| grimpirate |
Jun 8th, 2006 12:22 AM |
3D Engine in PHP
Well I originally started this project in Flash but it didn't work out for some reason there. I actually posted that problem here. After looking at the code for a while I realized that my math worked out and that something was wrong with the way Flash accesses arrays, at least in my humble opinion. Nevertheless, I succeeded in creating the 3d engine in PHP and it outputs itself as a still image of the view of a cube with white lines for visible edges and gray lines for non-visible ones. It turned out pretty nice I think. It uses pretty much every matrix operation you can think of and that's how it's structured. Test it out and tell me what you think. I still have to make modifications to it in order to be able to import some other formats for objects like in the VRML language, but for now enjoy the cube, change some of the numbers around and see the result.
:
<?php
// Creates a virtual cube composed of polygons
$cube = new VirtualObject();
// Viewable screen dimensions
$width = 640;
$height = 480;
// Sets up the camera's orientation and its focal distance
$x = sqrt(2) / 2;
$focal = 500;
$a = array(
array(-$x, 0.5, -0.5, 0),
array($x, 0.5, -0.5, 0),
array(0, -$x, -$x, 0),
array(0, 0, 0, 1)
);
// The vector normal to the viewing plane and parallel to the viewing direction
$n2[0][0] = $a[0][2];
$n2[1][0] = $a[1][2];
$n2[2][0] = $a[2][2];
$n2[3][0] = 0;
// Invert camera's plane to allow for change of basis
$a = inverse($a);
// Displace camera to its position in the world
$a[0][3] = 150;
$a[1][3] = 150;
$a[2][3] = 300 * $x;
// Create two images
$image1 = imagecreate($width, $height);
$image2 = imagecreate($width, $height);
// Set the line thickness of each image
imagesetthickness($image1, 1);
imagesetthickness($image2, 1);
// Set the background colors for each image
$blue1 = imagecolorallocate($image1, 0, 0, 127);
$blue2 = imagecolorallocate($image2, 0, 0, 127);
// Define some colors for each image
$white = imagecolorallocate($image1, 255, 255, 255);
$gray = imagecolorallocate($image2, 127, 127, 127);
// Draw each polygon
for($i = 0; $i < $cube->getPolyCount(); $i++){
// Retrieve the polygons from the cube object
$points = $cube->getPolygon($i);
// Modify each point of the polygon so that its now distanced to the camera's center point
for($j = 0; $j < 3; $j++){
$points[0][$j][0] -= $a[$j][3];
$points[1][$j][0] -= $a[$j][3];
$points[2][$j][0] -= $a[$j][3];
}
// Determine the normal vector for each polygon
$n1 = crossProduct(addition($points[0], scale(-1, $points[1])), addition($points[2], scale(-1, $points[1])));
// Write each vector in terms of the camera's orientation (change of basis)
$points[0] = multiply($a, $points[0]);
$points[1] = multiply($a, $points[1]);
$points[2] = multiply($a, $points[2]);
// Determine the distortion due to perspective
$focus[0] = $focal / magnitude($points[0]);
$focus[1] = $focal / magnitude($points[1]);
$focus[2] = $focal / magnitude($points[2]);
// Transform the 3d points into 2d coordinates for drawing
$x0 = $width / 2 + ($points[0][0][0] * $focus[0]);
$x1 = $width / 2 + ($points[1][0][0] * $focus[1]);
$x2 = $width / 2 + ($points[2][0][0] * $focus[2]);
$y0 = $height / 2 + ($points[0][1][0] * $focus[0]);
$y1 = $height / 2 + ($points[1][1][0] * $focus[1]);
$y2 = $height / 2 + ($points[2][1][0] * $focus[2]);
// Determine the cosine of the angle between the viewing plane and the polygonal plane
$cosTheta = dotProduct($n1, $n2) / (magnitude($n1) * magnitude($n2));
if($cosTheta <= 0){
// White lines because these are the planes at the front of the object
imageline($image1, $x0, $y0, $x1, $y1, $white);
imageline($image1, $x1, $y1, $x2, $y2, $white);
// imageline($image1, $x2, $y2, $x0, $y0, $white); // Draws the final polygon line
}else{
// Gray lines because these are the planes at the reare of the object
imageline($image2, $x0, $y0, $x1, $y1, $gray);
imageline($image2, $x1, $y1, $x2, $y2, $gray);
// imageline($image2, $x2, $y2, $x0, $y0, $gray); // Draws the final polygon line
}
}
// Make transparent the background of the viewable polygon drawing
imagecolortransparent($image1, $blue1);
// Layer it over the non-viewable polygon drawing
imagecopymerge($image2, $image1, 0, 0, 0, 0, 640, 480, 100);
// Output the finished image
imagepng($image2);
// Destroy the images
imagedestroy($image1);
imagedestroy($image2);
// Computes the cross product of two vectors
// $a: a row or column vector
// $b: a row or column vector
function crossProduct($a, $b){
if(sizeof($a) == 1){ // Cross product of row vectors
return array( // Return the cross product of row vectors as a row vector
array( $a[0][1] * $b[0][2] - $a[0][2] * $b[0][1],
-1 * ($a[0][0] * $b[0][2] - $a[0][2] * $b[0][0]),
$a[0][0] * $b[0][1] - $a[0][1] * $b[0][0],
0
)
);
}else{ // Cross product of column vectors
return array( // Return the cross product of column vectors as a column vector
array($a[1][0] * $b[2][0] - $a[2][0] * $b[1][0]),
array(-1 * ($a[0][0] * $b[2][0] - $a[2][0] * $b[0][0])),
array($a[0][0] * $b[1][0] - $a[1][0] * $b[0][0]),
array(0)
);
}
}
// Computes the addition of two matrices
// $A: an m x n matrix
// $B: an m x n matrix
function addition($A, $B){
$M; // The added matrix
// Add the two matrices together
for($i = 0; $i < sizeof($A); $i++){
for($j = 0; $j < sizeof($A[0]); $j++){
$M[$i][$j] = $A[$i][$j] + $B[$i][$j];
}
}
return $M; // Return the added matrix
}
// Computes a random matrix
// $m: the row dimension
// $n: the column dimension
// $lower: the lower bound of random numbers
// $upper: the upper bound of random numbers
function randMatrix($m, $n, $lower, $upper){
$M; // A random matrix
// Generate the random matrix
for($i = 0; $i < $m; $i++){
for($j = 0; $j < $n; $j++){
$M[$i][$j] = rand($lower, $upper);
}
}
return $M; // Return the random matrix
}
// Computes the magnitude of a vector
// $a: a row or column vector
function magnitude($a){
$result = 0; // The magnitude of the vector
// Use the Pythagorean theorem to determine the vector's magnitude
for($i = 0; $i < sizeof($a); $i++){
for($j = 0; $j < sizeof($a[0]); $j++){
$result += pow($a[$i][$j], 2);
}
}
return sqrt($result); // Return the magnitude of the vector
}
// Computes the adjoint of a matrix
// $A: a square matrix
function adjoint($A){
return scale(determinant($A), inverse($A)); // Returns the adjoint of a matrix
}
// Computes a scaled matrix
// $k: a scalar number
// $A: an m x n matrix
function scale($k, $A){
// Scale the matrix by multiplying each entry by k
for($i = 0; $i < sizeof($A); $i++){
for($j = 0; $j < sizeof($A[0]); $j++){
$A[$i][$j] *= $k;
}
}
return $A; // Returns the scaled matrix
}
// Computes the determinant of a matrix
// $A: a square matrix
function determinant($A){
$determinant = 1; // The determinant of the matrix
$n = sizeof($A); // Dimension of the square matrix
$E = identity($n); // An elementary matrix
// Find the determinant of a matrix by performing elementary matrix operations
// on an augmented matrix through the use of elementary matrices in order
// to convert the matrix into upper triangular form
// Multiply the diagonals of the upper triangular matrix to arrive at the
// determinant, only elementary operations I and III are allowed on the matrix.
// If operation I is performed at any time, multiply the determinant by a -1.
for($j = 0; $j < $n; $j++){
for($i = $j; $i < $n; $i++){
if($i == $j){
if($A[$i][$j] == 0){
for($k = $j + 1; $k < $n; $k++){
if($A[$k][$j] != 0){
$swap = $A[$i];
$A[$i] = $A[$k];
$A[$k] = $swap;
$determinant *= -1;
$k = $n;
}
}
}
$determinant *= $A[$i][$j];
}else{
$E[$i][$j] = -$A[$i][$j] / $A[$j][$j];
}
$A = multiply($E, $A);
$E = identity($n);
}
}
return $determinant; // Return the determinant of the matrix
}
// Computes the inverse of a matrix
// $A: a square matrix
function inverse($A){
$n = sizeof($A); // Dimension of the square matrix
$V = identity($n); // The inverse of the matrix
$E = identity($n); // An elementary matrix
// Find the inverse matrix by performing elementary matrix operations
// on an augmented matrix through the use of elementary matrices
for($j = 0; $j < $n; $j++){
for($i = $j; $i < $n; $i++){
if($i == $j){
if($A[$i][$j] == 0){
for($k = $j + 1; $k < $n; $k++){
if($A[$k][$j] != 0){
$swap = $A[$i];
$A[$i] = $A[$k];
$A[$k] = $swap;
$swap = $V[$i];
$V[$i] = $V[$k];
$V[$k] = $swap;
$k = $n;
}
}
}
$E[$i][$j] = 1 / $A[$i][$j];
}else{
$E[$i][$j] = -$A[$i][$j];
}
$A = multiply($E, $A);
$V = multiply($E, $V);
$E = identity($n);
}
}
for($j = $n - 1; $j > 0; $j--){
for($i = $j - 1; $i > -1; $i--){
$E[$i][$j] = -$A[$i][$j];
$A = multiply($E, $A);
$V = multiply($E, $V);
$E = identity($n);
}
}
return $V; // Return the inverse matrix
}
// Computes the product of a matrix multiplication
// $A: An m x n matrix
// $B: An n x o matrix
function multiply($A, $B){
$M; // The product of the multiplication
$B = transpose($B); // Transpose the second matrix
// Do the multiplication by making use of the dot product
for($i = 0; $i < sizeof($A); $i++){
for($j = 0; $j < sizeof($B); $j++){
$M[$i][$j] = dotProduct(array($A[$i]), array($B[$j]));
}
}
return $M; // Return the product
}
// Computes the dot product of two vectors
// $a: a row or column vector
// $b; a row or column vector
function dotProduct($a, $b){
$result = 0; // The result of the dot product
// The dot product
for($i = 0; $i < sizeof($a); $i++){
for($j = 0; $j < sizeof($a[0]); $j++){
$result += $a[$i][$j] * $b[$i][$j];
}
}
return $result; // Return the dot product
}
// Computes the transpose of a matrix
// $A: the matrix to be transposed
function transpose($A){
$T; // Transpose matrix
// Switch the i,jth entry to the j,ith entry
for($i = 0; $i < sizeof($A); $i++){
for($j = 0; $j < sizeof($A[0]); $j++){
$T[$j][$i] = $A[$i][$j];
}
}
return $T; // Return the transpose matrix
}
// Computes an identity matrix of size n x n
// $n: size of the matrix wanted
function identity($n){
$I; // Identity matrix
// Fill the matrices diagonal with 1's and everything else with 0's
for($i = 0; $i < $n; $i++){
for($j = 0; $j < $n; $j++){
if($i == $j){
$I[$i][$j] = 1;
}else{
$I[$i][$j] = 0;
}
}
}
return $I; // Return the identity matrix
}
class VirtualObject{
// Member declarations
private $polyObject;
// Constructor
function __construct(){
$this->polyObject =
array( // object
array( // polygon
array( // vector
array(50),
array(-50),
array(50),
array(0)
),
array( // vector
array(-50),
array(-50),
array(50),
array(0)
),
array( // vector
array(-50),
array(50),
array(50),
array(0)
)
),
array( // polygon
array( // vector
array(-50),
array(50),
array(50),
array(0)
),
array( // vector
array(50),
array(50),
array(50),
array(0)
),
array( // vector
array(50),
array(-50),
array(50),
array(0)
)
),
array( // polygon
array( // vector
array(50),
array(-50),
array(-50),
array(0)
),
array( // vector
array(50),
array(-50),
array(50),
array(0)
),
array( // vector
array(50),
array(50),
array(50),
array(0)
)
),
array( // polygon
array( // vector
array(50),
array(50),
array(50),
array(0)
),
array( // vector
array(50),
array(50),
array(-50),
array(0)
),
array( // vector
array(50),
array(-50),
array(-50),
array(0)
)
),
array( // polygon
array( // vector
array(50),
array(50),
array(-50),
array(0)
),
array( // vector
array(50),
array(50),
array(50),
array(0)
),
array( // vector
array(-50),
array(50),
array(50),
array(0)
)
),
array( // polygon
array( // vector
array(-50),
array(50),
array(50),
array(0)
),
array( // vector
array(-50),
array(50),
array(-50),
array(0)
),
array( // vector
array(50),
array(50),
array(-50),
array(0)
)
),
array( // polygon
array( // vector
array(-50),
array(-50),
array(-50),
array(0)
),
array( // vector
array(50),
array(-50),
array(-50),
array(0)
),
array( // vector
array(50),
array(50),
array(-50),
array(0)
)
),
array( // polygon
array( // vector
array(50),
array(50),
array(-50),
array(0)
),
array( // vector
array(-50),
array(50),
array(-50),
array(0)
),
array( // vector
array(-50),
array(-50),
array(-50),
array(0)
)
),
array( // polygon
array( // vector
array(-50),
array(50),
array(-50),
array(0)
),
array( // vector
array(-50),
array(50),
array(50),
array(0)
),
array( // vector
array(-50),
array(-50),
array(50),
array(0)
)
),
array( // polygon
array( // vector
array(-50),
array(-50),
array(50),
array(0)
),
array( // vector
array(-50),
array(-50),
array(-50),
array(0)
),
array( // vector
array(-50),
array(50),
array(-50),
array(0)
)
),
array( // polygon
array( // vector
array(-50),
array(-50),
array(-50),
array(0)
),
array( // vector
array(-50),
array(-50),
array(50),
array(0)
),
array( // vector
array(50),
array(-50),
array(50),
array(0)
)
),
array( // polygon
array( // vector
array(50),
array(-50),
array(50),
array(0)
),
array( // vector
array(50),
array(-50),
array(-50),
array(0)
),
array( // vector
array(-50),
array(-50),
array(-50),
array(0)
)
)
);
}
// Method declarations
// Returns the polygon count of the object
public function getPolyCount(){
return sizeof($this->polyObject);
}
// Returns a particular polygon of the object
public function getPolygon($n){
return $this->polyObject[$n];
}
}
?>
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