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3D Rotation

Click HERE to start a real-time loop, simulating a 3D engine/loop. The cube is the "world", where other objects could be put inside and be affected by the same formula by looping through all the vector points, forcing them to move in the same dimension. I've just used circles to mark the edges of the matrix. This is something I made just to illustrate and learn the basics of a 3D rotation formula.

The next time you play a 3D game you may look at it differently. Realizing that the world and all its objects spins around you every time you turn around, even though your head may tell you it's standing still. This should give better understanding for a good graphics card being important for gaming/3D work as well, since every time you move in a 3D world, it has to turn billions, maybe trillions of vectors to your wanted viewpoint, as well as showing e.g. particle explosions, lighting algorithms and more at the same time. GPUs are specialized for those kind of calculations, much more than CPUs.

Understanding the formula below and the example code (view source) should be able to inspire multiple creative uses as well as give a basic understanding of how 3D rotation and dynamic worlds works. Game development today largely happens in already-made 3D engines where all of this and SO much more is already covered, but it's still fun to learn and understand some of the lowest basics behind the curtains.

In the 2D script I used this formula to rotate around a center point:
x = Math.cos(Radian)*XRadius(hypotenus)
y = Math.sin(Radian)*YRadius(hypotenus)

By changing the radius, you can simulate x,y rotation as well, but only on a 2D plane. It would be simulation, not real 3D. Add depth with another z calculation with the above formula, and you would have the 3D effect available.

This formula gives the coordinate/vector 'x' and/or 'y' new values based on finding the radian x and y ratio through cosine and sine multiplied by hypotenus to keep the proper center point (radius). This formula could be used for simulating 3D on a 2D plane. It could even be used for real 3D calculation, but you would need to find a way to make the rotation radius/hypotenus values work together when e.g crossing a z-axis with a y-axis rotation. When e.g. the z-rotation closes in on the y-axis, the radius of z-rotation (x) would have to be shortened accordingly, start a x-rotation as well and you may have a challenge.

Here is a formula that does the proper job of 3D calculation. It respects every axis radius/hypotenus in regards to each other.

// Rotation around the X axis
xy = Math.cos(RadianX)*y - Math.sin(RadianX)*z
xz = Math.sin(RadianX)*y + Math.cos(RadianX)*z

// Rotation around the Y axis
yz = Math.cos(RadianY)*xz - Math.sin(RadianY)*x
yx = Math.sin(RadianY)*xz + Math.cos(RadianY)*x

// Rotation around the Z axis
zx = Math.cos(RadianZ)*yx - Math.sin(RadianZ)*xy
zy = Math.sin(RadianZ)*yx + Math.cos(RadianZ)*xy

Where zx(X), zy(Y), yz(Z) are the new values to use after every calculation in a 3D space of coordinated rotation.

By changing the RadianX, RadianY, RadianZ up to 2π(6.28)(360°) in this formula you control every axis rotation, and each rotation radius/hypotenus should work in harmony.

It will help to repeat the chapter in a math book on trigonometry (especially sine and cosine) to get a full grasp on the formula above. I had to myself, and will likely need to again when/if using this formula later.

// General variables
var center = 250;
var smooth = 0;

// Math.cos and Math.sin takes its value in radian, not degrees
var anglex = 0.00;
var angley = 0.00;
var anglez = 0.00;

// This array with vector points should make our cube move in 100x100x100 pixel space
var Points = new Array(
	new Array(-50,-50,-50),
	new Array(50,-50,-50),
	new Array(50,-50,50),
	new Array(-50,-50,50),
	new Array(-50,50,-50),
	new Array(50,50,-50),
	new Array(50,50,50),
	new Array(-50,50,50));

// Function for each cube calculation
// Loops itself once started - makes the cube move in real-time.
function runCube() {

	// Temporary function variables
	var xy, xz, yz, yx, zx, zy;
	var loop_a, x, y, z;
	// Loop through the vector points for new transform
	for (loop_a = 0; loop_a <= 7; loop_a++)  {
		// Original data from last transform calculation
		x = Points[loop_a][0];
		y = Points[loop_a][1];
		z = Points[loop_a][2];
		// Rotation around the X axis
		xy = Math.cos(anglex)*y - Math.sin(anglex)*z;
		xz = Math.sin(anglex)*y + Math.cos(anglex)*z;
		// Rotation around the Y axis
		yz = Math.cos(angley)*xz - Math.sin(angley)*x;
		yx = Math.sin(angley)*xz + Math.cos(angley)*x;
		// Rotation around the Z axis
		zx = Math.cos(anglez)*yx - Math.sin(anglez)*xy;
		zy = Math.sin(anglez)*yx + Math.cos(anglez)*xy;
		// Give points new data
		Points[loop_a][0] = zx;
		Points[loop_a][1] = zy;
		Points[loop_a][2] = yz;
		// Update screen representation elements with new data
		var maxScale = (Points[loop_a][2]/100)*48;
		maxScale = (maxScale<18?18:maxScale);
		document.getElementById('satellite'+loop_a) = (center-70) + Points[loop_a][1]+'px';
		document.getElementById('satellite'+loop_a).style.left = center + Points[loop_a][0]+'px';
		document.getElementById('satellite'+loop_a).style.fontSize = maxScale+'px';
		document.getElementById('satellite'+loop_a).style.display = "block";
	// Update screen information
	var c = document.getElementById('container');
	c.innerHTML = 'Radian X = '+Math.round(100*anglex)/100+'<br>';
	c.innerHTML += 'Radian Y = '+Math.round(100*angley)/100+'<br>';
	c.innerHTML += 'Radian Z = '+Math.round(100*anglez)/100+'<br><br>';
	c.innerHTML += 'Smoothness = '+smooth+'<br>(setTimeout, 0=best)<br ><br>';
	c.innerHTML += '1 = '+Math.round(Points[0][0])+','+Math.round(Points[0][1])+','+Math.round(Points[0][2])+'<br>';
	c.innerHTML += '2 = '+Math.round(Points[1][0])+','+Math.round(Points[1][1])+','+Math.round(Points[1][2])+'<br>';
	c.innerHTML += '3 = '+Math.round(Points[2][0])+','+Math.round(Points[2][1])+','+Math.round(Points[2][2])+'<br>';
	c.innerHTML += '4 = '+Math.round(Points[3][0])+','+Math.round(Points[3][1])+','+Math.round(Points[3][2])+'<br>';
	c.innerHTML += '5 = '+Math.round(Points[4][0])+','+Math.round(Points[4][1])+','+Math.round(Points[4][2])+'<br>';
	c.innerHTML += '6 = '+Math.round(Points[5][0])+','+Math.round(Points[5][1])+','+Math.round(Points[5][2])+'<br>';
	c.innerHTML += '7 = '+Math.round(Points[6][0])+','+Math.round(Points[6][1])+','+Math.round(Points[6][2])+'<br>';
	c.innerHTML += '8 = '+Math.round(Points[7][0])+','+Math.round(Points[7][1])+','+Math.round(Points[7][2])+'<br>';
	// Loop the function, let the cube move in real time

// Controls the execution rate of the function, less is faster with more smooth results.
function Smooth(smoothVal) {
	if (smooth > 0) { 
		smooth += smoothVal;
	} else if (smooth == 0 && smoothVal > 0) {
		smooth += smoothVal;
	} else { 
		smooth = 0; 

// X, Y, Z functions deciding rotation, both speed(angle rate) and direction(positive or negative numbers)
function X(newVal) {
	anglex += newVal;
function Y(newVal) {
	angley += newVal;
function Z(newVal) {
	anglez += newVal;

Original Post: Jan 26th, '22 13:06 CET.
Updated: Jan 27th, '22 15:51 CET.

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