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