of sines to find 2nd & 3rd angles
+// sin A / a = sin B / b
+// sin B = (sin A / a) * b
+// B = arcsin( sin A * (b / a))
+// ??? ==> B = A * arcsin(b / a)
+/*
+Well, look here:
+sin B = sin A * (b / a)
+sin B / sin A = b / a
+arcsin( sin B / sin A ) = arcsin( b / a )
+
+hmm... dunno...
+*/
+
+ double angle2 = asin(s2 * (sin(angle1) / s1));
+ double angle3 = asin(s3 * (sin(angle1) / s1));
+
+ if (a1)
+ *a1 = angle1;
+
+ if (a2)
+ *a2 = angle2;
+
+ if (a3)
+ *a3 = angle3;
+}
+
+
+Point Geometry::GetPointForParameter(Object * obj, double t)
+{
+ if (obj->type == OTLine)
+ {
+ // Translate line vector to the origin, then add the scaled vector to
+ // initial point of the line.
+ Vector v = obj->p[1] - obj->p[0];
+ return obj->p[0] + (v * t);
+ }
+
+ return Point(0, 0);
+}
+
+
+Point Geometry::Midpoint(Line * line)
+{
+ return Point((line->p[0].x + line->p[1].x) / 2.0,
+ (line->p[0].y + line->p[1].y) / 2.0);
+}
+
+
+/*
+How to find the tangent of a point off a circle:
+
+ • Calculate the midpoint on the point and the center of the circle
+ • Get the length of the line segment from the and the center divided by two
+ • Use that length to construct a circle with the point at the center and the
+ radius equal to that length
+ • The intersection of the two circles are the tangent points
+
+*/
+
projects / architektonas / blobdiff