Documentation
¶
Overview ¶
Package s1 implements types and functions for working with geometry in S¹ (circular geometry).
See ../s2 for a more detailed overview.
Index ¶
- Constants
- type Angle
- type ChordAngle
- func (c ChordAngle) Add(other ChordAngle) ChordAngle
- func (c ChordAngle) Angle() Angle
- func (c ChordAngle) Cos() float64
- func (c ChordAngle) Expanded(e float64) ChordAngle
- func (c ChordAngle) IsInfinity() bool
- func (c ChordAngle) MaxAngleError() float64
- func (c ChordAngle) MaxPointError() float64
- func (c ChordAngle) Predecessor() ChordAngle
- func (c ChordAngle) Sin() float64
- func (c ChordAngle) Sin2() float64
- func (c ChordAngle) Sub(other ChordAngle) ChordAngle
- func (c ChordAngle) Successor() ChordAngle
- func (c ChordAngle) Tan() float64
- type Interval
- func (i Interval) AddPoint(p float64) Interval
- func (i Interval) ApproxEqual(other Interval) bool
- func (i Interval) Center() float64
- func (i Interval) Complement() Interval
- func (i Interval) ComplementCenter() float64
- func (i Interval) Contains(p float64) bool
- func (i Interval) ContainsInterval(oi Interval) bool
- func (i Interval) DirectedHausdorffDistance(y Interval) Angle
- func (i Interval) Expanded(margin float64) Interval
- func (i Interval) InteriorContains(p float64) bool
- func (i Interval) InteriorContainsInterval(oi Interval) bool
- func (i Interval) InteriorIntersects(oi Interval) bool
- func (i Interval) Intersection(oi Interval) Interval
- func (i Interval) Intersects(oi Interval) bool
- func (i Interval) Invert() Interval
- func (i Interval) IsEmpty() bool
- func (i Interval) IsFull() bool
- func (i Interval) IsInverted() bool
- func (i Interval) IsValid() bool
- func (i Interval) Length() float64
- func (i Interval) Project(p float64) float64
- func (i Interval) String() string
- func (i Interval) Union(oi Interval) Interval
- Bugs
Examples ¶
Constants ¶
const ( Radian Angle = 1 Degree = (math.Pi / 180) * Radian E5 = 1e-5 * Degree E6 = 1e-6 * Degree E7 = 1e-7 * Degree )
Angle units.
const ( // NegativeChordAngle represents a chord angle smaller than the zero angle. // The only valid operations on a NegativeChordAngle are comparisons, // Angle conversions, and Successor/Predecessor. NegativeChordAngle = ChordAngle(-1) // RightChordAngle represents a chord angle of 90 degrees (a "right angle"). RightChordAngle = ChordAngle(2) // StraightChordAngle represents a chord angle of 180 degrees (a "straight angle"). // This is the maximum finite chord angle. StraightChordAngle = ChordAngle(4) )
Variables ¶
This section is empty.
Functions ¶
This section is empty.
Types ¶
type Angle ¶
type Angle float64
Angle represents a 1D angle. The internal representation is a double precision value in radians, so conversion to and from radians is exact. Conversions between E5, E6, E7, and Degrees are not always exact. For example, Degrees(3.1) is different from E6(3100000) or E7(31000000).
The following conversions between degrees and radians are exact:
Degree*180 == Radian*math.Pi Degree*(180/n) == Radian*(math.Pi/n) for n == 0..8
These identities hold when the arguments are scaled up or down by any power of 2. Some similar identities are also true, for example,
Degree*60 == Radian*(math.Pi/3)
But be aware that this type of identity does not hold in general. For example,
Degree*3 != Radian*(math.Pi/60)
Similarly, the conversion to radians means that (Angle(x)*Degree).Degrees() does not always equal x. For example,
(Angle(45*n)*Degree).Degrees() == 45*n for n == 0..8
but
(60*Degree).Degrees() != 60
When testing for equality, you should allow for numerical errors (ApproxEqual) or convert to discrete E5/E6/E7 values first.
func (Angle) ApproxEqual ¶
ApproxEqual reports whether the two angles are the same up to a small tolerance.
func (Angle) Normalized ¶
Normalized returns an equivalent angle in (-π, π].
type ChordAngle ¶
type ChordAngle float64
ChordAngle represents the angle subtended by a chord (i.e., the straight line segment connecting two points on the sphere). Its representation makes it very efficient for computing and comparing distances, but unlike Angle it is only capable of representing angles between 0 and π radians. Generally, ChordAngle should only be used in loops where many angles need to be calculated and compared. Otherwise it is simpler to use Angle.
ChordAngle loses some accuracy as the angle approaches π radians. There are several different ways to measure this error, including the representational error (i.e., how accurately ChordAngle can represent angles near π radians), the conversion error (i.e., how much precision is lost when an Angle is converted to an ChordAngle), and the measurement error (i.e., how accurate the ChordAngle(a, b) constructor is when the points A and B are separated by angles close to π radians). All of these errors differ by a small constant factor.
For the measurement error (which is the largest of these errors and also the most important in practice), let the angle between A and B be (π - x) radians, i.e. A and B are within "x" radians of being antipodal. The corresponding chord length is
r = 2 * sin((π - x) / 2) = 2 * cos(x / 2)
For values of x not close to π the relative error in the squared chord length is at most 4.5 * dblEpsilon (see MaxPointError below). The relative error in "r" is thus at most 2.25 * dblEpsilon ~= 5e-16. To convert this error into an equivalent angle, we have
|dr / dx| = sin(x / 2)
and therefore
|dx| = dr / sin(x / 2)
= 5e-16 * (2 * cos(x / 2)) / sin(x / 2)
= 1e-15 / tan(x / 2)
The maximum error is attained when
x = |dx| = 1e-15 / tan(x / 2) ~= 1e-15 / (x / 2) ~= sqrt(2e-15)
In summary, the measurement error for an angle (π - x) is at most
dx = min(1e-15 / tan(x / 2), sqrt(2e-15)) (~= min(2e-15 / x, sqrt(2e-15)) when x is small)
On the Earth's surface (assuming a radius of 6371km), this corresponds to the following worst-case measurement errors:
Accuracy: Unless antipodal to within: --------- --------------------------- 6.4 nanometers 10,000 km (90 degrees) 1 micrometer 81.2 kilometers 1 millimeter 81.2 meters 1 centimeter 8.12 meters 28.5 centimeters 28.5 centimeters
The representational and conversion errors referred to earlier are somewhat smaller than this. For example, maximum distance between adjacent representable ChordAngle values is only 13.5 cm rather than 28.5 cm. To see this, observe that the closest representable value to r^2 = 4 is r^2 = 4 * (1 - dblEpsilon / 2). Thus r = 2 * (1 - dblEpsilon / 4) and the angle between these two representable values is
x = 2 * acos(r / 2) = 2 * acos(1 - dblEpsilon / 4) ~= 2 * asin(sqrt(dblEpsilon / 2) ~= sqrt(2 * dblEpsilon) ~= 2.1e-8
which is 13.5 cm on the Earth's surface.
The worst case rounding error occurs when the value halfway between these two representable values is rounded up to 4. This halfway value is r^2 = (4 * (1 - dblEpsilon / 4)), thus r = 2 * (1 - dblEpsilon / 8) and the worst case rounding error is
x = 2 * acos(r / 2) = 2 * acos(1 - dblEpsilon / 8) ~= 2 * asin(sqrt(dblEpsilon / 4) ~= sqrt(dblEpsilon) ~= 1.5e-8
which is 9.5 cm on the Earth's surface.
func ChordAngleFromAngle ¶
func ChordAngleFromAngle(a Angle) ChordAngle
ChordAngleFromAngle returns a ChordAngle from the given Angle.
func ChordAngleFromSquaredLength ¶
func ChordAngleFromSquaredLength(length2 float64) ChordAngle
ChordAngleFromSquaredLength returns a ChordAngle from the squared chord length. Note that the argument is automatically clamped to a maximum of 4 to handle possible roundoff errors. The argument must be non-negative.
func InfChordAngle ¶
func InfChordAngle() ChordAngle
InfChordAngle returns a chord angle larger than any finite chord angle. The only valid operations on an InfChordAngle are comparisons, Angle conversions, and Successor/Predecessor.
func (ChordAngle) Add ¶
func (c ChordAngle) Add(other ChordAngle) ChordAngle
Add adds the other ChordAngle to this one and returns the resulting value. This method assumes the ChordAngles are not special.
func (ChordAngle) Angle ¶
func (c ChordAngle) Angle() Angle
Angle converts this ChordAngle to an Angle.
func (ChordAngle) Cos ¶
func (c ChordAngle) Cos() float64
Cos returns the cosine of this chord angle. This method is more efficient than converting to Angle and performing the computation.
func (ChordAngle) Expanded ¶
func (c ChordAngle) Expanded(e float64) ChordAngle
Expanded returns a new ChordAngle that has been adjusted by the given error bound (which can be positive or negative). Error should be the value returned by either MaxPointError or MaxAngleError. For example:
a := ChordAngleFromPoints(x, y) a1 := a.Expanded(a.MaxPointError())
func (ChordAngle) IsInfinity ¶
func (c ChordAngle) IsInfinity() bool
IsInfinity reports whether this ChordAngle is infinite.
func (ChordAngle) MaxAngleError ¶
func (c ChordAngle) MaxAngleError() float64
MaxAngleError returns the maximum error for a ChordAngle constructed as an Angle distance.
func (ChordAngle) MaxPointError ¶
func (c ChordAngle) MaxPointError() float64
MaxPointError returns the maximum error size for a ChordAngle constructed from 2 Points x and y, assuming that x and y are normalized to within the bounds guaranteed by s2.Point.Normalize. The error is defined with respect to the true distance after the points are projected to lie exactly on the sphere.
func (ChordAngle) Predecessor ¶
func (c ChordAngle) Predecessor() ChordAngle
Predecessor returns the largest representable ChordAngle less than this one.
Note the following special cases:
InfChordAngle.Predecessor == StraightChordAngle ChordAngle(0).Predecessor == NegativeChordAngle NegativeChordAngle.Predecessor == NegativeChordAngle
func (ChordAngle) Sin ¶
func (c ChordAngle) Sin() float64
Sin returns the sine of this chord angle. This method is more efficient than converting to Angle and performing the computation.
func (ChordAngle) Sin2 ¶
func (c ChordAngle) Sin2() float64
Sin2 returns the square of the sine of this chord angle. It is more efficient than Sin.
func (ChordAngle) Sub ¶
func (c ChordAngle) Sub(other ChordAngle) ChordAngle
Sub subtracts the other ChordAngle from this one and returns the resulting value. This method assumes the ChordAngles are not special.
func (ChordAngle) Successor ¶
func (c ChordAngle) Successor() ChordAngle
Successor returns the smallest representable ChordAngle larger than this one. This can be used to convert a "<" comparison to a "<=" comparison.
Note the following special cases:
NegativeChordAngle.Successor == 0 StraightChordAngle.Successor == InfChordAngle InfChordAngle.Successor == InfChordAngle
func (ChordAngle) Tan ¶
func (c ChordAngle) Tan() float64
Tan returns the tangent of this chord angle.
type Interval ¶
type Interval struct {
Lo, Hi float64
}
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.
func IntervalFromEndpoints ¶
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 IntervalFromPointPair ¶
IntervalFromPointPair returns the minimal interval containing the two given points. Both arguments must be in [-π,π].
func (Interval) AddPoint ¶
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 (Interval) ApproxEqual ¶
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 (Interval) Center ¶
Center returns the midpoint of the interval. It is undefined for full and empty intervals.
func (Interval) Complement ¶
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 (Interval) ComplementCenter ¶
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 (Interval) ContainsInterval ¶
ContainsInterval returns true iff the interval contains oi.
func (Interval) DirectedHausdorffDistance ¶
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.
Example ¶
package main
import (
"fmt"
"math"
"github.com/golang/geo/s1"
)
func main() {
// Small interval around the midpoints between quadrants, such that
// the center of each interval is offset slightly CCW from the midpoint.
mid := s1.IntervalFromEndpoints(math.Pi/2-0.01, math.Pi/2+0.02)
fmt.Println("empty to empty: ", s1.EmptyInterval().DirectedHausdorffDistance(s1.EmptyInterval()))
fmt.Println("empty to mid12: ", s1.EmptyInterval().DirectedHausdorffDistance(mid))
fmt.Println("mid12 to empty: ", mid.DirectedHausdorffDistance(s1.EmptyInterval()))
// Quadrant pair.
quad2 := s1.IntervalFromEndpoints(0, -math.Pi)
// Quadrant triple.
quad3 := s1.IntervalFromEndpoints(0, -math.Pi/2)
fmt.Println("quad12 to quad123 ", quad2.DirectedHausdorffDistance(quad3))
// An interval whose complement center is 0.
in := s1.IntervalFromEndpoints(3, -3)
ivs := []s1.Interval{s1.IntervalFromEndpoints(-0.1, 0.2), s1.IntervalFromEndpoints(0.1, 0.2), s1.IntervalFromEndpoints(-0.2, -0.1)}
for _, iv := range ivs {
fmt.Printf("dist from %v to in: %f\n", iv, iv.DirectedHausdorffDistance(in))
}
}
Output: empty to empty: 0.0000000 empty to mid12: 0.0000000 mid12 to empty: 180.0000000 quad12 to quad123 0.0000000 dist from [-0.1000000, 0.2000000] to in: 3.000000 dist from [0.1000000, 0.2000000] to in: 2.900000 dist from [-0.2000000, -0.1000000] to in: 2.900000
func (Interval) Expanded ¶
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 (Interval) InteriorContains ¶
InteriorContains returns true iff the interior of the interval contains p. Assumes p ∈ [-π,π].
func (Interval) InteriorContainsInterval ¶
InteriorContainsInterval returns true iff the interior of the interval contains oi.
func (Interval) InteriorIntersects ¶
InteriorIntersects returns true iff the interior of the interval contains any points in common with oi, including the latter's boundary.
func (Interval) Intersection ¶
Intersection returns the smallest interval that contains the intersection of the interval and oi.
func (Interval) Intersects ¶
Intersects returns true iff the interval contains any points in common with oi.
func (Interval) IsInverted ¶
IsInverted reports whether the interval is inverted; that is, whether Lo > Hi.
func (Interval) Length ¶
Length returns the length of the interval. The length of an empty interval is negative.
Notes ¶
Bugs ¶
The major differences from the C++ version are:
- no unsigned E5/E6/E7 methods
Source Files