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