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pdquat.lisp
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pdquat.lisp
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(in-package :option-9)
#+(or (not option-9-optimize-pvec) option-9-debug)
(declaim (optimize (safety 3) (space 0) (speed 0) (debug 3)))
;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; An implementation of dual quaternions.
;;; The reason I chose not to use two quaternions in a structure and instead
;;; reimplement this stuff as a new structure with new math operations is that
;;; I can more reliably believe how much space this stuff takes while
;;; making it easier to pass to opengl shaders.
;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; Note: Dual numbers do not have an explicit representation in this code.
(eval-when (:compile-toplevel :load-toplevel :execute)
(deftype pdquat () `(simple-array double-float (8)))
(defstruct (pdquat
(:type (vector double-float))
(:constructor make-pdquat)
(:constructor pdquat (&optional rw rx ry rz dw dx dy dz)))
;; The default value represents 0 radians of rotation and 0 units of
;; translation.
;; NOTE: This is NOT a normalized dual quaternion!!!!
(rw 1d0 :type double-float)
(rx 0d0 :type double-float)
(ry 0d0 :type double-float)
(rz 0d0 :type double-float)
(dw 1d0 :type double-float)
(dx 0d0 :type double-float)
(dy 0d0 :type double-float)
(dz 0d0 :type double-float))
(defmacro with-pdquat-accessors ((prefix-symbol pdquat) &body body)
`(with-accessors
((,(make-accessor-symbol prefix-symbol "RW") pdquat-rw)
(,(make-accessor-symbol prefix-symbol "RX") pdquat-rx)
(,(make-accessor-symbol prefix-symbol "RY") pdquat-ry)
(,(make-accessor-symbol prefix-symbol "RZ") pdquat-rz)
(,(make-accessor-symbol prefix-symbol "DW") pdquat-dw)
(,(make-accessor-symbol prefix-symbol "DX") pdquat-dx)
(,(make-accessor-symbol prefix-symbol "DY") pdquat-dy)
(,(make-accessor-symbol prefix-symbol "DZ") pdquat-dz))
,pdquat
,@body))
(defmacro with-multiple-pdquat-accessors (sbinds &body body)
(if (null sbinds)
`(progn ,@body)
`(with-pdquat-accessors ,(car sbinds)
(with-multiple-pdquat-accessors ,(cdr sbinds) ,@body))))
;; NOTE: I must use the pprint-dispatch table to emit nicely formatted
;; pvecs because they aren't a CLASS due to the defstruct definition
;; I am using. So PRINT-OBJECT doesn't work on PVEC types.
(set-pprint-dispatch
'pdquat #'(lambda (str pobj)
(with-pdquat-accessors (p pobj)
(print-unreadable-object (pobj str)
(format str
"[~A + ~Ai + ~Aj + ~Ak] + [~A + ~Ai + ~Aj + ~Ak]e"
prw prx pry prz pdw pdx pdy pdz))))))
;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(declaim (ftype (function (pdquat pdquat) pdquat) dquat-copy-into))
(declaim (inline dquat-copy-into))
(defun dquat-copy-into (dst src)
"Copy the SRC dual quaternion into the DST dual quaternion."
#+option-9-optimize-pvec (declare (optimize (speed 3) (safety 0)))
(with-multiple-pdquat-accessors ((d dst) (s src))
(psetf
;; real part
drw (as-double-float srw)
drx (as-double-float srx)
dry (as-double-float sry)
drz (as-double-float srz)
;; dual part
ddw (as-double-float sdw)
ddx (as-double-float sdx)
ddy (as-double-float sdy)
ddz (as-double-float sdz)))
dst)
(defun dquat-copy (src)
#+option-9-optimize-pvec (declare (optimize (speed 3) (safety 0)))
"Shortname for DQUAT-COPY-INTO."
(dquat-copy-into (pdquat) src))
;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defun dquat-scale-into (dst src scale-factor)
(with-multiple-pdquat-accessors ((d dst) (s src))
(psetf
;; real part
drw (as-double-float (* srw scale-factor))
drx (as-double-float (* srx scale-factor))
dry (as-double-float (* sry scale-factor))
drz (as-double-float (* srz scale-factor))
;; dual part
ddw (as-double-float (* sdw scale-factor))
ddx (as-double-float (* sdx scale-factor))
ddy (as-double-float (* sdy scale-factor))
ddz (as-double-float (* sdz scale-factor))))
dst)
(defun dquat-scale (src scale-factor)
(dquat-scale-into (pdquat) src scale-factor))
;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defun dquat-add-into (dst dquat-a dquat-b)
(with-multiple-pdquat-accessors ((d dst) (a dquat-a) (b dquat-b))
(psetf
;; real part
drw (as-double-float (+ arw brw))
drx (as-double-float (+ arx brx))
dry (as-double-float (+ ary bry))
drz (as-double-float (+ arz brz))
;; dual part
ddw (as-double-float (+ adw bdw))
ddx (as-double-float (+ adx bdx))
ddy (as-double-float (+ ady bdy))
ddz (as-double-float (+ adz bdz))))
dst)
(defun dquat-add (dquat-a dquat-b)
(dquat-add-into (pdquat) dquat-a dquat-b))
;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; TODO: I think this is wrong, verify in octave.
(defun dquat-mul-into (dst dquat-a dquat-b)
(with-multiple-pdquat-accessors ((d dst) (a dquat-a) (b dquat-b))
;; Slow and uses memory.
(let* ((qar (pquat arw arx ary arz))
(qad (pquat adw adx ady adz))
(qbr (pquat brw brx bry brz))
(qbd (pquat bdw bdx bdy bdz))
(qdr (quat-mul qar qbr))
(qdd (quat-add (quat-mul qar qbd)
(quat-mul qad qbr))))
(with-multiple-pquat-accessors ((qdr- qdr) (qdd- qdd))
(psetf
drw (as-double-float qdr-w)
drx (as-double-float qdr-x)
dry (as-double-float qdr-y)
drz (as-double-float qdr-z)
ddw (as-double-float qdd-w)
ddx (as-double-float qdd-x)
ddy (as-double-float qdd-y)
ddz (as-double-float qdd-z)))))
dst)
(defun dquat-mul (dquat-a dquat-b)
(dquat-mul-into (pdquat) dquat-a dquat-b))
;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defun dquat-conjugate-quaternion-into (dst src)
"Q* = Q0* + Q1*e"
(with-multiple-pdquat-accessors ((d dst) (s src))
(psetf
drw (as-double-float srw)
drx (as-double-float (- srx))
dry (as-double-float (- sry))
drz (as-double-float (- srz))
ddw (as-double-float sdw)
ddx (as-double-float (- sdx))
ddy (as-double-float (- sdy))
ddz (as-double-float (- sdz))))
dst)
(defun dquat-conjugate-quaternion (src)
(dquat-conjugate-quaternion-into (pdquat) src))
;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defun dquat-conjugate-dual-into (dst src)
"Qbar = Q0 - Q1e"
(with-multiple-pdquat-accessors ((d dst) (s src))
(psetf
drw (as-double-float srw)
drx (as-double-float srx)
dry (as-double-float sry)
drz (as-double-float srz)
ddw (as-double-float (- sdw))
ddx (as-double-float (- sdx))
ddy (as-double-float (- sdy))
ddz (as-double-float (- sdz))))
dst)
(defun dquat-conjugate-dual (src)
(dquat-conjugate-dual-into (pdquat) src))
;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defun dquat-conjugate-dual-quaternion-into (dst src)
"Qbar* = Q0* - Q1*e"
(with-multiple-pdquat-accessors ((d dst) (s src))
(psetf
drw (as-double-float srw)
drx (as-double-float (- srx))
dry (as-double-float (- sry))
drz (as-double-float (- srz))
ddw (as-double-float (- sdw))
ddx (as-double-float sdx)
ddy (as-double-float sdy)
ddz (as-double-float sdz)))
dst)
(defun dquat-conjugate-dual-quaternion (src)
(dquat-conjugate-dual-quaternion-into (pdquat) src))
;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; This must return a dual number as a values with the real part first.
;; The norm of q is sqrt(q time q*).
;; After a bunch of simplification of that equation, it ends up being:
;; sqrt( Qw^2 + Qx^2 + Qy^2 + Qz^2 ) where Qw Qz Qy Qz are the dual numbers of
;; SRC.
;; Then, note that the square root of a dual number is:
;; sqrt(A + De) = (sqrt A) + (/ B (* 2 (sqrt A)))e
(defun dquat-norm (src)
(with-pdquat-accessors (s src)
(let ((tmp (pdquat)))
;; First we compute the squared partial result
(with-pdquat-accessors (a tmp)
(psetf arw (* srw srw)
adw (* 2d0 srw sdw)
arx (* srx srx)
adx (* 2d0 srx sdx)
ary (* sry sry)
ady (* 2d0 sry sdy)
arz (* srz srz)
adz (* 2d0 srz sdz))
;; Now, compute the summation of the squared dual terms
(let ((r 0d0)
(d 0d0))
(psetf r (+ arw arx ary arz)
d (+ adw adx ady adz))
;; Now compute the square root
;; Not all dual quternions have defined norms....
(if (zerop d)
NIL
(let ((sqrt-r (sqrt r)))
(values sqrt-r (/ d (* 2d0 sqrt-r))))))))))
;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defun dual-number-divide (a b c d)
"Perform the dual number operation: (A + Be) / (C + De) and return
as values the real and dual part. Return NIL if the division cannot be
performed."
(if (zerop c)
NIL
(values (as-double-float (/ a c))
(as-double-float (/ (- (* b c) (* a d))
(* c c))))))
;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defun dquat-normalize-into (dst src)
(with-multiple-pdquat-accessors ((d dst) (s src))
(multiple-value-bind (nr nd) (dquat-norm src)
(when (null nr)
;; Some dual quaternions cannot be normalized: like degenerate ones.
(return-from dquat-normalize-into NIL))
(multiple-value-bind (r0 d0) (dual-number-divide srw sdw nr nd)
(multiple-value-bind (r1 d1) (dual-number-divide srx sdx nr nd)
(multiple-value-bind (r2 d2) (dual-number-divide sry sdy nr nd)
(multiple-value-bind (r3 d3) (dual-number-divide srz sdz nr nd)
(psetf drw r0
ddw d0
drx r1
ddx d1
dry r2
ddy d2
drz r3
ddz d3)))))))
dst)
(defun dquat-normalize (src)
(dquat-normalize-into (pdquat) src))
;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defun dquat-pure-rotation-into (dst angle axis)
(let ((naxis (vect-normalize axis))
(cos-term (as-double-float (cos (/ angle 2d0))))
(sin-term (as-double-float (sin (/ angle 2d0)))))
(with-pvec-accessors (n naxis)
(with-pdquat-accessors (d dst)
(psetf
drw (as-double-float cos-term)
drx (as-double-float (* nx sin-term))
dry (as-double-float (* ny sin-term))
drz (as-double-float (* nz sin-term))
ddw 0d0
ddx 0d0
ddy 0d0
ddz 0d0))))
dst)
(defun dquat-pure-rotation (angle axis)
(dquat-pure-rotation-into (pdquat) angle axis))
;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defun dquat-pure-translation-into (dst translation)
(with-pvec-accessors (tv translation)
(with-pdquat-accessors (d dst)
(psetf
drw 1d0
drx 0d0
dry 0d0
drz 0d0
ddw 0d0
ddx (as-double-float (/ tvx 2d0))
ddy (as-double-float (/ tvy 2d0))
ddz (as-double-float (/ tvz 2d0)))))
dst)
(defun dquat-pure-translation (translation)
(dquat-pure-translation-into (pdquat) translation))
;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defun dquat-create-point-into (dst point)
"Convert the pvec 3d POINT into a dual quaternion. Note, this cannot be used
as a translation, use dquat-pure-translation for that."
(with-pvec-accessors (p point)
(with-pdquat-accessors (d dst)
(psetf drw 1d0
drx 0d0
dry 0d0
drz 0d0
ddw 0d0
ddx (as-double-float px)
ddy (as-double-float py)
ddz (as-double-float pz))))
dst)
(defun dquat-create-point (point)
(dquat-create-point-into (pdquat) point))
;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defun dquat-extract-point-into (point src-dquat)
"Extract the 3D point represented in the SRC-DQUAT and store into the pvec
POINT."
(with-pvec-accessors (p point)
(with-pdquat-accessors (s src-dquat)
(psetf px (as-double-float sdx)
py (as-double-float sdy)
pz (as-double-float sdz))))
point)
(defun dquat-extract-point (src-dquat)
(dquat-extract-point-into (pvec) src-dquat))
;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; TODO: Figure out more detail about how the inverse and conjugate relate
;; in dual quaternion theory. They relate over normalization like in quaternions
;; but I don't know what normalized dual quaternions mean in terms of rigid
;; mody representation.
(defun dquat-apply-into (dst-point transform src-point)
"Apply a dual quaternion TRANSFORM to the 3D SRC-POINT and put the
result into DST-POST."
(let ((dquat-result
(dquat-mul transform
(dquat-mul (dquat-create-point src-point)
(dquat-conjugate-dual-quaternion transform)))))
(dquat-extract-point-into dst-point dquat-result)))
(defun dquat-apply (transform src-point)
(dquat-apply-into (pvec) transform src-point))