-type private SearchExtremum = Minimum | Maximum
-
-let private goldenSectionSearch (f: float -> float) (nbIter: int) (xmin: float) (xmax: float) (searchExtremum: SearchExtremum) : (float * float) =
- let gr = 1. / 1.6180339887498948482
- let mutable a = xmin
- let mutable b = xmax
- let mutable c = b - gr * (b - a)
- let mutable d = a + gr * (b - a)
-
- for i in 1 .. nbIter do
- let mutable fc = f c
- let mutable fd = f d
-
- if searchExtremum = Maximum
- then
- let tmp = fc
- fc <- fd
- fd <- tmp
-
- if fc < fd
- then
- b <- d
- d <- c
- c <- b - gr * (b - a)
- else
- a <- c
- c <- d
- d <- a + gr * (b - a)
-
- let x = (b + a) / 2.
- x, f x
-
-// Ellipse.A is always equal or greater than Ellipse.B.
-// Ellipse.Alpha is between 0 and Pi.
-let ellipse (p1x: float) (p1y: float) (m1: float) (p2x: float) (p2y: float) (m2: float) (p3x: float) (p3y: float) : Types.Ellipse option =
- let accuracy_extremum_search_1 = 10 // 3
- let accuracy_extremum_search_2 = 10 // 4
-
- // p3 as the referencial.
- let p1x = p1x - p3x
- let p1y = p1y - p3y
-
- let p2x = p2x - p3x
- let p2y = p2y - p3y
-
- // Convert to polar coordinates.
- let alpha1 = atan m1
- let alpha2 = atan m2
-
- let r1 = sqrt (p1x ** 2. + p1y ** 2.)
- let theta1 = atan2 p1y p1x
-
- let r2 = sqrt (p2x ** 2. + p2y ** 2.)
- let theta2 = atan2 p2y p2x
-
- let valid =
- 4. * sin (alpha1 - theta1) * (-r1 * sin (alpha1 - theta1) + r2 * sin (alpha1 - theta2)) *
- sin (alpha2 - theta2) * (-r1 * sin (alpha2 - theta1) + r2 * sin (alpha2 - theta2)) +
- r1 * r2 * sin (alpha1 - alpha2) ** 2. * sin (theta1 - theta2) ** 2. < 0.
-
- if valid
- then
- let r theta =
- (r1 * r2 * (r1 * (cos (alpha2 + theta - theta1 - theta2) - cos (alpha2 - theta) * cos (theta1 - theta2)) * sin (alpha1 - theta1) + r2 * (-cos (alpha1 + theta - theta1 - theta2) + cos (alpha1 - theta) * cos (theta1 - theta2)) * sin (alpha2 - theta2)) * sin (theta1 - theta2)) /
- (sin (alpha1 - theta1) * sin (alpha2 - theta2) * (r1 * sin (theta - theta1) - r2 * sin (theta - theta2)) ** 2. - r1 * r2 * sin (alpha1 - theta) * sin (alpha2 - theta) * sin (theta1 - theta2) ** 2.)
-
- let rabs = r >> abs
-
- // We search for an interval [theta_a, theta_b] and assume the function is unimodal in this interval.
- let thetaTan, _ = goldenSectionSearch rabs accuracy_extremum_search_1 0. Math.PI Maximum
- let rTan = r thetaTan
-
- let PTanx = rTan * cos thetaTan
- let PTany = rTan * sin thetaTan
-
- let d1a = tan alpha1
- let d1b = -d1a * p1x + p1y
-
- let d2a = tan alpha2
- let d2b = -d2a * p2x + p2y
-
- let d3a = -1. / tan thetaTan
- let d3b = -d3a * PTanx + PTany
-
- let Ux = -(d1b - d2b) / (d1a - d2a)
- let Uy = -(d2a * d1b - d1a * d2b) / (d1a - d2a)
-
- let Vx = -(d1b - d3b) / (d1a - d3a)
- let Vy = -(d3a * d1b - d1a * d3b) / (d1a - d3a)
-
- let Wx = p1x + (p2x - p1x) / 2.
- let Wy = p1y + (p2y - p1y) / 2.