3d Math functions

//increase or decrease the length of vector by size
Vector3 addVectorLength(Vector3 vector, float size){
 
	//get the vector length
	float magnitude = Vector3.Magnitude(vector);
 
	//change the length
	magnitude += size;
 
	//normalize the vector
	Vector3 vectorNormalized = Vector3.Normalize(vector);
 
	//scale the vector
	return Vector3.Scale(vectorNormalized, new Vector3(magnitude, magnitude, magnitude));		
}
 
//create a vector of direction "vector" with length "size"
Vector3 setVectorLength(Vector3 vector, float size){
 
	//normalize the vector
	Vector3 vectorNormalized = Vector3.Normalize(vector);
 
	//scale the vector
//	return Vector3.Scale(vectorNormalized, new Vector3(size, size, size));	
	return vectorNormalized *= size;
}
 
 
//calculate the rotational difference from A to B
Quaternion subtractRotation(Quaternion B, Quaternion A){		
	Quaternion C = Quaternion.Inverse(A) * B;		
	return C;
}
 
 
//Find the line of intersection between two planes.
//The inputs are two game objects which represent the planes. The plane is assumed to be made
//up of the x and z axis of the game object.
//The outputs are a point on the line and a vector which indicates it's direction.
void planePlaneIntersection(out Vector3 linePoint, out Vector3 lineVec, Vector3 plane1Normal, Vector3 plane1Position, Vector3 plane2Normal, Vector3 plane2Position){
 
	linePoint = Vector3.zero;
	lineVec = Vector3.zero;
 
	//We can get the direction of the line of intersection of the two planes by calculating the 
	//cross product of the normals of the two planes. Note that this is just a direction and the line
	//is not fixed in space yet. We need a point for that to go with the line vector.
	lineVec = Vector3.Cross(plane1Normal, plane2Normal);
 
	//Next is to calculate a point on the line to fix it's position in space. This is done by finding a vector from
	//the plane2 location, moving parallel to it's plane, and intersecting plane1. To prevent rounding
	//errors, this vector also has to be perpendicular to lineDirection. To get this vector, calculate
	//the cross product of the normal of plane2 and the lineDirection.		
	Vector3 ldir = Vector3.Cross(plane2Normal, lineVec);		
 
	float denominator = Vector3.Dot(plane1Normal, ldir);
 
	//Prevent divide by zero and rounding errors by requiring about 5 degrees angle between the planes.
	if(Mathf.Abs(denominator) > 0.006f){
 
		Vector3 plane1ToPlane2 = plane1Position - plane2Position;
		float t = Vector3.Dot(plane1Normal, plane1ToPlane2) / denominator;
		linePoint = plane2Position + t * ldir;
	}
}	
 
//Get the coordinates (world space) of the intersection between a line and a plane
Vector3 linePlaneIntersection(Vector3 linePoint, Vector3 lineVec, Vector3 planeNormal, Vector3 planePoint){
 
	float length;
	float dotNumerator;
	float dotDenominator;
	Vector3 vector;
 
	//calculate the distance between the linePoint and the line-plane intersection point
	dotNumerator = Vector3.Dot((planePoint - linePoint), planeNormal);
	dotDenominator = Vector3.Dot(lineVec, planeNormal);
 
	//line and plane are not parallel
	if(dotDenominator != 0.0f){
		length =  dotNumerator / dotDenominator;
 
		//create a vector from the linePoint to the intersection point
		vector = setVectorLength(lineVec, length);
 
		//get the coordinates of the line-plane intersection point
		return linePoint + vector;	
	}
 
	else{
		return Vector3.zero;
	}
 
}
 
//Two non-parallel lines which may or may not touch each other have a point on each line which lays closest
//to each other. This function finds those two points.
void closestPointsOnTwoLines(out Vector3 closestPointLine1, out Vector3 closestPointLine2, Vector3 linePoint1, Vector3 lineVec1, Vector3 linePoint2, Vector3 lineVec2){
 
	closestPointLine1 = Vector3.zero;
	closestPointLine2 = Vector3.zero;
 
	float a = Vector3.Dot(lineVec1, lineVec1);
	float b = Vector3.Dot(lineVec1, lineVec2);
	float e = Vector3.Dot(lineVec2, lineVec2);
 
	float d = a*e - b*b;
 
	//lines are not parallel
	if(d != 0.0f){
 
		Vector3 r = linePoint1 - linePoint2;
		float c = Vector3.Dot(lineVec1, r);
		float f = Vector3.Dot(lineVec2, r);
 
		float s = (b*f - c*e) / d;
		float t = (a*f - c*b) / d;
 
		closestPointLine1 = linePoint1 + lineVec1 * s;
		closestPointLine2 = linePoint2 + lineVec2 * t;
	}
 
	else{
		closestPointLine1 = new Vector3(float.MaxValue, float.MaxValue, float.MaxValue);
		closestPointLine2 = new Vector3(float.MaxValue, float.MaxValue, float.MaxValue);
	}
}	
 
//This function returns a 3d point in space which is a projection from point "point" to a line, consisting
//of a vector (lineVec) and a point on that line (linePoint).
Vector3 projectPointOnLine(Vector3 linePoint, Vector3 lineVec, Vector3 point){		
 
	//get vector from point on line to point in space
	Vector3 linePointToPoint = point - linePoint;
 
	float t = Vector3.Dot(linePointToPoint, lineVec);
 
	return linePoint + lineVec * t;
}	
 
Vector3 projectPointOnPlane(Vector3 planeNormal, Vector3 planePoint, Vector3 point){
 
	float distance;
	Vector3 translationVector;
 
	//First calculate the distance from the point to the plane:
	distance = signedDistancePlanePoint(planeNormal, planePoint, point);
 
	//Reverse the sign of the distance
	distance *= -1;
 
	//Get a translation vector
	translationVector = setVectorLength(planeNormal, distance);
 
	//Translate the point to form a projection
	return point + translationVector;
}
 
//output is not normalized
Vector3 ProjectVectorOnPlane(Vector3 planeNormal, Vector3 vector){
 
	return vector - (Vector3.Dot(vector, planeNormal) * planeNormal);
}	
 
//Get the shortest distance between a point and a plane
float signedDistancePlanePoint(Vector3 planeNormal, Vector3 planePoint, Vector3 point){
 
	return Vector3.Dot(planeNormal, (point - planePoint));
}	
 
//This function calculates a signed (+ or - sign instead of being ambiguous) dot product. It is basically used
//to figure out whether a vector is positioned to the left or right of another vector. The way this is done is
//by calculating a vector perpendicular to one of the vectors and using that as a reference. This is because
//the result of a dot product only has signed information when an angle is transitioning between more or less
//then 90 degrees.
float signedDotProduct(Vector3 vectorA, Vector3 vectorB, Vector3 normal){
	Vector3 perpVector;
	float dot;
 
	//Use the geometry object normal and one of the input vectors to calculate the perpendicular vector
	perpVector = Vector3.Cross(normal, vectorA);
 
	//Now calculate the dot product between the perpendicular vector (perpVector) and the other input vector
	dot = Vector3.Dot(perpVector, vectorB);
 
	return dot;
}
 
//Calculate the angle between a vector and a plane. The plane is made by a normal vector.
//Output is in radians.
float angleVectorPlane(Vector3 vector, Vector3 normal){
 
	float dot;
	float angle;
 
	//calculate the the dot product between the two input vectors. This gives the cosine between the two vectors
	dot = Vector3.Dot(vector, normal);
 
	//this is in radians
	angle = (float)Math.Acos(dot);
 
	return 1.570796326794897f - angle; //90
}
 
//Calculate the dot product as an angle
float dotProductAngle(Vector3 vec1, Vector3 vec2){
 
	double dot;
	double angle;
 
	//get the dot product
	dot = Vector3.Dot(vec1, vec2);
 
	//Clamp to prevent NaN error. Shouldn't need this in the first place, but there could be a rounding error issue.
	if(dot < -1.0f){
		dot = -1.0f;
	}							
	if(dot > 1.0f){
		dot =1.0f;
	}
 
	//Calculate the angle. The output is in radians
	//This step can be skipped for optimization...
	angle = Math.Acos(dot);
 
	return (float)angle;
}
 
//Calculate the instersection point of two lines. Returns true if lines intersect, otherwise false.
bool LineLineIntersection(out Vector3 intersection, Vector3 linePoint1, Vector3 lineVec1, Vector3 linePoint2, Vector3 lineVec2){
 
    intersection = Vector3.zero;
 
    Vector3 lineVec3 = linePoint2 - linePoint1;
    Vector3 crossVec1and2 = Vector3.Cross(lineVec1, lineVec2);
    Vector3 crossVec3and2 = Vector3.Cross(lineVec3, lineVec2);
 
    float planarFactor = Vector3.Dot(lineVec3, crossVec1and2);
 
    //Lines are not coplanar. Take into account rounding errors.
    if((planarFactor >= 0.00001f) || (planarFactor <= -0.00001f)){
 
        return false;
    }
 
    float s = Vector3.Dot(crossVec3and2, crossVec1and2) / crossVec1and2.sqrMagnitude;
 
    if((s >= 0.0f) && (s <= 1.0f)){
 
        intersection = linePoint1 + (lineVec1 * s);
        return true;
    }
 
    else{
        return false;       
    }
}
 
//Convert 3 points to a plane defined as a vector and a point. 
//The plane point is the middle of the triangle defined by the 3 points.
void PlaneFrom3Points(out Vector3 planeNormal, out Vector3 planePoint, Vector3 pointA, Vector3 pointB, Vector3 pointC){
 
    planeNormal = Vector3.zero;
    planePoint = Vector3.zero;
 
    //Make two vectors from the 3 input points, originating from point A
    Vector3 AB = pointB - pointA;
    Vector3 AC = pointC - pointA;
 
    //Calculate the normal
    planeNormal = Vector3.Normalize(Vector3.Cross(AB, AC));
 
    //Get the points in the middle AB and AC
    Vector3 middleAB = pointA + (AB / 2.0f);
    Vector3 middleAC = pointA + (AC / 2.0f);
 
    //Get vectors from the middle of AB and AC to the point which is not on that line.
    Vector3 middleABtoC = pointC - middleAB;
    Vector3 middleACtoB = pointB - middleAC;
 
    //Calculate the intersection between the two lines. This will be the center 
    //of the triangle defined by the 3 points.
    LineLineIntersection(out planePoint, middleAB, middleABtoC, middleAC, middleACtoB);
}