How to check a Point inside a Triangle

When i started to learn vector mathematics , i had found many methods to determine whether a point is inside a triangle or not.Easy method is not the fastest. This type of calculations are very important for making a fast graphics library.
Following are the methods which i remember now.
1. Using Cross Products

Take cross product between each triangle edge and the point to check. If the direction of cross product result is for the three edges that point is inside the triangle , otherwise not

2. Using the angle between the vectors made by point and triangle coordinates.If the sum of angle is 360 the point is inside , otherwise not.

3. Using Point and Plane test.See the figure. You need to know plane equation to understand it.

4. Using Bary centric Coordinates

This is the fastest of above. It involves checking the point in Bary centric coordinates system.

Collision detection using SAT

Following is the SAT code for triangle - Box intersection , I couldn't find any c++ code in internet ( Muller's is there , it is cryptic for me as well as it is not c++)

template

<typename T>

Interfaces3D::InterSection CBBox<T>::IsInsideBox(

const Triangle<T>& refTri ) const

{

// If three points are inside the box, definitily the triangle is inside the ABBBox.

   if( IsInsideBox( refTri.m_Pt1 ) &&  IsInsideBox( refTri.m_Pt2 ) &&  IsInsideBox( refTri.m_Pt3 ) )
   return Interfaces3D::InterSectionIn;

   Math3D::vector<T> xAxis(1,0,0);

   Math3D::vector<T> yAxis(0,1,0);

   Math3D::vector<T> zAxis(0,0,1);

  //..............................................................

  // Intersection test using SAT (separating Axis test) algorithm.

  //..............................................................

  float fMin,fMax;

  float fBoxMin,fBoxMax;

  float fProjTri[3] = { 0,0,0 };

  float fProjBox[2] = { 0,0 };

   

 // First test is agnist the AABBox's cardianl axis.

 fProjTri[0] = refTri.m_Pt1.x;

 fProjTri[1] = refTri.m_Pt2.x;

 fProjTri[2] = refTri.m_Pt3.x; 

 FindMinMax( fProjTri,3,fMin,fMax );

 if( (fMin < m_XL && fMax < m_XL ) || (fMin > m_XR && fMax > m_XR ) )

     return Interfaces3D::InterSectionOut; // the triangle is outside.

 fProjTri[0] = refTri.m_Pt1.y;

 fProjTri[1] = refTri.m_Pt2.y;

 fProjTri[2] = refTri.m_Pt3.y;

 FindMinMax( fProjTri,3,fMin,fMax );

 if( (fMin < m_YB && fMax < m_YB ) || (fMin > m_YT && fMax > m_YT ) )

    return Interfaces3D::InterSectionOut; // the triangle is outside.

 fProjTri[0] = refTri.m_Pt1.z;

 fProjTri[1] = refTri.m_Pt2.z;

 fProjTri[2] = refTri.m_Pt3.z;

 FindMinMax( fProjTri,3,fMin,fMax );

 if( (fMin < m_ZF && fMax < m_ZF ) || (fMin > m_ZN && fMax > m_ZN ) )

    return Interfaces3D::InterSectionOut; // the triangle is outside.

 // Second test is agnist triangles normal. 

 Math3D::vector<T> vtNormal = refTri.m_Normal;

 // project the triangle coordinates on this normal as well as the AABB corners.

 // projecting Box's coordinates

 Math3D::vector<T> vtRadius = GetTopLeftNear() - GetPosition();

 float fExtent = vtNormal.Dot( vtRadius );

 float fCenterBox = vtNormal.Dot( GetPosition() );

 fProjBox[0] = fCenterBox + fExtent;

 fProjBox[1] = fCenterBox - fExtent; 

 FindMinMax( fProjBox, 2,fBoxMin,fBoxMax );

 // projecting triangle coordinates

 fProjTri[0] = vtNormal.Dot( refTri.m_Pt1 );

 fProjTri[1] = vtNormal.Dot( refTri.m_Pt2 );

 fProjTri[2] = vtNormal.Dot( refTri.m_Pt3 );

 FindMinMax( fProjTri, 3,fMin,fMax );

 // Check for overlapping between pronected coordinates.

 // if not overlapping , no intersection.

 if( (fMin < fBoxMin && fMax < fBoxMin ) || (fMin > fBoxMax && fMax > fBoxMax ) )

    return Interfaces3D::InterSectionOut; // the triangle is outside.

  

 // Next possible SAT axis are the Cross product of each tri edges with

 // box's cardinal axis.

 Math3D::vector<T> satAxis[9];

 Math3D::vector<T> vTmp = (refTri.m_Pt1 - refTri.m_Pt2);

 // edge 0 and X,Y,Z axis 

 satAxis[0] = vTmp.Cross( xAxis).normalize() ; 

 satAxis[1] = vTmp.Cross( yAxis).normalize() ;

 satAxis[2] = vTmp.Cross( zAxis).normalize() ;

 // edge 1 and X,Y,Z axis 

 vTmp = (refTri.m_Pt3 - refTri.m_Pt2);

 satAxis[3] = vTmp.Cross( xAxis).normalize() ;

 satAxis[4] = vTmp.Cross( yAxis).normalize() ;

 satAxis[5] = vTmp.Cross( zAxis).normalize() ;

 //  edge 2 and X,Y,Z axis

 vTmp = (refTri.m_Pt1 - refTri.m_Pt3);

 satAxis[6] = vTmp.Cross( xAxis).normalize() ;

 satAxis[7] = vTmp.Cross( yAxis).normalize() ;

 satAxis[8] = vTmp.Cross( zAxis).normalize() ;

 for( int i = 0; i < 9 ; i ++ )

  {

fExtent = satAxis[i].Dot( vtRadius );

fCenterBox = satAxis[i].Dot( GetPosition() );

fProjBox[0] = fCenterBox + fExtent;

fProjBox[1] = fCenterBox - fExtent;

FindMinMax( fProjBox, 2,fBoxMin,fBoxMax ); 

 // projecting triangle coordinates

            fProjTri[0] = satAxis[i].Dot( refTri.m_Pt1 );

            fProjTri[1] = satAxis[i].Dot( refTri.m_Pt2 );

            fProjTri[2] = satAxis[i].Dot( refTri.m_Pt3 );

            FindMinMax( fProjTri, 3,fMin,fMax );

           // check for overlapping on the SAT axis

           if( (fMin < fBoxMin && fMax < fBoxMin ) || (fMin > fBoxMax && fMax > fBoxMax ) )

               return Interfaces3D::InterSectionOut; // the triangle is outside.

  } 

return Interfaces3D::InterSectionIntersect;

}