Definitions

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$overline{q} = q_w - q_x i - q_y k - q_z k = q_w - q_v$ .
$vert q vert = sqrt{{q_w}^2 + {q_x}^2 + {q_y}^2 + {q_z}^2}$ .
$vert q_v vert = sqrt{{q_x}^2 + {q_y}^2 + {q_z}^2}$ .
$e^q = e^{q_w} (cos vert q_v vert + {{q_v}/{vert q_v vert}} sin vert q_v vert)$ .
$ln q = ln vert q vert + {{q_v}/{vert q_v vert}} acos {{q_w}/{vert q vert}}$ .
$q^p = e^{(ln q)p}$ .
- For real p: ; $q^p = {{vert q vert}^{p}} (cos p alpha + {{q_v}/{vert q_v vert}} sin p alpha)$ , where $p alpha = delim{|}{ p arctan{{vert q_v vert}/q_w} }{|}$ .
- See here for quaternion p.

Mandelbulb Formulæ

For real p.

$p alpha = delim{|}{ p arctan{{vert s_v vert}/s_w} }{|}$
$p beta= delim{|}{ p arctan{{vert t_v vert}/t_w} }{|}$
$q^{½p} = r^{½p} s^{½p} t^{½p}$ .

$s^{½p}t^{½p}$ = ${{vert s vert}^{½p}} (cos {p alpha}/2 + {{s_v}/{vert s_v vert}} sin {p alpha}/2)$ ${{vert t vert}^{½p}} (cos {p beta }/2 + {{t_v}/{vert t_v vert}} sin {p beta }/2)$

= $({vert s vert}{vert t vert})^{½p}$ $(cos {p alpha}/2 + {{s_v}/{vert s_v vert}} sin {p alpha}/2)$ $(cos {p beta }/2 + {{t_v}/{vert t_v vert}} sin {p beta }/2)$

Quaternions

Definitions

Mandelbulb Formulæ

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