Jade

Transformations & Coordinate Systems

Transformations

Intro

  • We can try and move objects by modifying vertices and re-configuring buffers per frame but this is costly and cumbersome
  • Instead we use matrices

Vectors & Matrices

  • If we want to visualize vectors as positions, we use the position vector that originates from the origin
  • The unit vector is of unit length in any direction. Calculating the unit vector of a given vector (dividing by magnitude) is called normalizing the vector
  • The dot product is the component wise multiplication of vectors and is also the product of magnitude of vectors with the cosine of the angle between them
  • The cross product is defined in the 3D space only and produces a orthogonal vector given 2 vectors. Refer to link for calculation
  • The dot product in matrices is basically common matrix multiplication where Aa,b⋅Bb,c=Ca,cA_{a, b} \cdot B_{b, c} = C_{a, c}
  • A vector is basically a N×1N \times 1 matrix and hence can be multiplied with matrices to give transformations
    • Identity matrix II does an identity transform
    • Scale matrix is basically II but each 1 can be a different value based on how much you want each axes to scale
    • Translation matrix is used to alter the position of the vector. For a 3D vector, it is the top 3 values of the 4th column as that results in each component being added to the corresponding value.
    • Rotation matrix
      • Rotations in 3D are specified with an angle and a rotation axis
      • Rotations are done with a combination of cosine and sine
      • An example of a rotation about X axis[10000cos⁡θ−sin⁡θ00sin⁡θcos⁡θ00001]⋅(xyz1)=(xcos⁡θ⋅y−sin⁡θ⋅zsin⁡θ⋅y+cos⁡θ⋅z1)\begin{bmatrix} \color{red}1 & \color{red}0 & \color{red}0 & \color{red}0 \\ \color{green}0 & \color{green}{\cos \theta} & - \color{green}{\sin \theta} & \color{green}0 \\ \color{blue}0 & \color{blue}{\sin \theta} & \color{blue}{\cos \theta} & \color{blue}0 \\ \color{purple}0 & \color{purple}0 & \color{purple}0 & \color{purple}1 \end{bmatrix} \cdot \begin{pmatrix} x \\ y \\ z \\ 1 \end{pmatrix} = \begin{pmatrix} x \\ \color{green}{\cos \theta} \cdot y - \color{green}{\sin \theta} \cdot z \\ \color{blue}{\sin \theta} \cdot y + \color{blue}{\cos \theta} \cdot z \\ 1 \end{pmatrix}
      • To rotate about a 3D axis, rotate around X, then Y and lastly Z, however this produces a gimbal lock. Another better way is to create an arbitrary unit axis and rotate about it [cos⁡θ+Rx2(1−cos⁡θ)RxRy(1−cos⁡θ)−Rzsin⁡θRxRz(1−cos⁡θ)+Rysin⁡θ0RyRx(1−cos⁡θ)+Rzsin⁡θcos⁡θ+Ry2(1−cos⁡θ)RyRz(1−cos⁡θ)−Rxsin⁡θ0RzRx(1−cos⁡θ)−Rysin⁡θRzRy(1−cos⁡θ)+Rxsin⁡θcos⁡θ+Rz2(1−cos⁡θ)00001]\begin{bmatrix} \cos \theta + \color{red}{R_x}^2(1 - \cos \theta) & \color{red}{R_x}\color{green}{R_y}(1 - \cos \theta) - \color{blue}{R_z} \sin \theta & \color{red}{R_x}\color{blue}{R_z}(1 - \cos \theta) + \color{green}{R_y} \sin \theta & 0 \\ \color{green}{R_y}\color{red}{R_x} (1 - \cos \theta) + \color{blue}{R_z} \sin \theta & \cos \theta + \color{green}{R_y}^2(1 - \cos \theta) & \color{green}{R_y}\color{blue}{R_z}(1 - \cos \theta) - \color{red}{R_x} \sin \theta & 0 \\ \color{blue}{R_z}\color{red}{R_x}(1 - \cos \theta) - \color{green}{R_y} \sin \theta & \color{blue}{R_z}\color{green}{R_y}(1 - \cos \theta) + \color{red}{R_x} \sin \theta & \cos \theta + \color{blue}{R_z}^2(1 - \cos \theta) & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}
      • This too does not solve the problem entirely and the solution is to use quaternions.
  • We can combine several transforms with matrix multiplication. The rightmost in multiplication is first applied to the vector. A recommended order is scaling -> rotation -> translation
  • All of these can be implemented using glm and the matrices can be sent to the shaders with uniform's with type mat4
  • If the uniform's have to be changed frequently, put them in the loop

Coordinate Systems

Intro

  • As we know, GL expects all vertices to be normalized. What we do in practice is send coords in a range we specify and normalize it in vertex shader - normalized device coords (NDC).
  • The process of creating NDC involves many intermediate coord system reprs due to ease of operation/calculation
  • The coord systems of importance are
    • Local space (or Object space)
    • World space
    • View space (or Eye space)
    • Clip space
    • Screen space

Transformation matrices

  • To transform to a new coord system, we use the model, view and the projection matrices.
  • These are then processed into world, view and clip coords and eventually to screen coords.
    1. Local coords - coords of object relative to local origin
    2. World space coords - transform the local to world space which involves a larger world, relative to world origin
    3. View space coords - transform to view space such that each object is as seen from the camera
    4. Clip coords - clip the screen from -1 to 1 and determine what is shown
    5. Screen coords - a viewport transform, converts -1 to 1 to range defined by glViewport

Local space

  • Coord space local to object
  • Example: The center of a cube - as shown in a new blender file

World space

  • All objects are positioned w.r.t world origin of (0,0,0)
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