Position and Orientation in geometry of virtual world

 

Position and Orientation

Translations

• Co ordinates of 3D TRIANGLE

                 ( (𝒳1,𝒚1,𝑧1),(𝒳2,𝒚2,𝑧2),(𝒳3,𝒚3,𝑧3) )

• Change of 𝒳1, y1,z1 amount in the traiangles' position x, y, and z axes respectively can be given by

                      (x1, y1,z1) ↦ (x1 + x1,y1 + y1,z1 +z1)
                      (x2,y2,z2) ↦ (x2 + x1,y2 + y1,z2 +z1)
                      (x3,y3,z3) ↦ (x3 + x1,y3 + y1,z3 + z1)
       

Triangle is translated by x1 = -8 and y1 = -7

Relativity

• The triangle moved in the virtual world because of translation.
• The coordinates of the virtual world is reassigned so that triangle is closer to origin. translation is applied to coordinates. It is called as negation.
• If we perceive ourselves as having moved, then VR sickness might increase, even though it was the object that moved.

Rotations

1. The orientation of the virtual world is changed through a operation called "rotation". Consider a 3D virtual world in which points have coordinates (x,y,z).
2. Consider a 3 ⤬ 3 matrix,

                                    [ m11     m12      m13 ]
                             M= [ m21     m22      m23 ]
                                    [m31      m32      m33 ]

     3.  By multiplication we obtain,

 

    4. Using simple algebra, the matrix multiplication yields

                      x' = m11x + m12y + m13z

                      y' = m21x + m22y + m23z

                      z' = m31x + m32y + m33z

     5.  M is a transformation for which

                     (x,y,z) ↦ (x' , y' , z' )

     6. All the transformations not necessary lead to rotation of points considered in the 3D virtual  world.

Among set of all possible transformation, certain rules are to be followed to achieve rotation.

•  No stretching of axes
• No shearing
• No mirror images

              
Yaw, pitch, and roll

Any three dimensional can be described as a sequence of yaw, pitch, and roll rotations.

Roll - A counter clockwise rotation of Ƴ about the z-axis.
          The rotation matrix is given by -

Pitch - A counter clockwise rotation of ꞵ about the x - axis.
            The rotation matrix is given by -

Yaw - A counter clockwise rotation of α about the y = axis.
           The rotation matrix is given by 

Combining rotation - The yaw, pitch and roll rotation are combined sequentially to attain possible 3D rotation .
                         R(α , ꞵ , Ƴ) = Rz(α) Rx(ꞵ) Rz(Ƴ)

Translation and rotation in one matrix 

To apply both rotation and translation in a single operation, 4 by 4 homogeneous transformation matrix is used.

Inverting transforms 

• For a translation xt , yt, zt inverse is -xt, -yt, -zt
• For rotation, R-1 = Rt
• Inverse of homogeneous transform matrix should be in correct order as the operations are not commutative.