Lecture 03: 3D to 2D projections

Homogeneous coordinates

Homogeneous coordinates - cartesian coordinates + 1 dimension

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

\[ X = x/w, Y = y/w, Z = z/w \] [1]

Transformations in homogeneous coordinates

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

where \[A = \left[\begin{array}{cccc} x1 & x2 & x3 & x4 \\ y1 & y2 & y3 & y4 \\ z1 & z2 & z3 & z4 \\ w1 & w2 & w3 & w4 \end{array}\right]\] [2]

Translation in homogeneous coordinates

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

\[T = \left[\begin{array}{cccc} 1 & 0 & 0 & tr.x \\ 0 & 1 & 0 & tr.y \\ 0 & 0 & 1 & tr.z \\ 0 & 0 & 0 & 1 \end{array}\right]\] [2]

Scaling in homogeneous coordinates

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

\[S = \left[\begin{array}{cccc} scale.x & 0 & 0 & 0 \\ 0 & scale.y & 0 & 0 \\ 0 & 0 & scale.z & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\] [2]

Rotation around X

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

\[R_x = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & cos(\alpha) & -sin(\alpha) & 0 \\ 0 & sin(\alpha) & cos(\alpha) & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\] [2]

Rotation around Y

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

\[R_y = \left[\begin{array}{cccc} cos(\beta) & 0 & sin(\beta) & 0 \\ 0 & 1 & 0 & 0 \\ -sin(\beta) & 0 & cos(\beta) & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\] [2]

Rotation around Z

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

\[R_z = \left[\begin{array}{cccc} cos(\gamma) & -sin(\gamma) & 0 & 0 \\ sin(\gamma) & cos(\gamma) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\] [2]

World transformations

\[World=TRS\]

First scale, then rotate, then translate

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 [3]

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 [4]

Camera model

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

View transformations

\[ \overrightarrow{z_{axis}} = \frac{\overrightarrow{eye} - \overrightarrow{at}}{||\overrightarrow{eye} - \overrightarrow{at}||} \] \[ \overrightarrow{x_{axis}} = \frac{\overrightarrow{up} \times \overrightarrow{z_{axis}}}{||\overrightarrow{up} \times \overrightarrow{z_{axis}}||}\] \[ \overrightarrow{y_{axis}} = \overrightarrow{z_{axis}} \times \overrightarrow{x_{axis}} \]

View transformations

\[View = \left[\begin{array}{cccc} x_{axis}.x & y_{axis}.x & z_{axis}.x & 0 \\ x_{axis}.y & y_{axis}.y & z_{axis}.y & 0 \\ x_{axis}.z & y_{axis}.z & z_{axis}.z & 0 \\ -\overrightarrow{x_{axis}} \cdot \overrightarrow{eye} & -\overrightarrow{y_{axis}} \cdot \overrightarrow{eye} & -\overrightarrow{z_{axis}} \cdot \overrightarrow{eye} & 1 \end{array}\right]\]

Camera model ideas

Look-at target could be replaced as camera location vector + unit camera direction vector.

Unit direction vector could be set in spherical coordinates.

Projection parameters

• \(fov\) - field of view angle
• \(ar\) - aspect ratio (width to height)
• \(Z_n\) - 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

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

Viewport

\[Viewport = \left[\begin{array}{cccc} width & 0 & 0 & 0 \\ 0 & -height & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\]

Chain transitions together

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

Dangerous zone

Remember about left-handed and right-handed operations!

Check correctness of each matrix, e.x. in DirecX documentation [3]