**Position and Orientation**

**Translations**

- Co ordinates of 3D TRIANGLE

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

(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.

- 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).
- Consider a 3 ⤬ 3 matrix,

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' )

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.

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