Descriptions and Examples for the POV-Ray Raytracer by Friedrich A. Lohmueller
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  - Analytical Geometry
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  - Basics
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    Points & Lines
    - Points in 3D
    - Line Sections, Straight,
      Vectors, Distance Markers
    - Surfaces & Planes
    - Output of Results,
      Captions
    - Points of Intersection
    - Circles
    Solids
    - Tetrahedron
    - Parallelepiped
    - Round Solids
    -
  - Overview by Table
      on "analytical_g.inc"
  - Vector Analysis
      with POV-Ray
  - Righthanded & Lefthanded
    Systems of Coordinates
    and the Cross Product

  - Samples from
    Analytical Geometry
    - Parallelogram of the
        Middles of the Edges
    - Trace Points of a Straight Line
    - Calculations about a Triangle
   > Area of a Parallelogram
        and Cross Product
    - Shadow of a Pyramid
    - Hit a plane || yz-plane
    - Angle of triangle & yz-plane
                                       

Analytical Geometry with POV-Ray

- Samples -

Area of a Parallelogram    
and the Cross Product
The Area of a Parallelogram
 
We have 3 points A, B and C: 
#declare A  = < 2.0, 2.5,-4.0>;
#declare B  = < 1.0, 4.0,-4.0>;
#declare C  = < 1.0, 3.0,-1.5>;
The edge vectors AB and AC of a parallelogram are caculated by 
#declare AB = B-A;
#declare AC = C-A;
The 4th corner D of a parallelogram with the edges AB and AC we get by 
#declare D = A + AB + AC;
The area of the parallelogram can be represented by two triangles: 
triangle{ A, B, C pigment{color Green transmit 0.5}}
triangle{ B, C, C+(B-A) pigment{color Green transmit 0.5}}


The size of the area of the parallelogram
 
The size of the area of the parallelogram is the length of the cross product AB x AC : 
#declare N = vcross(AB, AC);
#declare Area_ABCD = vlength(N)
In a lefthanded system of coordinates the cross product is defined lefthanded! (For more about this see here: Righthanded and Lefthanded Systems of Coordinates and the Cross Product.)
Area of a Parallelogram and Cross Product
This scene for POV-Ray: ".txt" file or ".pov" file
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© Friedrich A. Lohmüller, 2007
www.f-lohmueller.de