gtsocial-umbx

edge_tessellator.go (14380B)

---

 // Copyright 2018 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 s2
 
 import (
 	"github.com/golang/geo/r2"
 	"github.com/golang/geo/s1"
 )
 
 // Tessellation is implemented by subdividing the edge until the estimated
 // maximum error is below the given tolerance. Estimating error is a hard
 // problem, especially when the only methods available are point evaluation of
 // the projection and its inverse. (These are the only methods that
 // Projection provides, which makes it easier and less error-prone to
 // implement new projections.)
 //
 // One technique that significantly increases robustness is to treat the
 // geodesic and projected edges as parametric curves rather than geometric ones.
 // Given a spherical edge AB and a projection p:S2->R2, let f(t) be the
 // normalized arc length parametrization of AB and let g(t) be the normalized
 // arc length parameterization of the projected edge p(A)p(B). (In other words,
 // f(0)=A, f(1)=B, g(0)=p(A), g(1)=p(B).)  We now define the geometric error as
 // the maximum distance from the point p^-1(g(t)) to the geodesic edge AB for
 // any t in [0,1], where p^-1 denotes the inverse projection. In other words,
 // the geometric error is the maximum distance from any point on the projected
 // edge (mapped back onto the sphere) to the geodesic edge AB. On the other
 // hand we define the parametric error as the maximum distance between the
 // points f(t) and p^-1(g(t)) for any t in [0,1], i.e. the maximum distance
 // (measured on the sphere) between the geodesic and projected points at the
 // same interpolation fraction t.
 //
 // The easiest way to estimate the parametric error is to simply evaluate both
 // edges at their midpoints and measure the distance between them (the "midpoint
 // method"). This is very fast and works quite well for most edges, however it
 // has one major drawback: it doesn't handle points of inflection (i.e., points
 // where the curvature changes sign). For example, edges in the Mercator and
 // Plate Carree projections always curve towards the equator relative to the
 // corresponding geodesic edge, so in these projections there is a point of
 // inflection whenever the projected edge crosses the equator. The worst case
 // occurs when the edge endpoints have different longitudes but the same
 // absolute latitude, since in that case the error is non-zero but the edges
 // have exactly the same midpoint (on the equator).
 //
 // One solution to this problem is to split the input edges at all inflection
 // points (i.e., along the equator in the case of the Mercator and Plate Carree
 // projections). However for general projections these inflection points can
 // occur anywhere on the sphere (e.g., consider the Transverse Mercator
 // projection). This could be addressed by adding methods to the S2Projection
 // interface to split edges at inflection points but this would make it harder
 // and more error-prone to implement new projections.
 //
 // Another problem with this approach is that the midpoint method sometimes
 // underestimates the true error even when edges do not cross the equator.
 // For the Plate Carree and Mercator projections, the midpoint method can
 // underestimate the error by up to 3%.
 //
 // Both of these problems can be solved as follows. We assume that the error
 // can be modeled as a convex combination of two worst-case functions, one
 // where the error is maximized at the edge midpoint and another where the
 // error is *minimized* (i.e., zero) at the edge midpoint. For example, we
 // could choose these functions as:
 //
 //    E1(x) = 1 - x^2
 //    E2(x) = x * (1 - x^2)
 //
 // where for convenience we use an interpolation parameter "x" in the range
 // [-1, 1] rather than the original "t" in the range [0, 1]. Note that both
 // error functions must have roots at x = {-1, 1} since the error must be zero
 // at the edge endpoints. E1 is simply a parabola whose maximum value is 1
 // attained at x = 0, while E2 is a cubic with an additional root at x = 0,
 // and whose maximum value is 2 * sqrt(3) / 9 attained at x = 1 / sqrt(3).
 //
 // Next, it is convenient to scale these functions so that the both have a
 // maximum value of 1. E1 already satisfies this requirement, and we simply
 // redefine E2 as
 //
 //   E2(x) = x * (1 - x^2) / (2 * sqrt(3) / 9)
 //
 // Now define x0 to be the point where these two functions intersect, i.e. the
 // point in the range (-1, 1) where E1(x0) = E2(x0). This value has the very
 // convenient property that if we evaluate the actual error E(x0), then the
 // maximum error on the entire interval [-1, 1] is bounded by
 //
 //   E(x) <= E(x0) / E1(x0)
 //
 // since whether the error is modeled using E1 or E2, the resulting function
 // has the same maximum value (namely E(x0) / E1(x0)). If it is modeled as
 // some other convex combination of E1 and E2, the maximum value can only
 // decrease.
 //
 // Finally, since E2 is not symmetric about the y-axis, we must also allow for
 // the possibility that the error is a convex combination of E1 and -E2. This
 // can be handled by evaluating the error at E(-x0) as well, and then
 // computing the final error bound as
 //
 //   E(x) <= max(E(x0), E(-x0)) / E1(x0) .
 //
 // Effectively, this method is simply evaluating the error at two points about
 // 1/3 and 2/3 of the way along the edges, and then scaling the maximum of
 // these two errors by a constant factor. Intuitively, the reason this works
 // is that if the two edges cross somewhere in the interior, then at least one
 // of these points will be far from the crossing.
 //
 // The actual algorithm implemented below has some additional refinements.
 // First, edges longer than 90 degrees are always subdivided; this avoids
 // various unusual situations that can happen with very long edges, and there
 // is really no reason to avoid adding vertices to edges that are so long.
 //
 // Second, the error function E1 above needs to be modified to take into
 // account spherical distortions. (It turns out that spherical distortions are
 // beneficial in the case of E2, i.e. they only make its error estimates
 // slightly more conservative.)  To do this, we model E1 as the maximum error
 // in a Plate Carree edge of length 90 degrees or less. This turns out to be
 // an edge from 45:-90 to 45:90 (in lat:lng format). The corresponding error
 // as a function of "x" in the range [-1, 1] can be computed as the distance
 // between the Plate Caree edge point (45, 90 * x) and the geodesic
 // edge point (90 - 45 * abs(x), 90 * sgn(x)). Using the Haversine formula,
 // the corresponding function E1 (normalized to have a maximum value of 1) is:
 //
 //   E1(x) =
 //     asin(sqrt(sin(Pi / 8 * (1 - x)) ^ 2 +
 //               sin(Pi / 4 * (1 - x)) ^ 2 * cos(Pi / 4) * sin(Pi / 4 * x))) /
 //     asin(sqrt((1 - 1 / sqrt(2)) / 2))
 //
 // Note that this function does not need to be evaluated at runtime, it
 // simply affects the calculation of the value x0 where E1(x0) = E2(x0)
 // and the corresponding scaling factor C = 1 / E1(x0).
 //
 // ------------------------------------------------------------------
 //
 // In the case of the Mercator and Plate Carree projections this strategy
 // produces a conservative upper bound (verified using 10 million random
 // edges). Furthermore the bound is nearly tight; the scaling constant is
 // C = 1.19289, whereas the maximum observed value was 1.19254.
 //
 // Compared to the simpler midpoint evaluation method, this strategy requires
 // more function evaluations (currently twice as many, but with a smarter
 // tessellation algorithm it will only be 50% more). It also results in a
 // small amount of additional tessellation (about 1.5%) compared to the
 // midpoint method, but this is due almost entirely to the fact that the
 // midpoint method does not yield conservative error estimates.
 //
 // For random edges with a tolerance of 1 meter, the expected amount of
 // overtessellation is as follows:
 //
 //                   Midpoint Method    Cubic Method
 //   Plate Carree               1.8%            3.0%
 //   Mercator                  15.8%           17.4%
 
 const (
 	// tessellationInterpolationFraction is the fraction at which the two edges
 	// are evaluated in order to measure the error between them. (Edges are
 	// evaluated at two points measured this fraction from either end.)
 	tessellationInterpolationFraction = 0.31215691082248312
 	tessellationScaleFactor           = 0.83829992569888509
 
 	// minTessellationTolerance is the minimum supported tolerance (which
 	// corresponds to a distance less than 1 micrometer on the Earth's
 	// surface, but is still much larger than the expected projection and
 	// interpolation errors).
 	minTessellationTolerance s1.Angle = 1e-13
 )
 
 // EdgeTessellator converts an edge in a given projection (e.g., Mercator) into
 // a chain of spherical geodesic edges such that the maximum distance between
 // the original edge and the geodesic edge chain is at most the requested
 // tolerance. Similarly, it can convert a spherical geodesic edge into a chain
 // of edges in a given 2D projection such that the maximum distance between the
 // geodesic edge and the chain of projected edges is at most the requested tolerance.
 //
 //   Method      | Input                  | Output
 //   ------------|------------------------|-----------------------
 //   Projected   | S2 geodesics           | Planar projected edges
 //   Unprojected | Planar projected edges | S2 geodesics
 type EdgeTessellator struct {
 	projection Projection
 
 	// The given tolerance scaled by a constant fraction so that it can be
 	// compared against the result returned by estimateMaxError.
 	scaledTolerance s1.ChordAngle
 }
 
 // NewEdgeTessellator creates a new edge tessellator for the given projection and tolerance.
 func NewEdgeTessellator(p Projection, tolerance s1.Angle) *EdgeTessellator {
 	return &EdgeTessellator{
 		projection:      p,
 		scaledTolerance: s1.ChordAngleFromAngle(maxAngle(tolerance, minTessellationTolerance)),
 	}
 }
 
 // AppendProjected converts the spherical geodesic edge AB to a chain of planar edges
 // in the given projection and returns the corresponding vertices.
 //
 // If the given projection has one or more coordinate axes that wrap, then
 // every vertex's coordinates will be as close as possible to the previous
 // vertex's coordinates. Note that this may yield vertices whose
 // coordinates are outside the usual range. For example, tessellating the
 // edge (0:170, 0:-170) (in lat:lng notation) yields (0:170, 0:190).
 func (e *EdgeTessellator) AppendProjected(a, b Point, vertices []r2.Point) []r2.Point {
 	pa := e.projection.Project(a)
 	if len(vertices) == 0 {
 		vertices = []r2.Point{pa}
 	} else {
 		pa = e.projection.WrapDestination(vertices[len(vertices)-1], pa)
 	}
 
 	pb := e.projection.Project(b)
 	return e.appendProjected(pa, a, pb, b, vertices)
 }
 
 // appendProjected splits a geodesic edge AB as necessary and returns the
 // projected vertices appended to the given vertices.
 //
 // The maximum recursion depth is (math.Pi / minTessellationTolerance) < 45
 func (e *EdgeTessellator) appendProjected(pa r2.Point, a Point, pbIn r2.Point, b Point, vertices []r2.Point) []r2.Point {
 	pb := e.projection.WrapDestination(pa, pbIn)
 	if e.estimateMaxError(pa, a, pb, b) <= e.scaledTolerance {
 		return append(vertices, pb)
 	}
 
 	mid := Point{a.Add(b.Vector).Normalize()}
 	pmid := e.projection.WrapDestination(pa, e.projection.Project(mid))
 	vertices = e.appendProjected(pa, a, pmid, mid, vertices)
 	return e.appendProjected(pmid, mid, pb, b, vertices)
 }
 
 // AppendUnprojected converts the planar edge AB in the given projection to a chain of
 // spherical geodesic edges and returns the vertices.
 //
 // Note that to construct a Loop, you must eliminate the duplicate first and last
 // vertex. Note also that if the given projection involves coordinate wrapping
 // (e.g. across the 180 degree meridian) then the first and last vertices may not
 // be exactly the same.
 func (e *EdgeTessellator) AppendUnprojected(pa, pb r2.Point, vertices []Point) []Point {
 	a := e.projection.Unproject(pa)
 	b := e.projection.Unproject(pb)
 
 	if len(vertices) == 0 {
 		vertices = []Point{a}
 	}
 
 	// Note that coordinate wrapping can create a small amount of error. For
 	// example in the edge chain "0:-175, 0:179, 0:-177", the first edge is
 	// transformed into "0:-175, 0:-181" while the second is transformed into
 	// "0:179, 0:183". The two coordinate pairs for the middle vertex
 	// ("0:-181" and "0:179") may not yield exactly the same S2Point.
 	return e.appendUnprojected(pa, a, pb, b, vertices)
 }
 
 // appendUnprojected interpolates a projected edge and appends the corresponding
 // points on the sphere.
 func (e *EdgeTessellator) appendUnprojected(pa r2.Point, a Point, pbIn r2.Point, b Point, vertices []Point) []Point {
 	pb := e.projection.WrapDestination(pa, pbIn)
 	if e.estimateMaxError(pa, a, pb, b) <= e.scaledTolerance {
 		return append(vertices, b)
 	}
 
 	pmid := e.projection.Interpolate(0.5, pa, pb)
 	mid := e.projection.Unproject(pmid)
 
 	vertices = e.appendUnprojected(pa, a, pmid, mid, vertices)
 	return e.appendUnprojected(pmid, mid, pb, b, vertices)
 }
 
 func (e *EdgeTessellator) estimateMaxError(pa r2.Point, a Point, pb r2.Point, b Point) s1.ChordAngle {
 	// See the algorithm description at the top of this file.
 	// We always tessellate edges longer than 90 degrees on the sphere, since the
 	// approximation below is not robust enough to handle such edges.
 	if a.Dot(b.Vector) < -1e-14 {
 		return s1.InfChordAngle()
 	}
 	t1 := tessellationInterpolationFraction
 	t2 := 1 - tessellationInterpolationFraction
 	mid1 := Interpolate(t1, a, b)
 	mid2 := Interpolate(t2, a, b)
 	pmid1 := e.projection.Unproject(e.projection.Interpolate(t1, pa, pb))
 	pmid2 := e.projection.Unproject(e.projection.Interpolate(t2, pa, pb))
 	return maxChordAngle(ChordAngleBetweenPoints(mid1, pmid1), ChordAngleBetweenPoints(mid2, pmid2))
 }