gtsocial-umbx

interval.go (14140B)

---

 // Copyright 2014 Google Inc. All rights reserved.
 //
 // Licensed under the Apache License, Version 2.0 (the "License");
 // you may not use this file except in compliance with the License.
 // You may obtain a copy of the License at
 //
 //     http://www.apache.org/licenses/LICENSE-2.0
 //
 // Unless required by applicable law or agreed to in writing, software
 // distributed under the License is distributed on an "AS IS" BASIS,
 // WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 // See the License for the specific language governing permissions and
 // limitations under the License.
 
 package s1
 
 import (
 	"math"
 	"strconv"
 )
 
 // An Interval represents a closed interval on a unit circle (also known
 // as a 1-dimensional sphere). It is capable of representing the empty
 // interval (containing no points), the full interval (containing all
 // points), and zero-length intervals (containing a single point).
 //
 // Points are represented by the angle they make with the positive x-axis in
 // the range [-π, π]. An interval is represented by its lower and upper
 // bounds (both inclusive, since the interval is closed). The lower bound may
 // be greater than the upper bound, in which case the interval is "inverted"
 // (i.e. it passes through the point (-1, 0)).
 //
 // The point (-1, 0) has two valid representations, π and -π. The
 // normalized representation of this point is π, so that endpoints
 // of normal intervals are in the range (-π, π]. We normalize the latter to
 // the former in IntervalFromEndpoints. However, we take advantage of the point
 // -π to construct two special intervals:
 //   The full interval is [-π, π]
 //   The empty interval is [π, -π].
 //
 // Treat the exported fields as read-only.
 type Interval struct {
 	Lo, Hi float64
 }
 
 // IntervalFromEndpoints constructs a new interval from endpoints.
 // Both arguments must be in the range [-π,π]. This function allows inverted intervals
 // to be created.
 func IntervalFromEndpoints(lo, hi float64) Interval {
 	i := Interval{lo, hi}
 	if lo == -math.Pi && hi != math.Pi {
 		i.Lo = math.Pi
 	}
 	if hi == -math.Pi && lo != math.Pi {
 		i.Hi = math.Pi
 	}
 	return i
 }
 
 // IntervalFromPointPair returns the minimal interval containing the two given points.
 // Both arguments must be in [-π,π].
 func IntervalFromPointPair(a, b float64) Interval {
 	if a == -math.Pi {
 		a = math.Pi
 	}
 	if b == -math.Pi {
 		b = math.Pi
 	}
 	if positiveDistance(a, b) <= math.Pi {
 		return Interval{a, b}
 	}
 	return Interval{b, a}
 }
 
 // EmptyInterval returns an empty interval.
 func EmptyInterval() Interval { return Interval{math.Pi, -math.Pi} }
 
 // FullInterval returns a full interval.
 func FullInterval() Interval { return Interval{-math.Pi, math.Pi} }
 
 // IsValid reports whether the interval is valid.
 func (i Interval) IsValid() bool {
 	return (math.Abs(i.Lo) <= math.Pi && math.Abs(i.Hi) <= math.Pi &&
 		!(i.Lo == -math.Pi && i.Hi != math.Pi) &&
 		!(i.Hi == -math.Pi && i.Lo != math.Pi))
 }
 
 // IsFull reports whether the interval is full.
 func (i Interval) IsFull() bool { return i.Lo == -math.Pi && i.Hi == math.Pi }
 
 // IsEmpty reports whether the interval is empty.
 func (i Interval) IsEmpty() bool { return i.Lo == math.Pi && i.Hi == -math.Pi }
 
 // IsInverted reports whether the interval is inverted; that is, whether Lo > Hi.
 func (i Interval) IsInverted() bool { return i.Lo > i.Hi }
 
 // Invert returns the interval with endpoints swapped.
 func (i Interval) Invert() Interval {
 	return Interval{i.Hi, i.Lo}
 }
 
 // Center returns the midpoint of the interval.
 // It is undefined for full and empty intervals.
 func (i Interval) Center() float64 {
 	c := 0.5 * (i.Lo + i.Hi)
 	if !i.IsInverted() {
 		return c
 	}
 	if c <= 0 {
 		return c + math.Pi
 	}
 	return c - math.Pi
 }
 
 // Length returns the length of the interval.
 // The length of an empty interval is negative.
 func (i Interval) Length() float64 {
 	l := i.Hi - i.Lo
 	if l >= 0 {
 		return l
 	}
 	l += 2 * math.Pi
 	if l > 0 {
 		return l
 	}
 	return -1
 }
 
 // Assumes p ∈ (-π,π].
 func (i Interval) fastContains(p float64) bool {
 	if i.IsInverted() {
 		return (p >= i.Lo || p <= i.Hi) && !i.IsEmpty()
 	}
 	return p >= i.Lo && p <= i.Hi
 }
 
 // Contains returns true iff the interval contains p.
 // Assumes p ∈ [-π,π].
 func (i Interval) Contains(p float64) bool {
 	if p == -math.Pi {
 		p = math.Pi
 	}
 	return i.fastContains(p)
 }
 
 // ContainsInterval returns true iff the interval contains oi.
 func (i Interval) ContainsInterval(oi Interval) bool {
 	if i.IsInverted() {
 		if oi.IsInverted() {
 			return oi.Lo >= i.Lo && oi.Hi <= i.Hi
 		}
 		return (oi.Lo >= i.Lo || oi.Hi <= i.Hi) && !i.IsEmpty()
 	}
 	if oi.IsInverted() {
 		return i.IsFull() || oi.IsEmpty()
 	}
 	return oi.Lo >= i.Lo && oi.Hi <= i.Hi
 }
 
 // InteriorContains returns true iff the interior of the interval contains p.
 // Assumes p ∈ [-π,π].
 func (i Interval) InteriorContains(p float64) bool {
 	if p == -math.Pi {
 		p = math.Pi
 	}
 	if i.IsInverted() {
 		return p > i.Lo || p < i.Hi
 	}
 	return (p > i.Lo && p < i.Hi) || i.IsFull()
 }
 
 // InteriorContainsInterval returns true iff the interior of the interval contains oi.
 func (i Interval) InteriorContainsInterval(oi Interval) bool {
 	if i.IsInverted() {
 		if oi.IsInverted() {
 			return (oi.Lo > i.Lo && oi.Hi < i.Hi) || oi.IsEmpty()
 		}
 		return oi.Lo > i.Lo || oi.Hi < i.Hi
 	}
 	if oi.IsInverted() {
 		return i.IsFull() || oi.IsEmpty()
 	}
 	return (oi.Lo > i.Lo && oi.Hi < i.Hi) || i.IsFull()
 }
 
 // Intersects returns true iff the interval contains any points in common with oi.
 func (i Interval) Intersects(oi Interval) bool {
 	if i.IsEmpty() || oi.IsEmpty() {
 		return false
 	}
 	if i.IsInverted() {
 		return oi.IsInverted() || oi.Lo <= i.Hi || oi.Hi >= i.Lo
 	}
 	if oi.IsInverted() {
 		return oi.Lo <= i.Hi || oi.Hi >= i.Lo
 	}
 	return oi.Lo <= i.Hi && oi.Hi >= i.Lo
 }
 
 // InteriorIntersects returns true iff the interior of the interval contains any points in common with oi, including the latter's boundary.
 func (i Interval) InteriorIntersects(oi Interval) bool {
 	if i.IsEmpty() || oi.IsEmpty() || i.Lo == i.Hi {
 		return false
 	}
 	if i.IsInverted() {
 		return oi.IsInverted() || oi.Lo < i.Hi || oi.Hi > i.Lo
 	}
 	if oi.IsInverted() {
 		return oi.Lo < i.Hi || oi.Hi > i.Lo
 	}
 	return (oi.Lo < i.Hi && oi.Hi > i.Lo) || i.IsFull()
 }
 
 // Compute distance from a to b in [0,2π], in a numerically stable way.
 func positiveDistance(a, b float64) float64 {
 	d := b - a
 	if d >= 0 {
 		return d
 	}
 	return (b + math.Pi) - (a - math.Pi)
 }
 
 // Union returns the smallest interval that contains both the interval and oi.
 func (i Interval) Union(oi Interval) Interval {
 	if oi.IsEmpty() {
 		return i
 	}
 	if i.fastContains(oi.Lo) {
 		if i.fastContains(oi.Hi) {
 			// Either oi ⊂ i, or i ∪ oi is the full interval.
 			if i.ContainsInterval(oi) {
 				return i
 			}
 			return FullInterval()
 		}
 		return Interval{i.Lo, oi.Hi}
 	}
 	if i.fastContains(oi.Hi) {
 		return Interval{oi.Lo, i.Hi}
 	}
 
 	// Neither endpoint of oi is in i. Either i ⊂ oi, or i and oi are disjoint.
 	if i.IsEmpty() || oi.fastContains(i.Lo) {
 		return oi
 	}
 
 	// This is the only hard case where we need to find the closest pair of endpoints.
 	if positiveDistance(oi.Hi, i.Lo) < positiveDistance(i.Hi, oi.Lo) {
 		return Interval{oi.Lo, i.Hi}
 	}
 	return Interval{i.Lo, oi.Hi}
 }
 
 // Intersection returns the smallest interval that contains the intersection of the interval and oi.
 func (i Interval) Intersection(oi Interval) Interval {
 	if oi.IsEmpty() {
 		return EmptyInterval()
 	}
 	if i.fastContains(oi.Lo) {
 		if i.fastContains(oi.Hi) {
 			// Either oi ⊂ i, or i and oi intersect twice. Neither are empty.
 			// In the first case we want to return i (which is shorter than oi).
 			// In the second case one of them is inverted, and the smallest interval
 			// that covers the two disjoint pieces is the shorter of i and oi.
 			// We thus want to pick the shorter of i and oi in both cases.
 			if oi.Length() < i.Length() {
 				return oi
 			}
 			return i
 		}
 		return Interval{oi.Lo, i.Hi}
 	}
 	if i.fastContains(oi.Hi) {
 		return Interval{i.Lo, oi.Hi}
 	}
 
 	// Neither endpoint of oi is in i. Either i ⊂ oi, or i and oi are disjoint.
 	if oi.fastContains(i.Lo) {
 		return i
 	}
 	return EmptyInterval()
 }
 
 // AddPoint returns the interval expanded by the minimum amount necessary such
 // that it contains the given point "p" (an angle in the range [-π, π]).
 func (i Interval) AddPoint(p float64) Interval {
 	if math.Abs(p) > math.Pi {
 		return i
 	}
 	if p == -math.Pi {
 		p = math.Pi
 	}
 	if i.fastContains(p) {
 		return i
 	}
 	if i.IsEmpty() {
 		return Interval{p, p}
 	}
 	if positiveDistance(p, i.Lo) < positiveDistance(i.Hi, p) {
 		return Interval{p, i.Hi}
 	}
 	return Interval{i.Lo, p}
 }
 
 // Define the maximum rounding error for arithmetic operations. Depending on the
 // platform the mantissa precision may be different than others, so we choose to
 // use specific values to be consistent across all.
 // The values come from the C++ implementation.
 var (
 	// epsilon is a small number that represents a reasonable level of noise between two
 	// values that can be considered to be equal.
 	epsilon = 1e-15
 	// dblEpsilon is a smaller number for values that require more precision.
 	dblEpsilon = 2.220446049e-16
 )
 
 // Expanded returns an interval that has been expanded on each side by margin.
 // If margin is negative, then the function shrinks the interval on
 // each side by margin instead. The resulting interval may be empty or
 // full. Any expansion (positive or negative) of a full interval remains
 // full, and any expansion of an empty interval remains empty.
 func (i Interval) Expanded(margin float64) Interval {
 	if margin >= 0 {
 		if i.IsEmpty() {
 			return i
 		}
 		// Check whether this interval will be full after expansion, allowing
 		// for a rounding error when computing each endpoint.
 		if i.Length()+2*margin+2*dblEpsilon >= 2*math.Pi {
 			return FullInterval()
 		}
 	} else {
 		if i.IsFull() {
 			return i
 		}
 		// Check whether this interval will be empty after expansion, allowing
 		// for a rounding error when computing each endpoint.
 		if i.Length()+2*margin-2*dblEpsilon <= 0 {
 			return EmptyInterval()
 		}
 	}
 	result := IntervalFromEndpoints(
 		math.Remainder(i.Lo-margin, 2*math.Pi),
 		math.Remainder(i.Hi+margin, 2*math.Pi),
 	)
 	if result.Lo <= -math.Pi {
 		result.Lo = math.Pi
 	}
 	return result
 }
 
 // ApproxEqual reports whether this interval can be transformed into the given
 // interval by moving each endpoint by at most ε, without the
 // endpoints crossing (which would invert the interval). Empty and full
 // intervals are considered to start at an arbitrary point on the unit circle,
 // so any interval with (length <= 2*ε) matches the empty interval, and
 // any interval with (length >= 2*π - 2*ε) matches the full interval.
 func (i Interval) ApproxEqual(other Interval) bool {
 	// Full and empty intervals require special cases because the endpoints
 	// are considered to be positioned arbitrarily.
 	if i.IsEmpty() {
 		return other.Length() <= 2*epsilon
 	}
 	if other.IsEmpty() {
 		return i.Length() <= 2*epsilon
 	}
 	if i.IsFull() {
 		return other.Length() >= 2*(math.Pi-epsilon)
 	}
 	if other.IsFull() {
 		return i.Length() >= 2*(math.Pi-epsilon)
 	}
 
 	// The purpose of the last test below is to verify that moving the endpoints
 	// does not invert the interval, e.g. [-1e20, 1e20] vs. [1e20, -1e20].
 	return (math.Abs(math.Remainder(other.Lo-i.Lo, 2*math.Pi)) <= epsilon &&
 		math.Abs(math.Remainder(other.Hi-i.Hi, 2*math.Pi)) <= epsilon &&
 		math.Abs(i.Length()-other.Length()) <= 2*epsilon)
 
 }
 
 func (i Interval) String() string {
 	// like "[%.7f, %.7f]"
 	return "[" + strconv.FormatFloat(i.Lo, 'f', 7, 64) + ", " + strconv.FormatFloat(i.Hi, 'f', 7, 64) + "]"
 }
 
 // Complement returns the complement of the interior of the interval. An interval and
 // its complement have the same boundary but do not share any interior
 // values. The complement operator is not a bijection, since the complement
 // of a singleton interval (containing a single value) is the same as the
 // complement of an empty interval.
 func (i Interval) Complement() Interval {
 	if i.Lo == i.Hi {
 		// Singleton. The interval just contains a single point.
 		return FullInterval()
 	}
 	// Handles empty and full.
 	return Interval{i.Hi, i.Lo}
 }
 
 // ComplementCenter returns the midpoint of the complement of the interval. For full and empty
 // intervals, the result is arbitrary. For a singleton interval (containing a
 // single point), the result is its antipodal point on S1.
 func (i Interval) ComplementCenter() float64 {
 	if i.Lo != i.Hi {
 		return i.Complement().Center()
 	}
 	// Singleton. The interval just contains a single point.
 	if i.Hi <= 0 {
 		return i.Hi + math.Pi
 	}
 	return i.Hi - math.Pi
 }
 
 // DirectedHausdorffDistance returns the Hausdorff distance to the given interval.
 // For two intervals i and y, this distance is defined by
 //     h(i, y) = max_{p in i} min_{q in y} d(p, q),
 // where d(.,.) is measured along S1.
 func (i Interval) DirectedHausdorffDistance(y Interval) Angle {
 	if y.ContainsInterval(i) {
 		return 0 // This includes the case i is empty.
 	}
 	if y.IsEmpty() {
 		return Angle(math.Pi) // maximum possible distance on s1.
 	}
 	yComplementCenter := y.ComplementCenter()
 	if i.Contains(yComplementCenter) {
 		return Angle(positiveDistance(y.Hi, yComplementCenter))
 	}
 
 	// The Hausdorff distance is realized by either two i.Hi endpoints or two
 	// i.Lo endpoints, whichever is farther apart.
 	hiHi := 0.0
 	if IntervalFromEndpoints(y.Hi, yComplementCenter).Contains(i.Hi) {
 		hiHi = positiveDistance(y.Hi, i.Hi)
 	}
 
 	loLo := 0.0
 	if IntervalFromEndpoints(yComplementCenter, y.Lo).Contains(i.Lo) {
 		loLo = positiveDistance(i.Lo, y.Lo)
 	}
 
 	return Angle(math.Max(hiHi, loLo))
 }
 
 // Project returns the closest point in the interval to the given point p.
 // The interval must be non-empty.
 func (i Interval) Project(p float64) float64 {
 	if p == -math.Pi {
 		p = math.Pi
 	}
 	if i.fastContains(p) {
 		return p
 	}
 	// Compute distance from p to each endpoint.
 	dlo := positiveDistance(p, i.Lo)
 	dhi := positiveDistance(i.Hi, p)
 	if dlo < dhi {
 		return i.Lo
 	}
 	return i.Hi
 }