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<?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 $focal = 500; array(- $x, 0. 5, - 0. 5, 0), ); // 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]), ); } } // 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(){ $this->polyObject = ), ), ) ), ), ), ) ), ), ), ) ), ), ), ) ), ), ), ) ), ), ), ) ), ), ), ) ), ), ), ) ), ), ), ) ), ), ), ) ), ), ), ) ), ), ), ) ) ); } // Method declarations // Returns the polygon count of the object public function getPolyCount(){ return sizeof($this-> polyObject); } // Returns a particular polygon of the object public function getPolygon($n){ return $this->polyObject[$n]; } } ?>
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