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// =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-==-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= //
//
//  Project:   Talina Gaming System (TgS) (∂)
//  File:      TgS Collision - Linear-Point.cpp
//  Author:    Andrew Aye (EMail: andrew.aye@gmail.com, Web: http://www.andrewaye.com) 
//  Version:   3.11
//
// ------------------------------------------------------------------------------------------------------------------------------ //
//
//  Copyright: © 2002-2008, Andrew Aye.  All Rights Reserved.
//
//  This software is free for non-commercial use. Redistribution and use in source and binary forms, with or without modification,
//  are permitted provided that the following conditions are met: 
//    Redistributions of source code must retain this copyright notice, this list of conditions and the following disclaimers. 
//    Redistributions in binary form must reproduce this copyright notice, this list of conditions and the following
//      disclaimers in the documentation and other materials provided with the distribution. 
//
//  Neither the names of the copyright owner nor the names of its contributors may be used to endorse or promote products derived
//  from this software without specific prior written permission. 
//
//  The intellectual property rights of the algorithms used reside with Andrew Aye.  You may not use this software, in whole or
//  in part, in support of any commercial product without the express written consent of the author.
//
//  There is no warranty or other guarantee of fitness of this software for any purpose. It is provided solely "as is".
//
// =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-==-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= //



//  The linear collision tests are grouped into three families.  The first returns only the minimal distance, the second and third
// additionally respectively return the points of closest contact, or the linear extrapolation values.  In each family of
// functions its possible to return either the distance or the distance squared.

// ============================================================================================================================== //

//
//  Segment Definition: G0(α) = P0 + α•D0 | α ε [ 0, 1]
//
// Derivation:
//  Let the points of closest contact be Q0 = P0+γ•D0, and v = P1-Q0
//  Geometrically we know that the vector connecting the two closest points of contact ( or the minimal distance
//    vector ) must be perpendicular to the line.  Thus, D0•v=0, D1•v=0, DS=P1-P0, v=P1-(P0+γ•D0)
//
//      0 = D0_(P1-P0-γ•D0,DIM)
//      0 = D0_(DS-γ•D0,DIM)
//      0 = D0•DS-γ•D0•D0
//      γ = D0•DS / D0•D0
//
// However, we know that γ ε [ 0, 1], generating three cases:
//
// [1] γ ε (-∞, 0) || γ = 0
//    Distance: The distance between P0 and P1
//       = (P1-P0)T_(P1-P0,DIM)
//       = DS•DS
//
// [2] γ ε [ 0, 1] || γ = (DS•D0) / (D0•D0)
//    Distance: The distance value would be || v ||.
//       = || v || = v•v = (DS-γ•D0)T_(DS-γ•D0,DIM)
//       = DS•DS + γ•γ•D0•D0 - 2•γ•DS•D0
//       = DS•DS + γ_(γ•D0•D0 - 2•DS•D0,DIM)
//       = DS•DS + γ_((DS•D0 / D0•D0,DIM)•D0•D0 - 2•DS•D0)
//       = DS•DS + γ_(DS•D0 - 2•DS•D0,DIM)
//       = DS•DS - γ_(DS•D0,DIM)
//
// [3] γ ε ( 1, ∞) || γ = 1
//    Distance: The distance between P0+D0 and P1
//       = (P1-P0-D0)T_(P1-P0-D0,DIM)
//       = (DS-D0)T_(DS-D0,DIM)
//       = DS•DS - 2•DS•D0 + D0•D0
//

// ============================================================================================================================== //

namespace TGS { // START TGS ///////////////////////////////////////////////////////////////////////////////////////////////////////
namespace COL { // START COL ///////////////////////////////////////////////////////////////////////////////////////////////////////

// ============================================================================================================================== //

    template struct TTgFSQ_LNPT<TgFLOAT32,4,0,0>;
    template struct TTgFSQ_LNPT<TgFLOAT32,4,1,0>;
    template struct TTgFSQ_LNPT<TgFLOAT32,4,1,1>;

    template struct TTgCSQ_LNPT<TgFLOAT32,4,0,0>;
    template struct TTgCSQ_LNPT<TgFLOAT32,4,1,0>;
    template struct TTgCSQ_LNPT<TgFLOAT32,4,1,1>;

// ============================================================================================================================== //

}; // END COL //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
}; // END TGS //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
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