Determine the angle needed to rotate a point in order to align it with another point in three-dimensional space

Question

Determine the angle needed to rotate a point in order to align it with another point in three-dimensional space

Given two vectors:

V1 = { x: 3.296372727813439, y: -14.497928014719344, z: 12.004105246875968 }

V2 = { x: 2.3652551657790695, y: -16.732085083053185, z: 8.945905454164146 }

How can I determine the angle at which V1 needs to be rotated in order to face directly towards V2?

To simplify, imagine knowing your exact position in space and that of another individual elsewhere... How could you calculate the angles necessary to point directly at them using mathematics?

Visual representation of my axis here

The formula I am currently using (which is incorrect)

v2.x -= v1.x;
v2.y -= v1.y;
v2.z -= v1.z;

v1.x = 0;
v1.y = 0;
v1.z = 0;

var r = Math.sqrt(Math.pow(v2.x,2) + Math.pow(v2.y,2) + Math.pow(v2.z,2));
var θ = Math.acos((Math.pow(v2.x,2) + Math.pow(v2.z,2))/(r*Math.sqrt(Math.pow(v2.x,2) + Math.pow(v2.z,2))));
var ϕ = Math.acos(v2.x/Math.sqrt(Math.pow(v2.x,2) + Math.pow(v2.z,2)));

My current process involves rotating V1 based on the calculated theta and phi angles.

v1.rotation.y = θ;
v2.rotation.x = ϕ;

However, this method results in inaccurate rotations.

θ = 0.6099683401012933

ϕ = 1.8663452274936656

Conversely, when utilizing THREE.js and the lookAt function, the following correct rotations are produced:

y/θ: -0.24106818240525682

x/ϕ: 2.5106584861123644

I appreciate all assistance provided! While THREE.js works well, I aim to achieve this functionality using pure vanilla JS for portability across different languages.

javascript 3d three.js rotation

Answer 1

Answer №1

Consider using a matrix for orientation calculation. The viewpoint v1 is where you are looking towards v2. A transformation matrix can help show orientation, while Euler Angle provides another interpretation.

To achieve what you want, you can build a transformation matrix from object space to world space in three steps:

Start with the camera at the origin of world space, with a rotation of (0,0,0), where world space aligns with object space: v1'(0,0,0).
Translate the camera to
```
v1(3.296372727813439,-14.497928014719344,12.004105246875968)
```
. Object space may have an offset from world space, but their axes remain parallel, and the camera rotation remains (0,0,0).
Adjust the camera to look at v2, causing a change in camera rotation.

By constructing a transformation matrix representing these actions, we can determine the orientation.

Create the translation matrix first: As translation is an affine transformation, a 4x4 matrix is required. The matrix can be obtained easily:

Use basis axes to obtain the rotation matrix.

You may need to set the camera's up vector, defaulting to (0,1,0). In object space, calculate the basis z axis by subtracting v1 from v2:

z = (v1.x-v2.x,v1.y-v2.y,v1.z-v2.z).normalize()

Determine the basis x vector by taking the cross product of up and z:

x = up.crossproduct(z)

The basis y vector, perpendicular to z-x plane, is found by crossing z and x:

y = z.product(x)

Construct the rotation matrix as a 3 x 3 matrix:

Next, derive the transformation matrix:

Different ways exist to convert between matrices, Euler Angles, and Quaternions; details can be explored in this book: 3D Math Primer for Graphics and Game Development

Three.js incorporates the LookAt() function similarly to this approach.

An example in three.js source code with added comments:

function lookAt( eye, target, up ) //eye : your camera position; target : which point you want to look at; up : camera up vector
{  
    if ( x === undefined ) {
        x = new Vector3();
        y = new Vector3();
        z = new Vector3();
    }
    var te = this.elements;

    z.subVectors( eye, target ).normalize();

    if ( z.lengthSq() === 0 ) {
        z.z = 1;
    }

    x.crossVectors( up, z ).normalize();

    if ( x.lengthSq() === 0 ) {
        z.z += 0.0001;
        x.crossVectors( up, z ).normalize();
    }

    y.crossVectors( z, x );

    te[ 0 ] = x.x; te[ 4 ] = y.x; te[ 8 ] = z.x; 
    te[ 1 ] = x.y; te[ 5 ] = y.y; te[ 9 ] = z.y;
    te[ 2 ] = x.z; te[ 6 ] = y.z; te[ 10 ] = z.z;

    return this;
};

Implement Matrix using a list structure like three.js does.

An alternative idea involves Spherical Polar Coordinates System. Utilizing coordinates expressed as (r,Θ,Φ), where Θ represents the heading angle and Φ denotes the pitch angle. Convert the Cartesian coordinates of v1 and v2 to spherical ones. Calculate the angular displacement between v1 and v2, then apply it to adjust the camera rotation.

A possible code snippet assuming the camera is at the world origin (0,0,0):

//convert v1 and v2 Cartesian coordinates to Spherical coordinates
var radiusV1 = Math.sqrt( Math.pow(v1.x) + Math.pow(v1.y) + Math.pow(v1.z));
var headingV1 = Math.atan2(v1.x , v1.z);
var pitchV1 = Math.asin(-(v1.y) / radiusV1);

var radiusV2 = Math.sqrt( Math.pow(v2.x) + Math.pow(v2.y) + Math.pow(v2.z));
var headingV2 = Math.atan2(v2.x , v2.z);
var pitchV2 = Math.asin(-(v2.y) / radiusV2);

//calculate angular displacement
var displacementHeading = headingV2 - headingV1;
var displacementPitch = pitchV2 - pitchV1;

//adjust camera rotation based on displacement
camera.rotation.x += displacementPitch;
camera.rotation.y += displacementHeading;

Knowledge of 3D math proves valuable and learning-worthy. All referenced formulas and concepts can be explored further in the recommended book.

Hopefully, this information aids your understanding.

Answer 2

Consider using a matrix for orientation calculation. The viewpoint v1 is where you are looking towards v2. A transformation matrix can help show orientation, while Euler Angle provides another interpretation.

To achieve what you want, you can build a transformation matrix from object space to world space in three steps:

Start with the camera at the origin of world space, with a rotation of (0,0,0), where world space aligns with object space: v1'(0,0,0).
Translate the camera to
```
v1(3.296372727813439,-14.497928014719344,12.004105246875968)
```
. Object space may have an offset from world space, but their axes remain parallel, and the camera rotation remains (0,0,0).
Adjust the camera to look at v2, causing a change in camera rotation.

By constructing a transformation matrix representing these actions, we can determine the orientation.

Create the translation matrix first: As translation is an affine transformation, a 4x4 matrix is required. The matrix can be obtained easily:

Use basis axes to obtain the rotation matrix.

You may need to set the camera's up vector, defaulting to (0,1,0). In object space, calculate the basis z axis by subtracting v1 from v2:

z = (v1.x-v2.x,v1.y-v2.y,v1.z-v2.z).normalize()

Determine the basis x vector by taking the cross product of up and z:

x = up.crossproduct(z)

The basis y vector, perpendicular to z-x plane, is found by crossing z and x:

y = z.product(x)

Construct the rotation matrix as a 3 x 3 matrix:

Next, derive the transformation matrix:

Different ways exist to convert between matrices, Euler Angles, and Quaternions; details can be explored in this book: 3D Math Primer for Graphics and Game Development

Three.js incorporates the LookAt() function similarly to this approach.

An example in three.js source code with added comments:

function lookAt( eye, target, up ) //eye : your camera position; target : which point you want to look at; up : camera up vector
{  
    if ( x === undefined ) {
        x = new Vector3();
        y = new Vector3();
        z = new Vector3();
    }
    var te = this.elements;

    z.subVectors( eye, target ).normalize();

    if ( z.lengthSq() === 0 ) {
        z.z = 1;
    }

    x.crossVectors( up, z ).normalize();

    if ( x.lengthSq() === 0 ) {
        z.z += 0.0001;
        x.crossVectors( up, z ).normalize();
    }

    y.crossVectors( z, x );

    te[ 0 ] = x.x; te[ 4 ] = y.x; te[ 8 ] = z.x; 
    te[ 1 ] = x.y; te[ 5 ] = y.y; te[ 9 ] = z.y;
    te[ 2 ] = x.z; te[ 6 ] = y.z; te[ 10 ] = z.z;

    return this;
};

Implement Matrix using a list structure like three.js does.

An alternative idea involves Spherical Polar Coordinates System. Utilizing coordinates expressed as (r,Θ,Φ), where Θ represents the heading angle and Φ denotes the pitch angle. Convert the Cartesian coordinates of v1 and v2 to spherical ones. Calculate the angular displacement between v1 and v2, then apply it to adjust the camera rotation.

A possible code snippet assuming the camera is at the world origin (0,0,0):

//convert v1 and v2 Cartesian coordinates to Spherical coordinates
var radiusV1 = Math.sqrt( Math.pow(v1.x) + Math.pow(v1.y) + Math.pow(v1.z));
var headingV1 = Math.atan2(v1.x , v1.z);
var pitchV1 = Math.asin(-(v1.y) / radiusV1);

var radiusV2 = Math.sqrt( Math.pow(v2.x) + Math.pow(v2.y) + Math.pow(v2.z));
var headingV2 = Math.atan2(v2.x , v2.z);
var pitchV2 = Math.asin(-(v2.y) / radiusV2);

//calculate angular displacement
var displacementHeading = headingV2 - headingV1;
var displacementPitch = pitchV2 - pitchV1;

//adjust camera rotation based on displacement
camera.rotation.x += displacementPitch;
camera.rotation.y += displacementHeading;

Knowledge of 3D math proves valuable and learning-worthy. All referenced formulas and concepts can be explored further in the recommended book.

Hopefully, this information aids your understanding.

Answer 3

Answer №2

Here are the corrected formulas:

 var dx = v2.x-v1.x; //-0.93
 var dy = v2.y-v1.y; //-31.22
 var dz = v2.z-v1.z;
 var rxy = Math.sqrt( Math.pow(dx,2) + Math.pow(dy,2) );
 var lambda = Math.atan(dy/dx);
 var phi = Math.atan(dz/rxy)

To ensure accuracy, adjust the formulas for phi and lambda based on the vector's quarter. Here is a simplified guide:

 //if the calculations are in degrees, add 180 instead of PI
 if (dx < 0) phi = phi + Math.PI;
 if (dz < 0) lambda = -1 * lambda;

Answer 4

Here are the corrected formulas:

 var dx = v2.x-v1.x; //-0.93
 var dy = v2.y-v1.y; //-31.22
 var dz = v2.z-v1.z;
 var rxy = Math.sqrt( Math.pow(dx,2) + Math.pow(dy,2) );
 var lambda = Math.atan(dy/dx);
 var phi = Math.atan(dz/rxy)

To ensure accuracy, adjust the formulas for phi and lambda based on the vector's quarter. Here is a simplified guide:

 //if the calculations are in degrees, add 180 instead of PI
 if (dx < 0) phi = phi + Math.PI;
 if (dz < 0) lambda = -1 * lambda;

Answer 5

Answer №3

When x/ϕ is rotated around the x-axis, it equals the angle between y and z. Similarly, for y/θ, we need to calculate the angle between x and z.

V1 = { x: 3.296372727813439, y: -14.497928014719344, z: 12.004105246875968 }
V2 = { x: 2.3652551657790695, y: -16.732085083053185, z: 8.945905454164146 }
var v={dx:V2.x-V1.x, dy:V2.y-V1.y, dz:V2.z-V1.z}
testVector(v);
 
function testVector(vec){
   console.log();
   var angles=calcAngles(vec);
   console.log("phi:"+angles.phi+" theta:"+angles.theta);
}
function calcAngles(vec){
   return {
      theta:(Math.PI/2)+Math.atan2(vec.dz, vec.dx),
      phi:(3*Math.PI/2)+Math.atan2(vec.dz, vec.dy)
   };
}

Answer 6

When x/ϕ is rotated around the x-axis, it equals the angle between y and z. Similarly, for y/θ, we need to calculate the angle between x and z.

V1 = { x: 3.296372727813439, y: -14.497928014719344, z: 12.004105246875968 }
V2 = { x: 2.3652551657790695, y: -16.732085083053185, z: 8.945905454164146 }
var v={dx:V2.x-V1.x, dy:V2.y-V1.y, dz:V2.z-V1.z}
testVector(v);
 
function testVector(vec){
   console.log();
   var angles=calcAngles(vec);
   console.log("phi:"+angles.phi+" theta:"+angles.theta);
}
function calcAngles(vec){
   return {
      theta:(Math.PI/2)+Math.atan2(vec.dz, vec.dx),
      phi:(3*Math.PI/2)+Math.atan2(vec.dz, vec.dy)
   };
}

Answer 7

Answer №4

After carefully analyzing the latest version of THREE.js (r84), I have extracted the relevant code that I believe will help you achieve your desired outcome.

// Here is the code extracted from the latest version of THREE.js (r84)
// Source: https://github.com/mrdoob/three.js/tree/master
// THREE.js is licensed under MIT (Copyright © 2010-2017 three.js authors)
// 
// Some modifications were made by K Scandrett to adapt the functions for this purpose, but not the calculations.
// Any errors are my responsibility and not those of the three.js authors. 
// I cannot guarantee that there are no bugs in the adapted code,
// so please use it at your own risk. Enjoy the pizza!



var v1 = {x: 3.296372727813439, y: -14.497928014719344, z: 12.004105246875968};
var v2 = {x: 2.3652551657790695, y: -16.732085083053185,z: 8.945905454164146};

var startVec = {x: v1.x, y: v1.y, z: v1.z, w: 0};
var endVec = {x: v2.x, y: v2.y, z: v2.z, w: 0};

var upVec = {x: 0, y: 1, z: 0}; // y up

var quat = lookAt(startVec, endVec, upVec);
var angles = eulerSetFromQuaternion(quat);

console.log(angles.x + " " + angles.y + " " + angles.z);

/* KS function */
function magnitude(v) {
  return Math.sqrt(v.x * v.x + v.y * v.y + v.z * v.z);
}

/* KS function */
function normalize(v) {
  var mag = magnitude(v);
  return {
    x: v.x / mag,
    y: v.y / mag,
    z: v.z / mag
  };
}

function subVectors(a, b) {
  return {
    x: a.x - b.x,
    y: a.y - b.y,
    z: a.z - b.z
  };
}

function crossVectors(a, b) {
  var ax = a.x,
    ay = a.y,
    az = a.z;
  var bx = b.x,
    by = b.y,
    bz = b.z;
  return {
    x: ay * bz - az * by,
    y: az * bx - ax * bz,
    z: ax * by - ay * bx
  };
}

function lengthSq(v) {
  return v.x * v.x + v.y * v.y + v.z * v.z;
}


function makeRotationFromQuaternion(q) {

  var matrix = new Float32Array([

    1, 0, 0, 0,
    0, 1, 0, 0,
    0, 0, 1, 0,
    0, 0, 0, 1

  ]);

  var te = matrix;

  var x = q.x,
    y = q.y,
    z = q.z,
    w = q.w;
  var x2 = x + x,
    y2 = y + y,
    z2 = z + z;
  var xx = x * x2,
    xy = x * y2,
    xz = x * z2;
  var yy = y * y2,
    yz = y * z2,
    zz = z * z2;
  var wx = w * x2,
    wy = w * y2,
    wz = w * z2;

  te[0] = 1 - (yy + zz);
  te[4] = xy - wz;
  te[8] = xz + wy;

  te[1] = xy + wz;
  te[5] = 1 - (xx + zz);
  te[9] = yz - wx;

  te[2] = xz - wy;
  te[6] = yz + wx;
  te[10] = 1 - (xx + yy);

  // last column
  te[3] = 0;
  te[7] = 0;
  te[11] = 0;

  // bottom row
  te[12] = 0;
  te[13] = 0;
  te[14] = 0;
  te[15] = 1;

  return te;

}

// More functions follow...

To view this code in action on Plunker, click here.

Answer 8

After carefully analyzing the latest version of THREE.js (r84), I have extracted the relevant code that I believe will help you achieve your desired outcome.

// Here is the code extracted from the latest version of THREE.js (r84)
// Source: https://github.com/mrdoob/three.js/tree/master
// THREE.js is licensed under MIT (Copyright © 2010-2017 three.js authors)
// 
// Some modifications were made by K Scandrett to adapt the functions for this purpose, but not the calculations.
// Any errors are my responsibility and not those of the three.js authors. 
// I cannot guarantee that there are no bugs in the adapted code,
// so please use it at your own risk. Enjoy the pizza!



var v1 = {x: 3.296372727813439, y: -14.497928014719344, z: 12.004105246875968};
var v2 = {x: 2.3652551657790695, y: -16.732085083053185,z: 8.945905454164146};

var startVec = {x: v1.x, y: v1.y, z: v1.z, w: 0};
var endVec = {x: v2.x, y: v2.y, z: v2.z, w: 0};

var upVec = {x: 0, y: 1, z: 0}; // y up

var quat = lookAt(startVec, endVec, upVec);
var angles = eulerSetFromQuaternion(quat);

console.log(angles.x + " " + angles.y + " " + angles.z);

/* KS function */
function magnitude(v) {
  return Math.sqrt(v.x * v.x + v.y * v.y + v.z * v.z);
}

/* KS function */
function normalize(v) {
  var mag = magnitude(v);
  return {
    x: v.x / mag,
    y: v.y / mag,
    z: v.z / mag
  };
}

function subVectors(a, b) {
  return {
    x: a.x - b.x,
    y: a.y - b.y,
    z: a.z - b.z
  };
}

function crossVectors(a, b) {
  var ax = a.x,
    ay = a.y,
    az = a.z;
  var bx = b.x,
    by = b.y,
    bz = b.z;
  return {
    x: ay * bz - az * by,
    y: az * bx - ax * bz,
    z: ax * by - ay * bx
  };
}

function lengthSq(v) {
  return v.x * v.x + v.y * v.y + v.z * v.z;
}


function makeRotationFromQuaternion(q) {

  var matrix = new Float32Array([

    1, 0, 0, 0,
    0, 1, 0, 0,
    0, 0, 1, 0,
    0, 0, 0, 1

  ]);

  var te = matrix;

  var x = q.x,
    y = q.y,
    z = q.z,
    w = q.w;
  var x2 = x + x,
    y2 = y + y,
    z2 = z + z;
  var xx = x * x2,
    xy = x * y2,
    xz = x * z2;
  var yy = y * y2,
    yz = y * z2,
    zz = z * z2;
  var wx = w * x2,
    wy = w * y2,
    wz = w * z2;

  te[0] = 1 - (yy + zz);
  te[4] = xy - wz;
  te[8] = xz + wy;

  te[1] = xy + wz;
  te[5] = 1 - (xx + zz);
  te[9] = yz - wx;

  te[2] = xz - wy;
  te[6] = yz + wx;
  te[10] = 1 - (xx + yy);

  // last column
  te[3] = 0;
  te[7] = 0;
  te[11] = 0;

  // bottom row
  te[12] = 0;
  te[13] = 0;
  te[14] = 0;
  te[15] = 1;

  return te;

}

// More functions follow...

To view this code in action on Plunker, click here.

Answer 9

Answer №5

If you're looking for a straightforward answer to your question, using Euler angles may not be the best approach. The Euler coordinate system is designed for coordination purposes and can lead to issues with orientation and rotation. While it may be easy for humans to understand, it can result in Gimbal lock, causing incorrect outcomes.

When it comes to orientation, there are two common systems used: Transform matrix and Quaternions. For example, three.js utilizes quaternions for its lookAt() function, while Crag.Li's recommendation involves using a Transform Matrix.

It's important to highlight the complexity of 3D transformation, as I once learned the hard way by attempting to simplify the process and ultimately wasting time on ineffective methods. Learning the math behind 3D transformations, such as through resources like the "3D Math Primer," is essential if you truly want to succeed in this area.

Answer 10

If you're looking for a straightforward answer to your question, using Euler angles may not be the best approach. The Euler coordinate system is designed for coordination purposes and can lead to issues with orientation and rotation. While it may be easy for humans to understand, it can result in Gimbal lock, causing incorrect outcomes.

When it comes to orientation, there are two common systems used: Transform matrix and Quaternions. For example, three.js utilizes quaternions for its lookAt() function, while Crag.Li's recommendation involves using a Transform Matrix.

It's important to highlight the complexity of 3D transformation, as I once learned the hard way by attempting to simplify the process and ultimately wasting time on ineffective methods. Learning the math behind 3D transformations, such as through resources like the "3D Math Primer," is essential if you truly want to succeed in this area.

Determine the angle needed to rotate a point in order to align it with another point in three-dimensional space

Answer №1

Answer №2

Answer №3

Answer №4

Answer №5

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