# 3d Math functions

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## Author

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## Description

A general description of what the script/code can do.

## Usage

How to use the script/code (please give clear precise instructions).

## Code

```//increase or decrease the length of vector by 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 vectorNormalized *= size;
}

//caclulate 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 planes are defined by a normal and a point on that plane.
//The outputs are a point on the line and a vector which indicates it's direction. If the planes are not parallel,
//the function outputs true, otherwise false.
bool 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;

return true;
}

//output not valid
else{
return false;
}
}

//Get the intersection between a line and a plane.
//If the line and plane are not parallel, the function outputs true, otherwise false.
bool LinePlaneIntersection(out Vector3 intersection, Vector3 linePoint, Vector3 lineVec, Vector3 planeNormal, Vector3 planePoint){

float length;
float dotNumerator;
float dotDenominator;
Vector3 vector;
intersection = Vector3.zero;

//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
intersection = linePoint + vector;

return true;
}

//output not valid
else{
return false;
}
}

//Calculate the instersection point of two lines. Returns true if lines intersect, otherwise false.
//Note that in 3d, two lines do not intersect most of the time. So if the two lines are not in the
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;
}

//Note: sqrMagnitude does x*x+y*y+z*z on the input vector.
float s = Vector3.Dot(crossVec3and2, crossVec1and2) / crossVec1and2.sqrMagnitude;

if((s >= 0.0f) && (s <= 1.0f)){

intersection = linePoint1 + (lineVec1 * s);
return true;
}

else{
return false;
}
}

//Two non-parallel lines which may or may not touch each other have a point on each line which are closest
//to each other. This function finds those two points. If the lines are not parallel, the function
//outputs true, otherwise false.
bool 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;

return true;
}

else{
return false;
}
}

//This function returns a point which is a projection from a point to a line.
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;
}

//This function returns a point which is a projection from a point to a plane.
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;
}

//Projects a vector onto a plane. The 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. The output is signed so it holds information
//as to which side of the plane normal the point is.
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.
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);

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;
}

//Convert a plane defined by 3 points to a plane defined by 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);
}```

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