+ (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))))