Lecture #3: Projections. Camera model

3D to 2D conversation pipeline

1. From model space to world space
2. From world space to view space (camera space)
3. From view space to projection space
4. From projection space to clipping space
5. From clipping space to homogeneous screen space
6. From homogeneous screen space to screen space [1]

Spaces

• Model space - coordinate system where the object was created
• World space - where all object located in proper locations
• View space - a space located around camera
• Projection space - a projected camera frustrum to cube space
• Screen space - a space of resulted image [2]

Homogeneous coordinates

Homogeneous coordinates - cartesian coordinates + 1 dimension

\[ (X, Y, Z) => (x, y, z, w) \]

\[ X = x/w, Y = y/w, Z = z/w \] [3]

Transformations in homogeneous coordinates

\[T(\overrightarrow{x}) = A\overrightarrow{x}\]

where \[A = \left[\begin{array}{cccc} x1 & x2 & x3 & x4 \\ y1 & y2 & y3 & y4 \\ z1 & z2 & z3 & z4 \\ w1 & w2 & w3 & w4 \end{array}\right]\] [4]

Translation in homogeneous coordinates

\(\overrightarrow{tr}\) - vector of translation

\[T = \left[\begin{array}{cccc} 1 & 0 & 0 & tr.x \\ 0 & 1 & 0 & tr.y \\ 0 & 0 & 1 & tr.z \\ 0 & 0 & 0 & 1 \end{array}\right]\] [4]

Scaling in homogeneous coordinates

\(\overrightarrow{scale}\) - vector of scaling

\[S = \left[\begin{array}{cccc} scale.x & 0 & 0 & 0 \\ 0 & scale.y & 0 & 0 \\ 0 & 0 & scale.z & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\] [4]

Rotation around X

\(\alpha\) - angle of rotation around axis X

\[R_x = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & cos(\alpha) & -sin(\alpha) & 0 \\ 0 & sin(\alpha) & cos(\alpha) & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\] [4]

Rotation around Y

\(\beta\) - angle of rotation around axis Y

\[R_y = \left[\begin{array}{cccc} cos(\beta) & 0 & sin(\beta) & 0 \\ 0 & 1 & 0 & 0 \\ -sin(\beta) & 0 & cos(\beta) & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\] [4]

Rotation around Z

\(\gamma\) - angle of rotation around axis Z

\[R_z = \left[\begin{array}{cccc} cos(\gamma) & -sin(\gamma) & 0 & 0 \\ sin(\gamma) & cos(\gamma) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\] [4]

World transformations

\[World=TRS\]

First scale, then rotate, then translate

Camera model

• \(\overrightarrow{Eye}\) - camera location vector
• \(\overrightarrow{At}\) - camera look-at target
• \(\overrightarrow{Up}\) - camera up vector (or world up direction)

View transformations

\[ \overrightarrow{zaxis} = \frac{\overrightarrow{Eye} - \overrightarrow{At}}{||\overrightarrow{Eye} - \overrightarrow{At}||} \] \[ \overrightarrow{xaxis} = \frac{\overrightarrow{Up} \times \overrightarrow{zaxis}}{||\overrightarrow{Up} \times \overrightarrow{zaxis}||}\] \[ \overrightarrow{yaxis} = \overrightarrow{zaxis} \times \overrightarrow{xaxis} \]

View transformations

\[View = \left[\begin{array}{cccc} xaxis.x & yaxis.x & zaxis.x & 0 \\ xaxis.y & yaxis.y & zaxis.y & 0 \\ xaxis.z & yaxis.z & zaxis.z & 0 \\ - \overrightarrow{xaxis} \cdot \overrightarrow{eye} & - \overrightarrow{yaxis} \cdot \overrightarrow{eye} & - \overrightarrow{zaxis} \cdot \overrightarrow{eye} & 1 \end{array}\right]\]

Projection parameters

• \(S_w\) - screen window width in camera space in near clipping plane
• \(S_h\) - screen window height in camera space in near clipping plane
• \(Z_n\) - distance to the near clipping plane along Z axes in camera space
• \(Z_f\) - distance to the far clipping plane along Z axes in camera space

Orthogonal projection

\[Projortho = \left[\begin{array}{cccc} \frac{2}{S_w} & 0 & 0 & 0 \\ 0 & \frac{2}{S_h} & 0 & 0 \\ 0 & 0 & \frac{1}{Z_f-Z_n} & 0 \\ 0 & 0 & \frac{-Z_n}{Z_f-Z_n} & 1 \end{array}\right]\]

Perspective projection

• \(fov_x\) - field of view angle by x-axis
• \(fov_y\) - field of view angle by y-axis
• \(Z_n\) - distance to the near clipping plane along Z axes in camera space
• \(Z_f\) - distance to the far clipping plane along Z axes in camera space

Perspective projection

\[Proj = \left[\begin{array}{cccc} \frac{1}{tan(fov_x/2)} & 0 & 0 & 0 \\ 0 & \frac{1}{tan(fov_y/2)} & 0 & 0 \\ 0 & 0 & \frac{Z_f +Z_n}{Z_f-Z_n} & -1 \\ 0 & 0 & \frac{-Z_n\cdot Z_f}{Z_f-Z_n} & 0 \end{array}\right]\]

Chain transitions together

\[\vec{X} = Proj \cdot View \cdot World \cdot \vec{x}\]

“Projection” experiment

Deduction phase

• All vertexes need proper transformation
• What about performance?

“Projection” experiment

Experiment

Let’s implement it together

“Projection” experiment

Reference

“Projection” experiment

What is the new knowledge?

• Proper way to map 3D vertees into image