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bezier.scm
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bezier.scm
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(define-module bezier
(use srfi-11)
(use ray)
(use gauche.array)
(use vec :prefix v:)
(use constant)
(use geometry :prefix g:)
(use math.const)
(export-all))
(select-module bezier)
(define (get-projection-mat ray)
(let* ((ray-origin (origin ray))
(ray-dir (dir ray))
(ox (- (v:x ray-origin)))
(oy (- (- (v:z ray-origin))))
(oz (- (v:y ray-origin)))
(ray-dir (v:unit ray-dir))
(lx (v:x ray-dir))
(ly (- (v:z ray-dir)))
(lz (v:y ray-dir))
(d (sqrt (+ (* lx lx) (* lz lz))))
(rotation (if (= d 0)
(let ((angle (if (>= ly 0)
(- pi/2)
pi/2)))
(array (shape 0 4 0 4)
1 0 0 0
0 (cos angle) (- (sin angle)) 0
0 (sin angle) (cos angle) 0
0 0 0 1))
(array (shape 0 4 0 4)
(/ lz d) (/ (* -1 lx ly) d) lx 0
0 d ly 0
(/ (- lx) d) (/ (* -1 ly lz) d) lz 0
0 0 0 1))))
(array-mul (array (shape 0 4 0 4)
1 0 0 0
0 1 0 0
0 0 1 0
ox oy oz 1)
rotation)))
(define (internally-divide a b t)
(v:sum (v:scale a (- 1 t))
(v:scale b t)))
(define (transform p mat)
(let ((t (array-mul (array (shape 0 1 0 4)
(v:x p) (- (v:z p)) (v:y p) 1)
mat)))
(v:vec3 (array-ref t 0 0)
(array-ref t 0 1)
(array-ref t 0 2))))
(define (dot2d v1 v2)
(v:dot (v:vec3 (v:x v1) (v:y v1) 0)
(v:vec3 (v:x v2) (v:y v2) 0)))
(define (make-bezier a b c d width material)
(letrec* ((cp (vector a b c d))
(n 4)
(width1 (/ width 2))
(width2 (* width1 width1))
(eps (/ width 20))
(bez-p
(lambda (t)
(let* ((t2 (* t t))
(t3 (* t2 t))
(1-t (- 1 t))
(1-t2 (* 1-t 1-t))
(1-t3 (* 1-t2 1-t)))
(v:sum (v:scale a 1-t3)
(v:scale b (* 3 1-t2 t))
(v:scale c (* 3 1-t t2))
(v:scale d t3)))))
(split
(lambda (t)
(let* ((split-point (bez-p t))
(nbc (internally-divide b c t))
(lb (internally-divide a b t))
(lc (internally-divide lb nbc t))
(rc (internally-divide c d t))
(rb (internally-divide nbc rc t)))
(values (make-bezier a lb lc split-point width material)
(make-bezier split-point rb rc d width material)))))
(bbox (let* ((min-vec (v:vec3 +max-float+ +max-float+ +max-float+))
(max-vec (v:vec3 (- +max-float+) (- +max-float+) (- +max-float+))))
(dotimes (i 4)
(dotimes (axis 3)
(v:vec3-set! min-vec axis
(min (- (v:vec3-ref (vector-ref cp i) axis) width1)
(v:vec3-ref min-vec axis)))
(v:vec3-set! max-vec axis
(max (+ (v:vec3-ref (vector-ref cp i) axis) width1)
(v:vec3-ref max-vec axis)))))
(g:make-aabb min-vec max-vec)))
(bezier-transform
(lambda (mat)
(make-bezier (transform a mat)
(transform b mat)
(transform c mat)
(transform d mat)
width material)))
(bez-tan-vec
(lambda (t)
(let ((t2 (* t t))
(coef-a (v:sum (v:scale b 3)
d
(v:scale c -3)
(v:scale a -1)))
(coef-b (v:scale (v:sum a (v:scale b -2) c) 3))
(coef-c (v:scale (v:diff b a) 3)))
(v:unit (v:sum (v:scale coef-a (* 3 t2))
(v:scale coef-b (* 2 t))
coef-c)))))
(to-string
(lambda ()
(format "~A ~%~A ~%~A ~%~A~%" a b c d)))
(converge
(lambda (depth c v0 vn t)
(let*-values (((valid? b) (g:bounding-box c 0 0))
((b-min) (g:aabb-min b))
((b-max) (g:aabb-max b)))
(cond ((or (>= (v:z b-min) t) (<= (v:z b-max) 0.000001)
(>= (v:x b-min) width1) (<= (v:x b-max) (- width1))
(>= (v:y b-min) width1) (<= (v:y b-max) (- width1)))
(values #f #f))
((< depth 0)
(let* ((dir (v:diff (bezier-cp c 3) (bezier-cp c 0)))
(dp0 (bezier-tan-vec c 0))
(dp0 (if (< (dot2d dir dp0) 0)
(v:scale dp0 -1)
dp0))
(dpn #f)
(w #f)
(v #f)
(p #f))
(if (< (dot2d dp0 (v:scale (bezier-cp c 0) -1)) 0)
(values #f #f)
(begin
(set! dpn (bezier-tan-vec c 1))
(set! dpn (if (< (dot2d dir dpn) 0)
(v:scale dpn -1)
dpn))
(if (< (dot2d dpn (bezier-cp c (- n 1))) 0)
(values #f #f)
(begin
(set! w (+ (* (v:x dir) (v:x dir))
(* (v:y dir) (v:y dir))))
(if (= w 0)
(values #f #f)
(begin
(set! w (/ (+ (* (v:x (bezier-cp c 0)) (v:x dir))
(* (v:y (bezier-cp c 0)) (v:y dir)))
(- w)))
(set! w (clamp w 0 1))
(set! v (+ (* v0 (- 1 w)) (* vn w)))
(set! p (bezier-point c v))
(if (or (>= (+ (* (v:x p) (v:x p))
(* (v:y p) (v:y p))) width2)
(<= (v:z p) 0.0001)
(< t (v:z p)))
(values #f #f)
(values #t (v:z p)))))))))))
(else (let*-values (((vm) (/ (+ v0 vn) 2))
((cl cr) (bezier-split c 0.5))
((hit-left? nt-left) (converge (- depth 1) cl v0 vm t))
((hit-right? nt-right) (converge (- depth 1) cr vm vn t)))
(if (and hit-left? (< nt-left t))
(set! t nt-left))
(if (and hit-right? (< nt-right t))
(set! t nt-right))
(values (or hit-left? hit-right?) t)))))))
(hit (lambda (r t-min t-max)
(let*-values (((tr) (get-projection-mat r))
((transformed) (bezier-transform tr))
((max-depth) (let ((l0 (let loop ((m (- +max-float+)) (i 0))
(if (= i (- n 2))
m
(let ((x (abs (+ (v:x (bezier-cp transformed i))
(* -2 (v:x (bezier-cp transformed (+ i 1))))
(v:x (bezier-cp transformed (+ i 2))))))
(y (abs (+ (v:y (bezier-cp transformed i))
(* -2 (v:y (bezier-cp transformed (+ i 1))))
(v:y (bezier-cp transformed (+ i 2)))))))
(loop (max x y m) (inc! i)))))))
(log (/ (* (sqrt 2) n (- n 1) l0) (* 8 eps)) 4)))
((max-depth) (if (= max-depth -inf.0)
0
(ceiling->exact max-depth)))
((hit? t) (converge max-depth transformed 0 1 t-max)))
;; (display (format "max-depth ~A~%" max-depth))
;; (display (format "dir ~A~%origin ~A~%" (dir r) (origin r)))
;; (display (format "~A~%" (transform (v:vec3 0 0 0) tr)))
;; (display (format "bezier~%~Atransformed~%~A"
;; (to-string)
;; (bezier-to-string transformed)))
;; (display (format "dist~% a-b ~A~% b-c ~A~% c-d ~A~%"
;; (v:length (v:diff a b))
;; (v:length (v:diff b c))
;; (v:length (v:diff c d))))
;; (display (format "transformed dist~% a-b ~A~% b-c ~A~% c-d ~A~%~%"
;; (bezier-compare transformed 0 1)
;; (bezier-compare transformed 1 2)
;; (bezier-compare transformed 2 3)))
(if (and hit? (< t-min t))
(values #t (make-hit-record t
(point-at-parameter r t)
(v:scale (dir r) -1)
material
0 0))
(values #f #f))))))
(vector hit
(lambda (t0 t1)
(values #t bbox))
cp
bez-p
split
bezier-transform
bez-tan-vec
to-string)))
(define (bezier-compare b i1 i2)
(v:length (v:diff (bezier-cp b i1) (bezier-cp b i2))))
(define (bezier-cp bezier index)
(vector-ref (vector-ref bezier 2) index))
(define (bezier-point bezier t)
((vector-ref bezier 3) t))
(define (bezier-split bezier t)
((vector-ref bezier 4) t))
(define (bezier-transform bezier mat)
((vector-ref bezier 5) mat))
(define (bezier-tan-vec bezier t)
((vector-ref bezier 6) t))
(define (bezier-to-string bezier)
((vector-ref bezier 7)))