428 lines
15 KiB
Go
428 lines
15 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 s2
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import (
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"math"
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"github.com/golang/geo/r3"
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)
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//
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// This file contains documentation of the various coordinate systems used
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// throughout the library. Most importantly, S2 defines a framework for
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// decomposing the unit sphere into a hierarchy of "cells". Each cell is a
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// quadrilateral bounded by four geodesics. The top level of the hierarchy is
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// obtained by projecting the six faces of a cube onto the unit sphere, and
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// lower levels are obtained by subdividing each cell into four children
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// recursively. Cells are numbered such that sequentially increasing cells
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// follow a continuous space-filling curve over the entire sphere. The
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// transformation is designed to make the cells at each level fairly uniform
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// in size.
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//
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////////////////////////// S2 Cell Decomposition /////////////////////////
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//
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// The following methods define the cube-to-sphere projection used by
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// the Cell decomposition.
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//
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// In the process of converting a latitude-longitude pair to a 64-bit cell
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// id, the following coordinate systems are used:
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//
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// (id)
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// An CellID is a 64-bit encoding of a face and a Hilbert curve position
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// on that face. The Hilbert curve position implicitly encodes both the
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// position of a cell and its subdivision level (see s2cellid.go).
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//
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// (face, i, j)
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// Leaf-cell coordinates. "i" and "j" are integers in the range
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// [0,(2**30)-1] that identify a particular leaf cell on the given face.
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// The (i, j) coordinate system is right-handed on each face, and the
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// faces are oriented such that Hilbert curves connect continuously from
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// one face to the next.
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//
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// (face, s, t)
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// Cell-space coordinates. "s" and "t" are real numbers in the range
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// [0,1] that identify a point on the given face. For example, the point
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// (s, t) = (0.5, 0.5) corresponds to the center of the top-level face
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// cell. This point is also a vertex of exactly four cells at each
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// subdivision level greater than zero.
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//
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// (face, si, ti)
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// Discrete cell-space coordinates. These are obtained by multiplying
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// "s" and "t" by 2**31 and rounding to the nearest unsigned integer.
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// Discrete coordinates lie in the range [0,2**31]. This coordinate
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// system can represent the edge and center positions of all cells with
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// no loss of precision (including non-leaf cells). In binary, each
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// coordinate of a level-k cell center ends with a 1 followed by
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// (30 - k) 0s. The coordinates of its edges end with (at least)
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// (31 - k) 0s.
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//
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// (face, u, v)
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// Cube-space coordinates in the range [-1,1]. To make the cells at each
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// level more uniform in size after they are projected onto the sphere,
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// we apply a nonlinear transformation of the form u=f(s), v=f(t).
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// The (u, v) coordinates after this transformation give the actual
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// coordinates on the cube face (modulo some 90 degree rotations) before
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// it is projected onto the unit sphere.
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//
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// (face, u, v, w)
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// Per-face coordinate frame. This is an extension of the (face, u, v)
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// cube-space coordinates that adds a third axis "w" in the direction of
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// the face normal. It is always a right-handed 3D coordinate system.
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// Cube-space coordinates can be converted to this frame by setting w=1,
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// while (u,v,w) coordinates can be projected onto the cube face by
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// dividing by w, i.e. (face, u/w, v/w).
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//
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// (x, y, z)
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// Direction vector (Point). Direction vectors are not necessarily unit
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// length, and are often chosen to be points on the biunit cube
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// [-1,+1]x[-1,+1]x[-1,+1]. They can be be normalized to obtain the
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// corresponding point on the unit sphere.
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//
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// (lat, lng)
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// Latitude and longitude (LatLng). Latitudes must be between -90 and
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// 90 degrees inclusive, and longitudes must be between -180 and 180
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// degrees inclusive.
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//
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// Note that the (i, j), (s, t), (si, ti), and (u, v) coordinate systems are
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// right-handed on all six faces.
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//
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//
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// There are a number of different projections from cell-space (s,t) to
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// cube-space (u,v): linear, quadratic, and tangent. They have the following
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// tradeoffs:
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//
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// Linear - This is the fastest transformation, but also produces the least
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// uniform cell sizes. Cell areas vary by a factor of about 5.2, with the
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// largest cells at the center of each face and the smallest cells in
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// the corners.
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//
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// Tangent - Transforming the coordinates via Atan makes the cell sizes
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// more uniform. The areas vary by a maximum ratio of 1.4 as opposed to a
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// maximum ratio of 5.2. However, each call to Atan is about as expensive
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// as all of the other calculations combined when converting from points to
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// cell ids, i.e. it reduces performance by a factor of 3.
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//
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// Quadratic - This is an approximation of the tangent projection that
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// is much faster and produces cells that are almost as uniform in size.
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// It is about 3 times faster than the tangent projection for converting
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// cell ids to points or vice versa. Cell areas vary by a maximum ratio of
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// about 2.1.
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//
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// Here is a table comparing the cell uniformity using each projection. Area
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// Ratio is the maximum ratio over all subdivision levels of the largest cell
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// area to the smallest cell area at that level, Edge Ratio is the maximum
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// ratio of the longest edge of any cell to the shortest edge of any cell at
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// the same level, and Diag Ratio is the ratio of the longest diagonal of
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// any cell to the shortest diagonal of any cell at the same level.
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//
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// Area Edge Diag
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// Ratio Ratio Ratio
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// -----------------------------------
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// Linear: 5.200 2.117 2.959
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// Tangent: 1.414 1.414 1.704
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// Quadratic: 2.082 1.802 1.932
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//
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// The worst-case cell aspect ratios are about the same with all three
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// projections. The maximum ratio of the longest edge to the shortest edge
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// within the same cell is about 1.4 and the maximum ratio of the diagonals
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// within the same cell is about 1.7.
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//
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// For Go we have chosen to use only the Quadratic approach. Other language
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// implementations may offer other choices.
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const (
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// maxSiTi is the maximum value of an si- or ti-coordinate.
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// It is one shift more than maxSize. The range of valid (si,ti)
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// values is [0..maxSiTi].
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maxSiTi = maxSize << 1
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)
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// siTiToST converts an si- or ti-value to the corresponding s- or t-value.
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// Value is capped at 1.0 because there is no DCHECK in Go.
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func siTiToST(si uint32) float64 {
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if si > maxSiTi {
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return 1.0
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}
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return float64(si) / float64(maxSiTi)
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}
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// stToSiTi converts the s- or t-value to the nearest si- or ti-coordinate.
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// The result may be outside the range of valid (si,ti)-values. Value of
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// 0.49999999999999994 (math.NextAfter(0.5, -1)), will be incorrectly rounded up.
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func stToSiTi(s float64) uint32 {
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if s < 0 {
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return uint32(s*maxSiTi - 0.5)
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}
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return uint32(s*maxSiTi + 0.5)
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}
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// stToUV converts an s or t value to the corresponding u or v value.
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// This is a non-linear transformation from [-1,1] to [-1,1] that
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// attempts to make the cell sizes more uniform.
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// This uses what the C++ version calls 'the quadratic transform'.
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func stToUV(s float64) float64 {
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if s >= 0.5 {
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return (1 / 3.) * (4*s*s - 1)
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}
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return (1 / 3.) * (1 - 4*(1-s)*(1-s))
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}
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// uvToST is the inverse of the stToUV transformation. Note that it
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// is not always true that uvToST(stToUV(x)) == x due to numerical
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// errors.
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func uvToST(u float64) float64 {
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if u >= 0 {
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return 0.5 * math.Sqrt(1+3*u)
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}
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return 1 - 0.5*math.Sqrt(1-3*u)
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}
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// face returns face ID from 0 to 5 containing the r. For points on the
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// boundary between faces, the result is arbitrary but deterministic.
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func face(r r3.Vector) int {
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f := r.LargestComponent()
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switch {
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case f == r3.XAxis && r.X < 0:
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f += 3
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case f == r3.YAxis && r.Y < 0:
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f += 3
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case f == r3.ZAxis && r.Z < 0:
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f += 3
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}
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return int(f)
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}
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// validFaceXYZToUV given a valid face for the given point r (meaning that
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// dot product of r with the face normal is positive), returns
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// the corresponding u and v values, which may lie outside the range [-1,1].
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func validFaceXYZToUV(face int, r r3.Vector) (float64, float64) {
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switch face {
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case 0:
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return r.Y / r.X, r.Z / r.X
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case 1:
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return -r.X / r.Y, r.Z / r.Y
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case 2:
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return -r.X / r.Z, -r.Y / r.Z
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case 3:
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return r.Z / r.X, r.Y / r.X
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case 4:
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return r.Z / r.Y, -r.X / r.Y
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}
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return -r.Y / r.Z, -r.X / r.Z
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}
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// xyzToFaceUV converts a direction vector (not necessarily unit length) to
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// (face, u, v) coordinates.
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func xyzToFaceUV(r r3.Vector) (f int, u, v float64) {
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f = face(r)
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u, v = validFaceXYZToUV(f, r)
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return f, u, v
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}
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// faceUVToXYZ turns face and UV coordinates into an unnormalized 3 vector.
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func faceUVToXYZ(face int, u, v float64) r3.Vector {
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switch face {
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case 0:
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return r3.Vector{1, u, v}
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case 1:
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return r3.Vector{-u, 1, v}
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case 2:
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return r3.Vector{-u, -v, 1}
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case 3:
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return r3.Vector{-1, -v, -u}
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case 4:
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return r3.Vector{v, -1, -u}
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default:
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return r3.Vector{v, u, -1}
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}
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}
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// faceXYZToUV returns the u and v values (which may lie outside the range
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// [-1, 1]) if the dot product of the point p with the given face normal is positive.
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func faceXYZToUV(face int, p Point) (u, v float64, ok bool) {
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switch face {
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case 0:
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if p.X <= 0 {
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return 0, 0, false
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}
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case 1:
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if p.Y <= 0 {
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return 0, 0, false
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}
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case 2:
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if p.Z <= 0 {
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return 0, 0, false
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}
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case 3:
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if p.X >= 0 {
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return 0, 0, false
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}
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case 4:
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if p.Y >= 0 {
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return 0, 0, false
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}
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default:
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if p.Z >= 0 {
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return 0, 0, false
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}
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}
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u, v = validFaceXYZToUV(face, p.Vector)
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return u, v, true
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}
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// faceXYZtoUVW transforms the given point P to the (u,v,w) coordinate frame of the given
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// face where the w-axis represents the face normal.
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func faceXYZtoUVW(face int, p Point) Point {
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// The result coordinates are simply the dot products of P with the (u,v,w)
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// axes for the given face (see faceUVWAxes).
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switch face {
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case 0:
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return Point{r3.Vector{p.Y, p.Z, p.X}}
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case 1:
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return Point{r3.Vector{-p.X, p.Z, p.Y}}
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case 2:
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return Point{r3.Vector{-p.X, -p.Y, p.Z}}
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case 3:
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return Point{r3.Vector{-p.Z, -p.Y, -p.X}}
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case 4:
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return Point{r3.Vector{-p.Z, p.X, -p.Y}}
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default:
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return Point{r3.Vector{p.Y, p.X, -p.Z}}
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}
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}
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// faceSiTiToXYZ transforms the (si, ti) coordinates to a (not necessarily
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// unit length) Point on the given face.
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func faceSiTiToXYZ(face int, si, ti uint32) Point {
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return Point{faceUVToXYZ(face, stToUV(siTiToST(si)), stToUV(siTiToST(ti)))}
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}
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// xyzToFaceSiTi transforms the (not necessarily unit length) Point to
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// (face, si, ti) coordinates and the level the Point is at.
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func xyzToFaceSiTi(p Point) (face int, si, ti uint32, level int) {
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face, u, v := xyzToFaceUV(p.Vector)
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si = stToSiTi(uvToST(u))
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ti = stToSiTi(uvToST(v))
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// If the levels corresponding to si,ti are not equal, then p is not a cell
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// center. The si,ti values of 0 and maxSiTi need to be handled specially
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// because they do not correspond to cell centers at any valid level; they
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// are mapped to level -1 by the code at the end.
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level = maxLevel - findLSBSetNonZero64(uint64(si|maxSiTi))
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if level < 0 || level != maxLevel-findLSBSetNonZero64(uint64(ti|maxSiTi)) {
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return face, si, ti, -1
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}
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// In infinite precision, this test could be changed to ST == SiTi. However,
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// due to rounding errors, uvToST(xyzToFaceUV(faceUVToXYZ(stToUV(...)))) is
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// not idempotent. On the other hand, the center is computed exactly the same
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// way p was originally computed (if it is indeed the center of a Cell);
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// the comparison can be exact.
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if p.Vector == faceSiTiToXYZ(face, si, ti).Normalize() {
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return face, si, ti, level
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}
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return face, si, ti, -1
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}
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// uNorm returns the right-handed normal (not necessarily unit length) for an
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// edge in the direction of the positive v-axis at the given u-value on
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// the given face. (This vector is perpendicular to the plane through
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// the sphere origin that contains the given edge.)
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func uNorm(face int, u float64) r3.Vector {
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switch face {
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case 0:
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return r3.Vector{u, -1, 0}
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case 1:
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return r3.Vector{1, u, 0}
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case 2:
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return r3.Vector{1, 0, u}
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case 3:
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return r3.Vector{-u, 0, 1}
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case 4:
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return r3.Vector{0, -u, 1}
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default:
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return r3.Vector{0, -1, -u}
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}
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}
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// vNorm returns the right-handed normal (not necessarily unit length) for an
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// edge in the direction of the positive u-axis at the given v-value on
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// the given face.
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func vNorm(face int, v float64) r3.Vector {
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switch face {
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case 0:
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return r3.Vector{-v, 0, 1}
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case 1:
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return r3.Vector{0, -v, 1}
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case 2:
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return r3.Vector{0, -1, -v}
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case 3:
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return r3.Vector{v, -1, 0}
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case 4:
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return r3.Vector{1, v, 0}
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default:
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return r3.Vector{1, 0, v}
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}
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}
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// faceUVWAxes are the U, V, and W axes for each face.
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var faceUVWAxes = [6][3]Point{
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{Point{r3.Vector{0, 1, 0}}, Point{r3.Vector{0, 0, 1}}, Point{r3.Vector{1, 0, 0}}},
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{Point{r3.Vector{-1, 0, 0}}, Point{r3.Vector{0, 0, 1}}, Point{r3.Vector{0, 1, 0}}},
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{Point{r3.Vector{-1, 0, 0}}, Point{r3.Vector{0, -1, 0}}, Point{r3.Vector{0, 0, 1}}},
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{Point{r3.Vector{0, 0, -1}}, Point{r3.Vector{0, -1, 0}}, Point{r3.Vector{-1, 0, 0}}},
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{Point{r3.Vector{0, 0, -1}}, Point{r3.Vector{1, 0, 0}}, Point{r3.Vector{0, -1, 0}}},
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{Point{r3.Vector{0, 1, 0}}, Point{r3.Vector{1, 0, 0}}, Point{r3.Vector{0, 0, -1}}},
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}
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// faceUVWFaces are the precomputed neighbors of each face.
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var faceUVWFaces = [6][3][2]int{
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{{4, 1}, {5, 2}, {3, 0}},
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{{0, 3}, {5, 2}, {4, 1}},
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{{0, 3}, {1, 4}, {5, 2}},
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{{2, 5}, {1, 4}, {0, 3}},
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{{2, 5}, {3, 0}, {1, 4}},
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{{4, 1}, {3, 0}, {2, 5}},
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}
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// uvwAxis returns the given axis of the given face.
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func uvwAxis(face, axis int) Point {
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return faceUVWAxes[face][axis]
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}
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// uvwFaces returns the face in the (u,v,w) coordinate system on the given axis
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// in the given direction.
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func uvwFace(face, axis, direction int) int {
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return faceUVWFaces[face][axis][direction]
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}
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// uAxis returns the u-axis for the given face.
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func uAxis(face int) Point {
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return uvwAxis(face, 0)
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}
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// vAxis returns the v-axis for the given face.
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func vAxis(face int) Point {
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return uvwAxis(face, 1)
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}
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// Return the unit-length normal for the given face.
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func unitNorm(face int) Point {
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return uvwAxis(face, 2)
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}
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