Original Post
This is my first post here on GameDev. I ran into a problem when I was trying to code a simple 3D Engine in PHP. The problem occurs with hidden planes or what I believe is referred to here as backface culling. I believe everything is in order but for some reason certain hidden faces are being drawn when they're not supposed to be. As of right now I have the code working with four VRML models you can get HERE. The problem is only noticeable on the sphere.wrl model. The cylinder.wrl model does not render. However, the cube.wrl and pyramid.wrl render fine. Here's the code:
<?php
$filename = 'sphere.wrl';
$vrml_str = file_get_contents($filename); // Convert the file into a string
$vrml_tok = str_split($vrml_str); // Split the string into an array of each character
// Traverse the array and eliminate anything that isn't of numeric relevance
// Also retain brackets to allow dissection of quantities
for($i = 0; $i < sizeof($vrml_tok); $i++){
switch($vrml_tok[$i]){
case '0':
case '1':
case '2':
case '3':
case '4':
case '5':
case '6':
case '7':
case '8':
case '9':
case '-':
case '.':
case ' ':
case '{':
case '}':
case '[':
case ']':
break;
default:
$vrml_tok[$i] = "";
break;
}
}
$vrml_tok = implode($vrml_tok); // Join the array into a string
$vrml_tok = explode(' 01 ', $vrml_tok); // Split the string into arrays
array_shift($vrml_tok); // Destroy the first entry in the array
// Extract the relevant pieces of info from the strings in each array
// $vrml_1: Object's center location
// $vrml_2: Object's rotational orientation
// $vrml_3: Object's vertices
// $vrml_4: Object's polygons
for($i = 0; $i < sizeof($vrml_tok); $i++){
$vrml_tok[$i] = substr($vrml_tok[$i], strpos($vrml_tok[$i], "{") + 2);
$vrml_1[$i] = substr($vrml_tok[$i], 0, strpos($vrml_tok[$i], "[") - 1);
$vrml_tok[$i] = substr($vrml_tok[$i], strpos($vrml_tok[$i], "[") + 1);
$vrml_tok[$i] = substr($vrml_tok[$i], strpos($vrml_tok[$i], "[") + 1);
$vrml_3[$i] = substr($vrml_tok[$i], 0, strpos($vrml_tok[$i], "]"));
$vrml_tok[$i] = substr($vrml_tok[$i], strpos($vrml_tok[$i], "[") + 1);
$vrml_4[$i] = substr($vrml_tok[$i], 0, strpos($vrml_tok[$i], "]"));
$vrml_tok[$i] = substr($vrml_tok[$i], strpos($vrml_tok[$i], "]") + 1);
$vrml_1[$i] = explode(" ", $vrml_1[$i]);
$vrml_2[$i] = array_slice($vrml_1[$i], 3);
$vrml_1[$i] = array_slice($vrml_1[$i], 0, 3);
$vrml_3[$i] = explode(" ", $vrml_3[$i]);
$vrml_4[$i] = explode(" ", $vrml_4[$i]);
$temp = array_slice($vrml_4[$i], 0, 3);
for($j = 4; $j < sizeof($vrml_4[$i]); $j += 4){
$temp = array_merge($temp, array_slice($vrml_4[$i], $j, 3));
}
$vrml_4[$i] = $temp;
}
// Swap the values of x z y to become x y z
for($i = 0; $i < sizeof($vrml_1); $i++){
$temp = $vrml_1[$i][1];
$vrml_1[$i][1] = $vrml_1[$i][2];
$vrml_1[$i][2] = $temp;
}
for($i = 0; $i < sizeof($vrml_2); $i++){
$temp = $vrml_2[$i][1];
$vrml_2[$i][1] = $vrml_2[$i][2];
$vrml_2[$i][2] = $temp;
}
for($i = 0; $i < sizeof($vrml_3); $i++){
for($j = 0; $j < sizeof($vrml_3[0]); $j += 3){
$temp = $vrml_3[$i][$j + 1];
$vrml_3[$i][$j + 1] = $vrml_3[$i][$j + 2];
$vrml_3[$i][$j + 2] = $temp;
}
}
// Compute the orientation of the object so as to find a rotational transform matrix
for($i = 0; $i < sizeof($vrml_2); $i++){
$theta_1 = asin(abs($vrml_2[$i][2]));
if($vrml_2[$i][1] >= 0 && $vrml_2[$i][2] >= 0){
$theta_1 = -(pi() / 2 - $theta_1);
}else{
if($vrml_2[$i][1] < 0 && $vrml_2[$i][2] >= 0){
$theta_1 = pi() / 2 - $theta_1;
}else{
if($vrml_2[$i][1] < 0 && $vrml_2[$i][2] < 0){
$theta_1 = pi() / 2 + $theta_1;
}else{
if($vrml_2[$i][1] >= 0 && $vrml_2[$i][2] < 0){
$theta_1 = -pi() / 2 - $theta_1;
}
}
}
}
$theta_2 = acos(abs($vrml_2[$i][0]) / sqrt(pow($vrml_2[$i][0], 2) + pow($vrml_2[$i][1], 2)));
if($vrml_2[$i][0] >= 0 && $vrml_2[$i][1] >= 0){
$theta_2 = -(pi() / 2 - $theta_2);
}else{
if($vrml_2[$i][0] < 0 && $vrml_2[$i][1] >= 0){
$theta_2 = pi() / 2 - $theta_2;
}else{
if($vrml_2[$i][0] < 0 && $vrml_2[$i][1] < 0){
$theta_2 = -(pi() / 2 - $theta_2);
}else{
if($vrml_2[$i][0] >= 0 && $vrml_2[$i][1] < 0){
$theta_2 = pi() / 2 - $theta_2;
}
}
}
}
$ini_orient[$i] = identity(3);
// Rotate the object about the world's X axis according to the computed $theta_1
$rotX = array(
array(1, 0, 0),
array(0, cos($theta_1), -sin($theta_1)),
array(0, sin($theta_1), cos($theta_1))
);
$ini_orient[$i] = multiply($rotX, $ini_orient[$i]);
// Rotate the object about the world's Z axis according to the computed $theta_2
$rotZ = array(
array(cos($theta_2), -sin($theta_2), 0),
array(sin($theta_2), cos($theta_2), 0),
array(0, 0, 1)
);
$ini_orient[$i] = multiply($rotZ, $ini_orient[$i]);
// Rotate the object about its own Z axis using the value given
$rotZ = array(
array(cos($vrml_2[$i][3]), -sin($vrml_2[$i][3]), 0),
array(sin($vrml_2[$i][3]), cos($vrml_2[$i][3]), 0),
array(0, 0, 1)
);
$ini_orient[$i] = multiply($ini_orient[$i], $rotZ);
}
// Create all the objects find in the world and store them in an array
// Initialize these as well
for($i = 0; $i < sizeof($vrml_1); $i++){
$worldContainer[$i] = new VirtualObject();
$temp = array(array());
$temp[0][0] = $vrml_1[$i][0];
$temp[1][0] = $vrml_1[$i][1];
$temp[2][0] = $vrml_1[$i][2];
$worldContainer[$i]->setCenter($temp);
$worldContainer[$i]->setOrientation($ini_orient[$i]);
for($j = 0; $j < sizeof($vrml_3[0]); $j += 3){
$temp[0][0] = $vrml_3[$i][$j];
$temp[1][0] = $vrml_3[$i][$j + 1];
$temp[2][0] = $vrml_3[$i][$j + 2];
$worldContainer[$i]->setVertex($temp);
}
$worldContainer[$i]->setPolygons($vrml_4[$i]);
}
// The class that stores virtual objects
class VirtualObject{
// Member declarations
private $center; // Object center
private $orientation; // Object orientation
private $vertices; // Object vertices
private $polyOrder; // Object polygons
// Constructor
function __construct(){
}
// Method declarations
public function setCenter($n){
$this->center = $n;
}
public function getCenter(){
return $this->center;
}
public function setOrientation($n){
$this->orientation = $n;
}
public function getOrientation(){
return $this->orientation;
}
public function setVertex($n){
$this->vertices[] = $n;
}
public function getVertex($n){
return $this->vertices[$n];
}
public function getVertexCount(){
return sizeof($this->vertices);
}
public function setPolygons($n){
$this->polyOrder = $n;
}
public function getPolygon($n){
$i = 3 * $n;
return array(
$this->getVertex($this->polyOrder[$i]),
$this->getVertex($this->polyOrder[$i + 1]),
$this->getVertex($this->polyOrder[$i + 2])
);
}
public function getPolyCount(){
return sizeof($this->polyOrder) / 3;
}
}
// Viewable screen dimensions
$width = 640;
$height = 480;
// Sets up the camera's orientation and its focal distance
$x = sqrt(2) / 2;
$focal = 300;
$a = array(
array(-$x, 0.5, -0.5),
array($x, 0.5, -0.5),
array(0, -$x, -$x)
);
// 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];
// Invert camera's plane to allow for change of basis
$a = inverse($a);
// 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 < $worldContainer[0]->getPolyCount(); $i++){
// Retrieve the polygons from the cube object
$points = $worldContainer[0]->getPolygon($i);
// Rotate each vertex vector according to the indicated object's orientation
$points[0] = multiply($worldContainer[0]->getOrientation(), $points[0]);
$points[1] = multiply($worldContainer[0]->getOrientation(), $points[1]);
$points[2] = multiply($worldContainer[0]->getOrientation(), $points[2]);
// Determine the normal vector for each polygon
$n1 = crossProduct(addition($points[0], scale(-1, $points[1])), addition($points[2], scale(-1, $points[1])));
// Modify each point of the polygon so that its now distanced to the camera's center point
for($j = 0; $j < 3; $j++){
$points[$j][0][0] -= 150;
$points[$j][1][0] -= 150;
$points[$j][2][0] -= 300 * $x;
}
// 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, $white);
}else{
// Gray lines because these are the planes at the reare of the object
imageline($image2, $x0, $y0, $x1, $y1, $gray);
imageline($image2, $x1, $y1, $x2, $y2, $gray);
imageline($image2, $x2, $y2, $x0, $y0, $gray);
}
}
// Make transparent the background of the viewable polygon drawing
imagecolortransparent($image1, $blue1);
// Layer it over the non-viewable polygon drawing
imagecopymerge($image2, $image1, 0, 0, 0, 0, 640, 480, 100);
// Flip the image horizontally
// Output the finished image
header("Content-type: image/png");
imagepng($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] * $b[0][0]
)
);
}else{ // Cross product of column vectors
return array( // Return the cross product of column vectors as a column vector
array($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])
);
}
}
// 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
}
?>