Feb
1
Perspective projection in Flash 10
Flash 10 added new features to render triangle meshes. However the vertices have to be converted to 2D first since Graphics.drawTriangles() is a 2D function. This can be done by using flash.geom.Utils3D.projectVectors() to project a vector of 3D vertices using a projection matrix. However I noticed that this function does not use standard projection matrix like the ones used by OpenGL or DirectX. For simple projections you can use PerspectiveProjection.toMatrix3D(). However this will create a matrix that disregards the projection center, so it can't create off-center projections. The matrix used by Flash 10 seems pretty powerful, however it is poorly documented, if at all. I was able to figure out how the Flash 10 projection matrix works, and how to use it to create off-center perspective projections. Feel free to use this code if you need an off-center projection matrix that is compatible with flash.geom.Utils3D.projectVectors().
/*
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~ Esoteric Framework ~
~ framework.esotericorp.com ~
~ ~
~ Crafted with care by Stephan Florquin ~
~ stephan.florquin@gmail.com ~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
----- -----
Copyright (c) 2010 Stephan Florquin
Permission is hereby granted, free of charge, to any person obtaining a
copy of this software and associated documentationfiles (the "Software"),
to deal in the Software without restriction, including without limitation
the rights to use, copy, modify, merge, publish, distribute, sublicense,
and / or sell copies of the Software, and to permit persons to whom the
Software is furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in
all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
DEALINGS IN THE SOFTWARE.
----- -----
*/
package com.esoteric.utils
{
import flash.geom.Matrix3D;
import flash.geom.PerspectiveProjection;
/**
* 3D math functions.
*
* @author Stephan Florquin
*/
public class Math3D
{
/**
* Creates a Matrix3D that can be used with
* flash.utils.Utils3D.projectVectors().
*
* This matrix works diffently than transorm matrices:
*
* sxx syx szx cfx
* sxy syy szy cfy
* sxz syz szz cfz
* dx dy dz cd
*
* The method flash.utils.Utils3D.projectVectors() will use this matrix
* to compute the screen coordinate of a points P using the matrix in
* the following way:
*
* X(x,y,z) = (sxx*x + sxy*y + sxz*z + dx)/(cfx*x + cfy*y + cfz*z + cd)
* Y(x,y,z) = (syx*x + syy*y + syz*z + dy)/(cfx*x + cfy*y + cfz*z + cd)
* Z(x,y,z) = (szx*x + szy*y + szz*z + dz)/(cfx*x + cfy*y + cfz*z + cd)
*
* See AyMaN comment on this page:
*
* http://help.adobe.com/en_US/AS3LCR/Flash_10.0/flash/geom/
* PerspectiveProjection.html
*
* @param fov the field of view
* @param cx projection center x
* @param cy projection center y
* @param pixelPerUnitRatio the pixel per unit ratio
*/
public static function perspectiveMatrix3D(
fov:Number,
cx:Number = 0, cy:Number = 0,
pixelPerUnitRatio:Number = 250
):Matrix3D
{
var sxx:Number = pixelPerUnitRatio / Math.tan(fov * Math.PI / 360);
var syy:Number = sxx;
var dx:Number = -cx * sxx;
var dy:Number = -cy * syy;
var m:Vector.<Number> = new Vector.<Number>();
m.push( sxx, 0, 0, 0 );
m.push( 0, syy, 0, 0 );
m.push( cx, cy, 1, 1 );
m.push( dx, dy, 0, 1 );
return new Matrix3D(m);
}
/**
* Converts a perspective projection to a pespective matrix, taking
* into account the projection center.
*
* @param pp the perspective projection
* @param pixelPerUnitRatio the pixel per unit ratio
* @return the perspective matrix
*/
public static function ppToMatrix3D(
pp:PerspectiveProjection,
pixelPerUnitRatio:Number = 250
):Matrix3D
{
var fov:Number = pp.fieldOfView;
var cx:Number = pp.projectionCenter.x;
var cy:Number = pp.projectionCenter.y;
return perspectiveMatrix3D(fov, cx, cy, pixelPerUnitRatio);
}
/**
* Converts a field of view to a focal length.
* @param fov the field of view
* @return the focal length
*/
public static function focalLengthFromFoV(fov:Number):Number
{
return -.5 * Math.cos(fov * .5) / Math.sin(fov * .5);
}
}
}
