+
+ (ly:make-stencil
+ (list 'circle radius thickness fill)
+ (cons (- out-radius) out-radius)
+ (cons (- out-radius) out-radius))))
+
+(define-public (make-oval-stencil x-radius y-radius thickness fill)
+ "Make an oval from two Bezier curves, of x@tie{}radius @var{x-radius},
+y@tie{}radius @code{y-radius}, and thickness @var{thickness} with fill
+defined by @code{fill}."
+ (let*
+ ((x-out-radius (+ x-radius (/ thickness 2.0)))
+ (y-out-radius (+ y-radius (/ thickness 2.0)))
+ (x-max x-radius)
+ (x-min (- x-radius))
+ (y-max y-radius)
+ (y-min (- y-radius))
+ (commands `(,(list 'moveto x-max 0)
+ ,(list 'curveto x-max y-max x-min y-max x-min 0)
+ ,(list 'curveto x-min y-min x-max y-min x-max 0)
+ ,(list 'closepath)))
+ (command-list (fold-right append '() commands)))
+ (ly:make-stencil
+ `(path ,thickness `(,@',command-list) 'round 'round ,fill)
+ (cons (- x-out-radius) x-out-radius)
+ (cons (- y-out-radius) y-out-radius))))
+
+(define-public
+ (make-partial-ellipse-stencil
+ x-radius y-radius start-angle end-angle thick connect fill)
+ "Create an elliptical arc
+@var{x-radius} is the X radius of the arc.
+@var{y-radius} is the Y radius of the arc.
+@var{start-angle} is the starting angle of the arc in degrees.
+@var{end-angle} is the ending angle of the arc in degrees.
+@var{thick} is the thickness of the line.
+@var{connect} is a boolean flag indicating if the end should
+be connected to the start by a line.
+@var{fill} is a boolean flag indicating if the shape should be filled."
+ (define (make-radius-list x-radius y-radius)
+ "Makes a list of angle/radius pairs at intervals of PI/2 for
+the partial ellipse until 7*PI/2. For example, in pseudo-code:
+> (make-radius-list 2 3)\
+\n((0.0 . 2) (PI/2 . 3) (PI . -2) (3*PI/2 . -3)\
+\n(2*PI . 2) (5*PI/2 . 3) (3*PI . -2) (7*PI/2 . -3))
+"
+ (append-map
+ (lambda (adder)
+ (map (lambda (quadrant)
+ (cons (+ adder (car quadrant))
+ (cdr quadrant)))
+ `((0.0 . (,x-radius . 0.0))
+ (,PI-OVER-TWO . (0.0 . ,y-radius))
+ (,PI . (,(- x-radius) . 0.0))
+ (,THREE-PI-OVER-TWO . (0.0 . ,(- y-radius))))))
+ `(0.0 ,TWO-PI)))
+
+ (define
+ (insert-in-ordered-list ordering-function value inlist cutl? cutr?)
+ "Insert @var{value} in ordered list @var{inlist}. If @var{cutl?}, we
+cut away any parts of @var{inlist} before @var{value}. @var{cutr?} works
+the same way but for the right side. For example:
+> (insert-in-ordered-list < 4 '(1 2 3 6 7) #f #f)
+'(1 2 3 4 6 7)
+> (insert-in-ordered-list < 4 '(1 2 3 6 7) #t #f)
+'(4 6 7)
+> (insert-in-ordered-list < 4 '(1 2 3 6 7) #f #t)
+'(1 2 3 4)
+"
+ (define
+ (helper ordering-function value left-list right-list cutl? cutr?)
+ (if (null? right-list)
+ (append
+ (if cutl? '() left-list)
+ (list value)
+ (if cutr? '() right-list))
+ (if (ordering-function value (car right-list))
+ (append
+ (if cutl? '() left-list)
+ (list value)
+ (if cutr? '() right-list))
+ (helper
+ ordering-function
+ value
+ (append left-list (list (car right-list)))
+ (cdr right-list)
+ cutl?
+ cutr?))))
+ (helper ordering-function value '() inlist cutl? cutr?))
+
+ (define (ordering-function-1 a b) (car< a b))
+
+ (define (ordering-function-2 a b) (car<= a b))
+
+ (define (min-max-crawler min-max side l)
+ "Apply function @var{side} to each member of list and
+then reduce using @var{min-max}:
+> (min-max-crawler min car '((0 . 3) (-1 . 4) (1 . 2)))
+-1
+> (min-max-crawler min cdr '((0 . 3) (-1 . 4) (1 . 2)))
+2
+"
+ (reduce min-max
+ (if (eq? min-max min) 100000 -100000)
+ (map side l)))
+
+ (let*
+ (;; the outside limit of the x-radius
+ (x-out-radius (+ x-radius (/ thick 2.0)))
+ ;; the outside limit of the y-radius
+ (y-out-radius (+ y-radius (/ thick 2.0)))
+ ;; end angle to radians
+ (new-end-angle (angle-0-2pi (degrees->radians end-angle)))
+ ;; length of the radius at the end angle
+ (end-radius (ellipse-radius x-out-radius y-out-radius new-end-angle))
+ ;; start angle to radians
+ (new-start-angle (angle-0-2pi (degrees->radians start-angle)))
+ ;; length of the radius at the start angle
+ (start-radius (ellipse-radius x-out-radius y-out-radius new-start-angle))
+ ;; points that the arc passes through at 90 degree intervals
+ (radius-list (make-radius-list x-out-radius y-out-radius))
+ ;; rectangular coordinates of arc endpoint
+ (rectangular-end-radius (polar->rectangular end-radius end-angle))
+ ;; rectangular coordinates of arc begin point
+ (rectangular-start-radius (polar->rectangular start-radius start-angle))
+ ;; we want the end angle to always be bigger than the start angle
+ ;; so we redefine it here just in case it is less
+ (new-end-angle
+ (if (<= new-end-angle new-start-angle)
+ (+ TWO-PI new-end-angle)
+ new-end-angle))
+ ;; all the points that may be extrema of the arc
+ ;; this is the 90 degree points plus the beginning and end points
+ ;; we use this to calculate extents
+ (possible-extrema
+ (insert-in-ordered-list
+ ordering-function-2
+ (cons new-end-angle rectangular-end-radius)
+ (insert-in-ordered-list
+ ordering-function-1
+ (cons new-start-angle rectangular-start-radius)
+ radius-list
+ #t
+ #f)
+ #f
+ #t)))
+ (ly:make-stencil
+ (list
+ 'partial-ellipse
+ x-radius
+ y-radius
+ start-angle
+ end-angle
+ thick
+ connect
+ fill)
+ ;; we know the extrema points by crawling through the
+ ;; list of possible extrema and finding the min and max
+ ;; for x and y
+ (cons (min-max-crawler min cadr possible-extrema)
+ (min-max-crawler max cadr possible-extrema))
+ (cons (min-max-crawler min cddr possible-extrema)
+ (min-max-crawler max cddr possible-extrema)))))
+
+(define (line-part-min-max x1 x2)
+ (list (min x1 x2) (max x1 x2)))
+
+(define (bezier-part-min-max x1 x2 x3 x4)
+ ((lambda (x) (list (reduce min 10000 x) (reduce max -10000 x)))
+ (map
+ (lambda (x)
+ (+ (* x1 (expt (- 1 x) 3))
+ (+ (* 3 (* x2 (* (expt (- 1 x) 2) x)))
+ (+ (* 3 (* x3 (* (- 1 x) (expt x 2))))
+ (* x4 (expt x 3))))))
+ (if (< (+ (expt x2 2) (+ (expt x3 2) (* x1 x4)))
+ (+ (* x1 x3) (+ (* x2 x4) (* x2 x3))))
+ (list 0.0 1.0)
+ (filter
+ (lambda (x) (and (>= x 0) (<= x 1)))
+ (append
+ (list 0.0 1.0)
+ (map (lambda (op)
+ (if (not (eqv? 0.0
+ (exact->inexact (- (+ x1 (* 3 x3)) (+ x4 (* 3 x2))))))
+ ;; Zeros of the bezier curve
+ (/ (+ (- x1 (* 2 x2))
+ (op x3
+ (sqrt (- (+ (expt x2 2)
+ (+ (expt x3 2) (* x1 x4)))
+ (+ (* x1 x3)
+ (+ (* x2 x4) (* x2 x3)))))))
+ (- (+ x1 (* 3 x3)) (+ x4 (* 3 x2))))
+ ;; Apply L'hopital's rule to get the zeros if 0/0
+ (* (op 0 1)
+ (/ (/ (- x4 x3) 2)
+ (sqrt (- (+ (* x2 x2)
+ (+ (* x3 x3) (* x1 x4)))
+ (+ (* x1 x3)
+ (+ (* x2 x4) (* x2 x3)))))))))
+ (list + -))))))))
+
+(define (bezier-min-max x1 y1 x2 y2 x3 y3 x4 y4)
+ (map (lambda (x)
+ (apply bezier-part-min-max x))
+ `((,x1 ,x2 ,x3 ,x4) (,y1 ,y2 ,y3 ,y4))))
+
+(define (line-min-max x1 y1 x2 y2)
+ (map (lambda (x)
+ (apply line-part-min-max x))
+ `((,x1 ,x2) (,y1 ,y2))))
+
+(define (path-min-max origin pointlist)
+
+ ((lambda (x)
+ (list
+ (reduce min +inf.0 (map caar x))
+ (reduce max -inf.0 (map cadar x))
+ (reduce min +inf.0 (map caadr x))
+ (reduce max -inf.0 (map cadadr x))))
+ (map (lambda (x)
+ (if (= (length x) 8)
+ (apply bezier-min-max x)
+ (apply line-min-max x)))
+ (map (lambda (x y)
+ (append (list (cadr (reverse x)) (car (reverse x))) y))
+ (append (list origin)
+ (reverse (cdr (reverse pointlist)))) pointlist))))
+
+(define-public (make-connected-path-stencil pointlist thickness
+ x-scale y-scale connect fill)
+ "Make a connected path described by the list @var{pointlist}, with
+thickness @var{thickness}, and scaled by @var{x-scale} in the X direction
+and @var{y-scale} in the Y direction. @var{connect} and @var{fill} are
+boolean arguments that specify if the path should be connected or filled,
+respectively."
+
+ ;; paths using this routine are designed to begin at point '(0 . 0)
+ (let* ((origin (list 0 0))
+ (boundlist (path-min-max origin pointlist))
+ ;; modify pointlist to scale the coordinates
+ (path (map (lambda (x)
+ (apply
+ (if (= 6 (length x))
+ (lambda (x1 x2 x3 x4 x5 x6)
+ (list 'curveto
+ (* x1 x-scale)
+ (* x2 y-scale)
+ (* x3 x-scale)
+ (* x4 y-scale)
+ (* x5 x-scale)
+ (* x6 y-scale)))
+ (lambda (x1 x2)
+ (list 'lineto
+ (* x1 x-scale)
+ (* x2 y-scale))))
+ x))
+ pointlist))
+ ;; a path must begin with a `moveto'
+ (prepend-origin (cons (cons 'moveto origin) path))
+ ;; if this path is connected, add closepath to the end
+ (final-path (if connect
+ (append prepend-origin (list '(closepath)))
+ prepend-origin))
+ (command-list (concatenate final-path)))
+ (ly:make-stencil
+ `(path ,thickness
+ `(,@',command-list)
+ 'round
+ 'round
+ ,(if fill #t #f))
+ (coord-translate
+ ((if (< x-scale 0) reverse-interval identity)
+ (cons (* x-scale (list-ref boundlist 0))
+ (* x-scale (list-ref boundlist 1))))
+ `(,(/ thickness -2) . ,(/ thickness 2)))
+ (coord-translate
+ ((if (< y-scale 0) reverse-interval identity)
+ (cons (* y-scale (list-ref boundlist 2))
+ (* y-scale (list-ref boundlist 3))))
+ `(,(/ thickness -2) . ,(/ thickness 2))))))
+
+(define-public (make-ellipse-stencil x-radius y-radius thickness fill)
+ "Make an ellipse of x@tie{}radius @var{x-radius}, y@tie{}radius
+@code{y-radius}, and thickness @var{thickness} with fill defined by
+@code{fill}."
+ (let*
+ ((x-out-radius (+ x-radius (/ thickness 2.0)))
+ (y-out-radius (+ y-radius (/ thickness 2.0))) )
+
+ (ly:make-stencil
+ (list 'ellipse x-radius y-radius thickness fill)
+ (cons (- x-out-radius) x-out-radius)
+ (cons (- y-out-radius) y-out-radius))))