WIP: Lobachevsky (hyperbolic) space ray tracer

The work on this article is in progress.

Theory

The approach was inspired by the Game of Life on Lobachevsky plane and extrapolated to 3-dimensional space.

Position

The position in 3-dimensional Lobachevsky space is represented by a quaternion of form:

\[\mathbf{p} = x + iy + jz\]

where \(x\), \(y\) and \(z\) are coordinates in Poincaré half-space model with \(z > 0\).

The point of origin in such representation is \(j\).

Transformation

Requirements

In Lobachevsky space we have only 2 kinds of similarity transformations: shift and rotation (and their combination). To implement them we need at least 3 following basic transformations:

  1. Shift along \(z\)-axis.
  2. Rotation around \(z\)-axis.
  3. Rotation around \(y\)-axis.

In Poincaré half-space model shift along \(z\)-axis is simply a scaling; and rotation around \(z\)-axis remains itself. The rotation around \(y\)-axis is more tricky - it’s like twisting around unit ring lying in \(x = 0\) plane.

Representation

Required transformations could be represented by the subset of quaternionic Möbius transformations:

\[f(\mathbf{q}) = (a \mathbf{q} + b)(c \mathbf{q} + d)^{-1}\]

where \(\mathbf{q}\) is a quaternion representing position, \(a\), \(b\), \(c\) and \(d\) - some complex numbers which satisfy \(ad - bc = 1\). Remember that quaternion multiplication is non-commutative.

We can also write this transformation in matrix form:

\[f(\mathbf{q}) = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \mathbf{q}\]

Interestingly, we can consider the composition of Möbius transformations as the matrix product.

Now let’s write down our basic transformations as Möbius transformations:

  1. Shift along \(z\)-axis by the distance \(L\):

    \[S_x = \begin{bmatrix} e^{L/2} & 0 \\ 0 & e^{-L/2} \end{bmatrix}\]

  2. Rotation around \(z\)-axis by the angle \(\varphi\):

    \[R_z = \begin{bmatrix} \cos{\frac{\varphi}{2}} + i\sin{\frac{\varphi}{2}} & 0 \\ 0 & \cos{\frac{\varphi}{2}} - i\sin{\frac{\varphi}{2}} \end{bmatrix}\]

  3. Rotation around \(y\)-axis by the angle \(\theta\):

    \[R_y = \begin{bmatrix} \cos{\frac{\theta}{2}} & -\sin{\frac{\theta}{2}} \\ \sin{\frac{\theta}{2}} & \cos{\frac{\theta}{2}} \end{bmatrix}\]

Derivation

We also need to have a derivative of the transformation, e.g. to apply the transformation to direction rather than point. The derivation of quaternionic functions isn’t easy because of their non-commutativity, and also the derivative is direction-dependent.

The derivative of quaternionic function \(f\) at the point \(\mathbf{p}\) along the direction \(\mathbf{v}\):

\[\frac{d f(\mathbf{q})}{d \mathbf{q}} \circ \mathbf{v} = \lim_{\varepsilon \to 0} \frac{f(\mathbf{q} + \varepsilon \mathbf{v}) - f(\mathbf{q})}{\varepsilon}\]

Substituting our Möbius transformation for \(f\) and slightly simplifying the formula we get:

\[\mathbf{P} = a \mathbf{q} + b\] \[\mathbf{Q} = c \mathbf{q} + d\] \[\frac{d f(\mathbf{q})}{d \mathbf{q}} \circ \mathbf{v} = a \mathbf{v} \mathbf{Q}^{-1} + \mathbf{P} (\overline{(c \mathbf{v})} - 2 [\mathbf{Q} \cdot (c \mathbf{v})] \mathbf{Q}^{-1}) |\mathbf{Q}|^{-2}\]

where \(\overline{\mathbf{x}}\) is quaternionic conjugation, and \([\mathbf{x} \cdot \mathbf{y}]\) is dot product of two quaternions as if they were 4-dimensional vectors.

To be continued …

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