# 3d Math functions

(Difference between revisions)

Tjeerd Schouten

## Description

This is a collection of generic 3d math functions such as line plane intersection, closest points on two lines, etc.

## Usage

-Place the Math3d.cs script in the scripts folder.

-To call a function from another script, place "Math3d." of the function, for example: Math3d.LookRotationExtended(...)

-If you want to use the TransformWithParent() function, you have to call Math3d.Init() first.

using UnityEngine; using System.Collections; using System;

public class Math3d : MonoBehaviour {

private static GameObject tempChild; private static GameObject tempParent;

public static void Init(){

tempChild = new GameObject("TempChild"); tempParent = new GameObject("TempParent");

//set the parent tempChild.transform.parent = tempParent.transform; }

//increase or decrease the length of vector by size public static 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" public static 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 public static 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. public static 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. public static 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 intersection 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 //same plane, use ClosestPointsOnTwoLines() instead. public static 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. public static 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. public static 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. public static 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. public static 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. public static 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. public static 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. public static 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 degrees - angle }

//Calculate the dot product as an angle public static 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. public static 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); }

//Returns the forward vector of a quaternion public static Vector3 GetForwardVector(Quaternion q){

return new Vector3(2.0f * (q.x * q.z + q.w * q.y), 2.0f * (q.y * q.x - q.w * q.x), 1.0f - 2.0f * (q.x * q.x + q.y * q.y)); }

//Returns the up vector of a quaternion public static Vector3 GetUpVector(Quaternion q){

return new Vector3(2.0f * (q.x * q.y - q.w * q.z), 1.0f - 2.0f * (q.x * q.x + q.z * q.z), 2.0f * (q.y * q.z + q.w * q.x)); }

//Returns the right vector of a quaternion public static Vector3 GetRightVector(Quaternion q){

return new Vector3(1.0f - 2.0f * (q.y * q.y + q.z * q.z), 2.0f * (q.x * q.y + q.w * q.z), 2.0f * (q.x * q.z - q.w * q.y)); }

//Gets a quaternion from a matrix public static Quaternion QuaternionFromMatrix(Matrix4x4 m){

return Quaternion.LookRotation(m.GetColumn(2), m.GetColumn(1)); }

//Gets a position from a matrix public static Vector3 PositionFromMatrix(Matrix4x4 m){

Vector4 vector4Position = m.GetColumn(3); return new Vector3(vector4Position.x, vector4Position.y, vector4Position.z); }

//This is an alternative for Quaternion.LookRotation. Instead of aligning the forward and up vector of the game //object with the input vectors, a custom direction can be used instead of the fixed forward and up vectors. //alignWithVector and alignWithNormal are in world space. //customForward and customUp are in object space. //Usage: use alignWithVector and alignWithNormal as if you are using the default LookRotation function. //Set customForward and customUp to the vectors you wish to use instead of the default forward and up vectors. public static void LookRotationExtended(ref GameObject gameObjectInOut, Vector3 alignWithVector, Vector3 alignWithNormal, Vector3 customForward, Vector3 customUp){

//Set the rotation of the destination Quaternion rotationA = Quaternion.LookRotation(alignWithVector, alignWithNormal);

//Set the rotation of the custom normal and up vectors. //When using the default LookRotation function, this would be hard coded to the forward and up vector. Quaternion rotationB = Quaternion.LookRotation(customForward, customUp);

//Calculate the rotation gameObjectInOut.transform.rotation = rotationA * Quaternion.Inverse(rotationB); }

//This function translates one object as if it was parented to the other. //Before using this function, the Init() function must be called //Input: parentRotation and parentPosition: the current parent transform. //Input: startParentRotation and startParentPosition: the transform of the parent object at the time the objects are parented. //Input: startChildRotation and startChildPosition: the transform of the child object at the time the objects are parented. //Output: childRotation and childPosition. //All transforms are in world space. public static void TransformWithParent(out Quaternion childRotation, out Vector3 childPosition, Quaternion parentRotation, Vector3 parentPosition, Quaternion startParentRotation, Vector3 startParentPosition, Quaternion startChildRotation, Vector3 startChildPosition){

childRotation = Quaternion.identity; childPosition = Vector3.zero;

//set the parent start transform tempParent.transform.rotation = startParentRotation; tempParent.transform.position = startParentPosition; tempParent.transform.localScale = Vector3.one; //to prevent scale wandering

//set the child start transform tempChild.transform.rotation = startChildRotation; tempChild.transform.position = startChildPosition; tempChild.transform.localScale = Vector3.one; //to prevent scale wandering

//translate and rotate the child by moving the parent tempParent.transform.rotation = parentRotation; tempParent.transform.position = parentPosition;

//get the child transform childRotation = tempChild.transform.rotation; childPosition = tempChild.transform.position; }

//With this function you can align a triangle of an object with any transform. //Usage: gameObjectInOut is the game object you want to transform. //alignWithVector, alignWithNormal, and alignWithPosition is the transform with which the triangle of the object should be aligned with. //triangleForward, triangleNormal, and trianglePosition is the transform of the triangle from the object. //alignWithVector, alignWithNormal, and alignWithPosition are in world space. //triangleForward, triangleNormal, and trianglePosition are in object space. //trianglePosition is the mesh position of the triangle. The effect of the scale of the object is handled automatically. //trianglePosition can be set at any position, it does not have to be at a vertex or in the middle of the triangle. public static void PreciseAlign(ref GameObject gameObjectInOut, Vector3 alignWithVector, Vector3 alignWithNormal, Vector3 alignWithPosition, Vector3 triangleForward, Vector3 triangleNormal, Vector3 trianglePosition){

//Set the rotation. LookRotationExtended(ref gameObjectInOut, alignWithVector, alignWithNormal, triangleForward, triangleNormal);

//Get the world space position of trianglePosition Vector3 trianglePositionWorld = gameObjectInOut.transform.TransformPoint(trianglePosition);

//Get a vector from trianglePosition to alignWithPosition Vector3 translateVector = alignWithPosition - trianglePositionWorld;

//Now transform the object so the triangle lines up correctly. gameObjectInOut.transform.Translate(translateVector, Space.World); } }

3d Math