+ (let*
+ ((x-out-radius (+ x-radius (/ thick 2.0)))
+ (y-out-radius (+ y-radius (/ thick 2.0)))
+ (new-end-angle (angle-0-2pi (degrees->radians end-angle)))
+ (end-radius (ellipse-radius x-out-radius y-out-radius new-end-angle))
+ (new-start-angle (angle-0-2pi (degrees->radians start-angle)))
+ (start-radius (ellipse-radius x-out-radius y-out-radius new-start-angle))
+ (radius-list (make-radius-list x-out-radius y-out-radius))
+ (rectangular-end-radius (polar->rectangular end-radius end-angle))
+ (rectangular-start-radius (polar->rectangular start-radius start-angle))
+ (new-end-angle
+ (if (<= new-end-angle new-start-angle)
+ (+ TWO-PI new-end-angle)
+ new-end-angle))
+ (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)
+ (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 (eq? (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 (eq? 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 (apply list (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 (fold-right append '() final-path)))