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view-transform.md

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title Transformation Matrix to Transform 3D Objects from World Space to View Space (View Transform)
summary One matrix transformation in the 3D to 2D transformation pipeline is the view transform, where objects are transformed from world space to view space using a transformation matrix. <br /> In this article, I cover the math behind the generation of this transformation matrix.
image /images/camera-transformation!camera-space.jpg
tags
computer graphics
transformation matrix
view transform
3d
2d
libraries
math
date 2016-02-13 03:59:56 -0800
references
Schaback, J. (2016). Camera Transformation and View Matrix. [online] Schabby.de. [Accessed 7 Mar. 2016].
Shirley, P. and Ashikhmin, M. (2005). Fundamentals of computer graphics. Wellesley, Mass.: AK Peters.

The objective of this step is to find a transformation matrix to transform points expressed in world space to view space. A camera can be imagined to exist from a known point of view that captures some objects in the space.

$$ \mathbf{v}_{view} = \mathbf{M}_{view} \mathbf{v}_{wld} $$

The construction of the transformation matrix to transform points from world space to view space needs three parameters:

  • $\mathbf{camera}$: a point expressed in world space defining the location of the point of view. Note that the $\mathbf{camera}$ is at the origin of the view space.
  • $\mathbf{at}$: the direction where the camera is aiming.
  • $\mathbf{up}$: denotes the upward orientation of the camera (typically coincides with the positive $y$-axis).

{{< figure src="/https/github.com/images/camera-transformation!camera-space.jpg" title="View Transform" >}}

Note that the camera is looking at the negative $z$-axis of the view space; this is a convention rather than a rule since the projection matrix will be constructed in a way so that points in the $-z$-axis in view space are transformed to the range $[-1,1]$.

Derivation of the View Transform Matrix

The process of transforming the vertices in world space to view space is given by:

  • Creation of a coordinate frame for the view space.
  • Application of the appropriate translation for the camera location (world space -> upright space).
  • Transformation of the points in world space to camera space (upright space -> object space).

Creation of a Coordinate Frame for the View Space

Given $\mathbf{camera}$, $\mathbf{at}$, and $\mathbf{up}$, the steps to compute the coordinate frame whose basis vectors are $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$ are as follows (note that since these are basis vectors, they need to be unit vectors):

  • Compute $\mathbf{w}$ trivially by normalizing the vector $\mathbf{camera - at}$.
$$ \mathbf{w} = \frac{\mathbf{camera - at}}{\norm{\mathbf{camera - at}}} $$

  • Next, $\mathbf{u}$ can be computed with the cross product of $\mathbf{w}$ and $\mathbf{up}$. Again, the resulting vector must be normalized.
$$ \mathbf{u} = \frac{\mathbf{w} \times \mathbf{up}}{\norm{ \mathbf{w} \times \mathbf{up} }} $$

  • Finally, $\mathbf{v}$ can be computed as:
$$ \mathbf{v} = \mathbf{w} \times \mathbf{u} $$

Camera Translation

The transformation matrix that moves all the points from world space to view's upright space is:

$$ \mathbf{T} = \begin{bmatrix} 1 & 0 & 0 & -camera_x \\ 0 & 1 & 0 & -camera_y \\ 0 & 0 & 1 & -camera_z \\ 0 & 0 & 0 & 1 \end{bmatrix} $$

Transformation of the Points from World Space to View Space

Given that the camera transformation basis vectors (encoded in a matrix) are:

$$ \mathbf{M}_{wld \leftarrow view} = \begin{bmatrix} \mathbf{u}_{3 \times 1} & \mathbf{v}_{3 \times 1} & \mathbf{w}_{3 \times 1} \end{bmatrix} $$

Expressed in a 4x4 matrix:

$$ \mathbf{M}_{wld \leftarrow view} = \begin{bmatrix} x_u & x_v & x_w & 0 \\ y_u & y_v & y_w & 0 \\ z_u & z_v & z_w & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$

This works as a transformation matrix to transform points from view space to world space. Therefore, the matrix that does the opposite operation (transformation from world space to view space) is the inverse of this matrix (the transpose is equivalent since the matrix is orthonormal).

$$ \mathbf{M}_{view \leftarrow wld} = \mathbf{M^{-1}}_{wld \leftarrow view} = \mathbf{M}^T_{wld \leftarrow view} = \begin{bmatrix} x_u & y_u & z_u & 0 \\ x_v & y_v & z_v & 0 \\ x_w & y_w & z_w & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$

The View Matrix

We can combine the translation and the rotation matrix in a single matrix called the view matrix, which has the form:

$$ \begin{align*} \mathbf{M}_{view} &= \mathbf{M}_{view \leftarrow wld} \mathbf{T} \\ &= \begin{bmatrix} x_u & y_u & z_u & 0 \\ x_v & y_v & z_v & 0 \\ x_w & y_w & z_w & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & -camera_x \\ 0 & 1 & 0 & -camera_y \\ 0 & 0 & 1 & -camera_z \\ 0 & 0 & 0 & 1 \end{bmatrix} \\ &= \begin{bmatrix} x_u & y_u & z_u & - \mathbf{camera} \cdot \mathbf{u} \\ x_v & y_v & z_v & - \mathbf{camera} \cdot \mathbf{v} \\ x_w & y_w & z_w & - \mathbf{camera} \cdot \mathbf{w} \\ 0 & 0 & 0 & 1 \end{bmatrix} \end{align*} $$