Quick Revision

Intro

  • We have some basis $B$ for a vector space in $\mathbb{R}^2$ with these basis vectors:

$$ B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} $$

  • Where the left column is the first basis vector $b_1$ and the second is the second basis vector $b_2$.

  • We also have a vector $v$ which is portrayed in that basis as:

$$[v]_B = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$$

  • This means:

$$[v]_S = 3b_1 + 4b_2$$

  • Where $S$ is the standard/canonical basis.
  • So to convert from a basis $B$ to $S$:

$$B \cdot [v]_B= [v]_S$$

  • To go from $S$ to $B$ instead:

$$B^{-1} \cdot [v]_S = [v]_B$$

  • The matrix $B$ can also be written as $P_{B \rightarrow S}$.
  • So, in general:

$$P_{A \rightarrow B} = (P_{B \rightarrow A})^{-1}$$

  • And vice-versa.

Change of Basis from $B$ to $C$

  • To go from basis $B$ to basis $C$, we can get from $B$ to $S$, then from $S$ to $C$.
  • We make use of the fact that we can go to and from $S$ at any basis here.
  • So:

$$P_{B \rightarrow C} = P_{S \rightarrow C} \cdot P_{B \rightarrow S}$$

  • Read from right to left, since the vector we're transforming is ultimately put on the far right.

Orthonormal Basis

  • An orthonormal basis (ONB) is one whose vectors are orthogonal and normalized (perpendicular and have a magnitude of 1)
  • Which means their basis change matrix $P_{B \rightarrow S}$ is an orthogonal matrix.
  • Orthogonal matrices are matrices whose columns and rows$^1$ are all orthonormal.
  • They have a very useful property:

$$B^T = B^{-1}$$

  • This can be derived easily by treating matrix multiplication as dot products.

  • This property makes finding the inverse of a basis change matrix for an ONB really easy.

  • Also, the product of two orthogonal matrices is also orthogonal. This can be easily checked with the property we just mentioned:

$$(BC)^T(BC)=C^TB^TBC=C^T(I)C=C^TC=I$$

  • So, if $B$ and $C$ are both ONBs, the change of basis matrix going from one to the other is an orthogonal matrix.

$1$: columns being orthonormal implies rows are orthonormal too, and vice versa.

Sampling From Any Normal

  • Back to raytracing.

  • We discussed how to sample random vectors in a hemisphere based on a joint PDF.

  • But we did this while assuming the normal was the positive $z$-axis.

  • Our ultimate goal is to sample for ANY possible normal, allowing us to reflect from anywhere we want.

  • Think of it like this: the $z$-axis is currently our normal, forming a full orthonormal basis with the $x$ and $y$ axes.

  • We can treat the new normal $n$ we are trying to orient the vectors around as the new $z$ axis, then create two other vectors $s$ and $t$ that form an ONB with it: $(s,t,n)$. Let's call it $B$.

  • If we form $B$, we can think of those directions we get as if they were written in terms of $B$, not the standard basis.

  • Then we would just need to convert them to standard form using the basis change matrix $P_{B \rightarrow S}$:

$$P_{B \rightarrow S} \cdot [\text{sampled}]_B = [\text{sampled}]_S$$

  • We can also forget the basis change idea and just think of it as weighting those new axes $(s, t, n)$ by the local directions we sample.
  • In game dev terms, we're basically just treating these points as if they were in the local space of that new normal vector $n$ as a trick, then finding their values global space through basis change.