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Diffstat (limited to 'vendor/github.com/golang/geo/s2/stuv.go')
| -rw-r--r-- | vendor/github.com/golang/geo/s2/stuv.go | 310 |
1 files changed, 310 insertions, 0 deletions
diff --git a/vendor/github.com/golang/geo/s2/stuv.go b/vendor/github.com/golang/geo/s2/stuv.go new file mode 100644 index 0000000..e669b7c --- /dev/null +++ b/vendor/github.com/golang/geo/s2/stuv.go @@ -0,0 +1,310 @@ +/* +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" +) + +const ( + // maxSiTi is the maximum value of an si- or ti-coordinate. + // It is one shift more than maxSize. + 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 uint64) 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) uint64 { + if s < 0 { + return uint64(s*maxSiTi - 0.5) + } + return uint64(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 { + abs := r.Abs() + id := 0 + value := r.X + if abs.Y > abs.X { + id = 1 + value = r.Y + } + if abs.Z > math.Abs(value) { + id = 2 + value = r.Z + } + if value < 0 { + id += 3 + } + return id +} + +// 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 uint64) 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 uint64, 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(si|maxSiTi) + if level < 0 || level != maxLevel-findLSBSetNonZero64(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) +} |