gtsocial-umbx

stuv.go (14900B)

---

 // Copyright 2014 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 (
 	"math"
 
 	"github.com/golang/geo/r3"
 )
 
 //
 // This file contains documentation of the various coordinate systems used
 // throughout the library. Most importantly, S2 defines a framework for
 // decomposing the unit sphere into a hierarchy of "cells". Each cell is a
 // quadrilateral bounded by four geodesics. The top level of the hierarchy is
 // obtained by projecting the six faces of a cube onto the unit sphere, and
 // lower levels are obtained by subdividing each cell into four children
 // recursively. Cells are numbered such that sequentially increasing cells
 // follow a continuous space-filling curve over the entire sphere. The
 // transformation is designed to make the cells at each level fairly uniform
 // in size.
 //
 ////////////////////////// S2 Cell Decomposition /////////////////////////
 //
 // The following methods define the cube-to-sphere projection used by
 // the Cell decomposition.
 //
 // In the process of converting a latitude-longitude pair to a 64-bit cell
 // id, the following coordinate systems are used:
 //
 //  (id)
 //    An CellID is a 64-bit encoding of a face and a Hilbert curve position
 //    on that face. The Hilbert curve position implicitly encodes both the
 //    position of a cell and its subdivision level (see s2cellid.go).
 //
 //  (face, i, j)
 //    Leaf-cell coordinates. "i" and "j" are integers in the range
 //    [0,(2**30)-1] that identify a particular leaf cell on the given face.
 //    The (i, j) coordinate system is right-handed on each face, and the
 //    faces are oriented such that Hilbert curves connect continuously from
 //    one face to the next.
 //
 //  (face, s, t)
 //    Cell-space coordinates. "s" and "t" are real numbers in the range
 //    [0,1] that identify a point on the given face. For example, the point
 //    (s, t) = (0.5, 0.5) corresponds to the center of the top-level face
 //    cell. This point is also a vertex of exactly four cells at each
 //    subdivision level greater than zero.
 //
 //  (face, si, ti)
 //    Discrete cell-space coordinates. These are obtained by multiplying
 //    "s" and "t" by 2**31 and rounding to the nearest unsigned integer.
 //    Discrete coordinates lie in the range [0,2**31]. This coordinate
 //    system can represent the edge and center positions of all cells with
 //    no loss of precision (including non-leaf cells). In binary, each
 //    coordinate of a level-k cell center ends with a 1 followed by
 //    (30 - k) 0s. The coordinates of its edges end with (at least)
 //    (31 - k) 0s.
 //
 //  (face, u, v)
 //    Cube-space coordinates in the range [-1,1]. To make the cells at each
 //    level more uniform in size after they are projected onto the sphere,
 //    we apply a nonlinear transformation of the form u=f(s), v=f(t).
 //    The (u, v) coordinates after this transformation give the actual
 //    coordinates on the cube face (modulo some 90 degree rotations) before
 //    it is projected onto the unit sphere.
 //
 //  (face, u, v, w)
 //    Per-face coordinate frame. This is an extension of the (face, u, v)
 //    cube-space coordinates that adds a third axis "w" in the direction of
 //    the face normal. It is always a right-handed 3D coordinate system.
 //    Cube-space coordinates can be converted to this frame by setting w=1,
 //    while (u,v,w) coordinates can be projected onto the cube face by
 //    dividing by w, i.e. (face, u/w, v/w).
 //
 //  (x, y, z)
 //    Direction vector (Point). Direction vectors are not necessarily unit
 //    length, and are often chosen to be points on the biunit cube
 //    [-1,+1]x[-1,+1]x[-1,+1]. They can be be normalized to obtain the
 //    corresponding point on the unit sphere.
 //
 //  (lat, lng)
 //    Latitude and longitude (LatLng). Latitudes must be between -90 and
 //    90 degrees inclusive, and longitudes must be between -180 and 180
 //    degrees inclusive.
 //
 // Note that the (i, j), (s, t), (si, ti), and (u, v) coordinate systems are
 // right-handed on all six faces.
 //
 //
 // There are a number of different projections from cell-space (s,t) to
 // cube-space (u,v): linear, quadratic, and tangent. They have the following
 // tradeoffs:
 //
 //   Linear - This is the fastest transformation, but also produces the least
 //   uniform cell sizes. Cell areas vary by a factor of about 5.2, with the
 //   largest cells at the center of each face and the smallest cells in
 //   the corners.
 //
 //   Tangent - Transforming the coordinates via Atan makes the cell sizes
 //   more uniform. The areas vary by a maximum ratio of 1.4 as opposed to a
 //   maximum ratio of 5.2. However, each call to Atan is about as expensive
 //   as all of the other calculations combined when converting from points to
 //   cell ids, i.e. it reduces performance by a factor of 3.
 //
 //   Quadratic - This is an approximation of the tangent projection that
 //   is much faster and produces cells that are almost as uniform in size.
 //   It is about 3 times faster than the tangent projection for converting
 //   cell ids to points or vice versa. Cell areas vary by a maximum ratio of
 //   about 2.1.
 //
 // Here is a table comparing the cell uniformity using each projection. Area
 // Ratio is the maximum ratio over all subdivision levels of the largest cell
 // area to the smallest cell area at that level, Edge Ratio is the maximum
 // ratio of the longest edge of any cell to the shortest edge of any cell at
 // the same level, and Diag Ratio is the ratio of the longest diagonal of
 // any cell to the shortest diagonal of any cell at the same level.
 //
 //               Area    Edge    Diag
 //              Ratio   Ratio   Ratio
 // -----------------------------------
 // Linear:      5.200   2.117   2.959
 // Tangent:     1.414   1.414   1.704
 // Quadratic:   2.082   1.802   1.932
 //
 // The worst-case cell aspect ratios are about the same with all three
 // projections. The maximum ratio of the longest edge to the shortest edge
 // within the same cell is about 1.4 and the maximum ratio of the diagonals
 // within the same cell is about 1.7.
 //
 // For Go we have chosen to use only the Quadratic approach. Other language
 // implementations may offer other choices.
 
 const (
 	// maxSiTi is the maximum value of an si- or ti-coordinate.
 	// It is one shift more than maxSize. The range of valid (si,ti)
 	// values is [0..maxSiTi].
 	maxSiTi = maxSize << 1
 )
 
 // siTiToST converts an si- or ti-value to the corresponding s- or t-value.
 // Value is capped at 1.0 because there is no DCHECK in Go.
 func siTiToST(si uint32) float64 {
 	if si > maxSiTi {
 		return 1.0
 	}
 	return float64(si) / float64(maxSiTi)
 }
 
 // stToSiTi converts the s- or t-value to the nearest si- or ti-coordinate.
 // The result may be outside the range of valid (si,ti)-values. Value of
 // 0.49999999999999994 (math.NextAfter(0.5, -1)), will be incorrectly rounded up.
 func stToSiTi(s float64) uint32 {
 	if s < 0 {
 		return uint32(s*maxSiTi - 0.5)
 	}
 	return uint32(s*maxSiTi + 0.5)
 }
 
 // stToUV converts an s or t value to the corresponding u or v value.
 // This is a non-linear transformation from [-1,1] to [-1,1] that
 // attempts to make the cell sizes more uniform.
 // This uses what the C++ version calls 'the quadratic transform'.
 func stToUV(s float64) float64 {
 	if s >= 0.5 {
 		return (1 / 3.) * (4*s*s - 1)
 	}
 	return (1 / 3.) * (1 - 4*(1-s)*(1-s))
 }
 
 // uvToST is the inverse of the stToUV transformation. Note that it
 // is not always true that uvToST(stToUV(x)) == x due to numerical
 // errors.
 func uvToST(u float64) float64 {
 	if u >= 0 {
 		return 0.5 * math.Sqrt(1+3*u)
 	}
 	return 1 - 0.5*math.Sqrt(1-3*u)
 }
 
 // face returns face ID from 0 to 5 containing the r. For points on the
 // boundary between faces, the result is arbitrary but deterministic.
 func face(r r3.Vector) int {
 	f := r.LargestComponent()
 	switch {
 	case f == r3.XAxis && r.X < 0:
 		f += 3
 	case f == r3.YAxis && r.Y < 0:
 		f += 3
 	case f == r3.ZAxis && r.Z < 0:
 		f += 3
 	}
 	return int(f)
 }
 
 // validFaceXYZToUV given a valid face for the given point r (meaning that
 // dot product of r with the face normal is positive), returns
 // the corresponding u and v values, which may lie outside the range [-1,1].
 func validFaceXYZToUV(face int, r r3.Vector) (float64, float64) {
 	switch face {
 	case 0:
 		return r.Y / r.X, r.Z / r.X
 	case 1:
 		return -r.X / r.Y, r.Z / r.Y
 	case 2:
 		return -r.X / r.Z, -r.Y / r.Z
 	case 3:
 		return r.Z / r.X, r.Y / r.X
 	case 4:
 		return r.Z / r.Y, -r.X / r.Y
 	}
 	return -r.Y / r.Z, -r.X / r.Z
 }
 
 // xyzToFaceUV converts a direction vector (not necessarily unit length) to
 // (face, u, v) coordinates.
 func xyzToFaceUV(r r3.Vector) (f int, u, v float64) {
 	f = face(r)
 	u, v = validFaceXYZToUV(f, r)
 	return f, u, v
 }
 
 // faceUVToXYZ turns face and UV coordinates into an unnormalized 3 vector.
 func faceUVToXYZ(face int, u, v float64) r3.Vector {
 	switch face {
 	case 0:
 		return r3.Vector{1, u, v}
 	case 1:
 		return r3.Vector{-u, 1, v}
 	case 2:
 		return r3.Vector{-u, -v, 1}
 	case 3:
 		return r3.Vector{-1, -v, -u}
 	case 4:
 		return r3.Vector{v, -1, -u}
 	default:
 		return r3.Vector{v, u, -1}
 	}
 }
 
 // faceXYZToUV returns the u and v values (which may lie outside the range
 // [-1, 1]) if the dot product of the point p with the given face normal is positive.
 func faceXYZToUV(face int, p Point) (u, v float64, ok bool) {
 	switch face {
 	case 0:
 		if p.X <= 0 {
 			return 0, 0, false
 		}
 	case 1:
 		if p.Y <= 0 {
 			return 0, 0, false
 		}
 	case 2:
 		if p.Z <= 0 {
 			return 0, 0, false
 		}
 	case 3:
 		if p.X >= 0 {
 			return 0, 0, false
 		}
 	case 4:
 		if p.Y >= 0 {
 			return 0, 0, false
 		}
 	default:
 		if p.Z >= 0 {
 			return 0, 0, false
 		}
 	}
 
 	u, v = validFaceXYZToUV(face, p.Vector)
 	return u, v, true
 }
 
 // faceXYZtoUVW transforms the given point P to the (u,v,w) coordinate frame of the given
 // face where the w-axis represents the face normal.
 func faceXYZtoUVW(face int, p Point) Point {
 	// The result coordinates are simply the dot products of P with the (u,v,w)
 	// axes for the given face (see faceUVWAxes).
 	switch face {
 	case 0:
 		return Point{r3.Vector{p.Y, p.Z, p.X}}
 	case 1:
 		return Point{r3.Vector{-p.X, p.Z, p.Y}}
 	case 2:
 		return Point{r3.Vector{-p.X, -p.Y, p.Z}}
 	case 3:
 		return Point{r3.Vector{-p.Z, -p.Y, -p.X}}
 	case 4:
 		return Point{r3.Vector{-p.Z, p.X, -p.Y}}
 	default:
 		return Point{r3.Vector{p.Y, p.X, -p.Z}}
 	}
 }
 
 // faceSiTiToXYZ transforms the (si, ti) coordinates to a (not necessarily
 // unit length) Point on the given face.
 func faceSiTiToXYZ(face int, si, ti uint32) Point {
 	return Point{faceUVToXYZ(face, stToUV(siTiToST(si)), stToUV(siTiToST(ti)))}
 }
 
 // xyzToFaceSiTi transforms the (not necessarily unit length) Point to
 // (face, si, ti) coordinates and the level the Point is at.
 func xyzToFaceSiTi(p Point) (face int, si, ti uint32, level int) {
 	face, u, v := xyzToFaceUV(p.Vector)
 	si = stToSiTi(uvToST(u))
 	ti = stToSiTi(uvToST(v))
 
 	// If the levels corresponding to si,ti are not equal, then p is not a cell
 	// center. The si,ti values of 0 and maxSiTi need to be handled specially
 	// because they do not correspond to cell centers at any valid level; they
 	// are mapped to level -1 by the code at the end.
 	level = maxLevel - findLSBSetNonZero64(uint64(si|maxSiTi))
 	if level < 0 || level != maxLevel-findLSBSetNonZero64(uint64(ti|maxSiTi)) {
 		return face, si, ti, -1
 	}
 
 	// In infinite precision, this test could be changed to ST == SiTi. However,
 	// due to rounding errors, uvToST(xyzToFaceUV(faceUVToXYZ(stToUV(...)))) is
 	// not idempotent. On the other hand, the center is computed exactly the same
 	// way p was originally computed (if it is indeed the center of a Cell);
 	// the comparison can be exact.
 	if p.Vector == faceSiTiToXYZ(face, si, ti).Normalize() {
 		return face, si, ti, level
 	}
 
 	return face, si, ti, -1
 }
 
 // uNorm returns the right-handed normal (not necessarily unit length) for an
 // edge in the direction of the positive v-axis at the given u-value on
 // the given face.  (This vector is perpendicular to the plane through
 // the sphere origin that contains the given edge.)
 func uNorm(face int, u float64) r3.Vector {
 	switch face {
 	case 0:
 		return r3.Vector{u, -1, 0}
 	case 1:
 		return r3.Vector{1, u, 0}
 	case 2:
 		return r3.Vector{1, 0, u}
 	case 3:
 		return r3.Vector{-u, 0, 1}
 	case 4:
 		return r3.Vector{0, -u, 1}
 	default:
 		return r3.Vector{0, -1, -u}
 	}
 }
 
 // vNorm returns the right-handed normal (not necessarily unit length) for an
 // edge in the direction of the positive u-axis at the given v-value on
 // the given face.
 func vNorm(face int, v float64) r3.Vector {
 	switch face {
 	case 0:
 		return r3.Vector{-v, 0, 1}
 	case 1:
 		return r3.Vector{0, -v, 1}
 	case 2:
 		return r3.Vector{0, -1, -v}
 	case 3:
 		return r3.Vector{v, -1, 0}
 	case 4:
 		return r3.Vector{1, v, 0}
 	default:
 		return r3.Vector{1, 0, v}
 	}
 }
 
 // faceUVWAxes are the U, V, and W axes for each face.
 var faceUVWAxes = [6][3]Point{
 	{Point{r3.Vector{0, 1, 0}}, Point{r3.Vector{0, 0, 1}}, Point{r3.Vector{1, 0, 0}}},
 	{Point{r3.Vector{-1, 0, 0}}, Point{r3.Vector{0, 0, 1}}, Point{r3.Vector{0, 1, 0}}},
 	{Point{r3.Vector{-1, 0, 0}}, Point{r3.Vector{0, -1, 0}}, Point{r3.Vector{0, 0, 1}}},
 	{Point{r3.Vector{0, 0, -1}}, Point{r3.Vector{0, -1, 0}}, Point{r3.Vector{-1, 0, 0}}},
 	{Point{r3.Vector{0, 0, -1}}, Point{r3.Vector{1, 0, 0}}, Point{r3.Vector{0, -1, 0}}},
 	{Point{r3.Vector{0, 1, 0}}, Point{r3.Vector{1, 0, 0}}, Point{r3.Vector{0, 0, -1}}},
 }
 
 // faceUVWFaces are the precomputed neighbors of each face.
 var faceUVWFaces = [6][3][2]int{
 	{{4, 1}, {5, 2}, {3, 0}},
 	{{0, 3}, {5, 2}, {4, 1}},
 	{{0, 3}, {1, 4}, {5, 2}},
 	{{2, 5}, {1, 4}, {0, 3}},
 	{{2, 5}, {3, 0}, {1, 4}},
 	{{4, 1}, {3, 0}, {2, 5}},
 }
 
 // uvwAxis returns the given axis of the given face.
 func uvwAxis(face, axis int) Point {
 	return faceUVWAxes[face][axis]
 }
 
 // uvwFaces returns the face in the (u,v,w) coordinate system on the given axis
 // in the given direction.
 func uvwFace(face, axis, direction int) int {
 	return faceUVWFaces[face][axis][direction]
 }
 
 // uAxis returns the u-axis for the given face.
 func uAxis(face int) Point {
 	return uvwAxis(face, 0)
 }
 
 // vAxis returns the v-axis for the given face.
 func vAxis(face int) Point {
 	return uvwAxis(face, 1)
 }
 
 // Return the unit-length normal for the given face.
 func unitNorm(face int) Point {
 	return uvwAxis(face, 2)
 }