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Diffstat (limited to 'vendor/github.com/golang/geo/s2/edge_clipping.go')
| -rw-r--r-- | vendor/github.com/golang/geo/s2/edge_clipping.go | 672 |
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diff --git a/vendor/github.com/golang/geo/s2/edge_clipping.go b/vendor/github.com/golang/geo/s2/edge_clipping.go new file mode 100644 index 000000000..57a53bf0f --- /dev/null +++ b/vendor/github.com/golang/geo/s2/edge_clipping.go @@ -0,0 +1,672 @@ +// Copyright 2017 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 + +// This file contains a collection of methods for: +// +// (1) Robustly clipping geodesic edges to the faces of the S2 biunit cube +// (see s2stuv), and +// +// (2) Robustly clipping 2D edges against 2D rectangles. +// +// These functions can be used to efficiently find the set of CellIDs that +// are intersected by a geodesic edge (e.g., see CrossingEdgeQuery). + +import ( + "math" + + "github.com/golang/geo/r1" + "github.com/golang/geo/r2" + "github.com/golang/geo/r3" +) + +const ( + // edgeClipErrorUVCoord is the maximum error in a u- or v-coordinate + // compared to the exact result, assuming that the points A and B are in + // the rectangle [-1,1]x[1,1] or slightly outside it (by 1e-10 or less). + edgeClipErrorUVCoord = 2.25 * dblEpsilon + + // edgeClipErrorUVDist is the maximum distance from a clipped point to + // the corresponding exact result. It is equal to the error in a single + // coordinate because at most one coordinate is subject to error. + edgeClipErrorUVDist = 2.25 * dblEpsilon + + // faceClipErrorRadians is the maximum angle between a returned vertex + // and the nearest point on the exact edge AB. It is equal to the + // maximum directional error in PointCross, plus the error when + // projecting points onto a cube face. + faceClipErrorRadians = 3 * dblEpsilon + + // faceClipErrorDist is the same angle expressed as a maximum distance + // in (u,v)-space. In other words, a returned vertex is at most this far + // from the exact edge AB projected into (u,v)-space. + faceClipErrorUVDist = 9 * dblEpsilon + + // faceClipErrorUVCoord is the maximum angle between a returned vertex + // and the nearest point on the exact edge AB expressed as the maximum error + // in an individual u- or v-coordinate. In other words, for each + // returned vertex there is a point on the exact edge AB whose u- and + // v-coordinates differ from the vertex by at most this amount. + faceClipErrorUVCoord = 9.0 * (1.0 / math.Sqrt2) * dblEpsilon + + // intersectsRectErrorUVDist is the maximum error when computing if a point + // intersects with a given Rect. If some point of AB is inside the + // rectangle by at least this distance, the result is guaranteed to be true; + // if all points of AB are outside the rectangle by at least this distance, + // the result is guaranteed to be false. This bound assumes that rect is + // a subset of the rectangle [-1,1]x[-1,1] or extends slightly outside it + // (e.g., by 1e-10 or less). + intersectsRectErrorUVDist = 3 * math.Sqrt2 * dblEpsilon +) + +// ClipToFace returns the (u,v) coordinates for the portion of the edge AB that +// intersects the given face, or false if the edge AB does not intersect. +// This method guarantees that the clipped vertices lie within the [-1,1]x[-1,1] +// cube face rectangle and are within faceClipErrorUVDist of the line AB, but +// the results may differ from those produced by FaceSegments. +func ClipToFace(a, b Point, face int) (aUV, bUV r2.Point, intersects bool) { + return ClipToPaddedFace(a, b, face, 0.0) +} + +// ClipToPaddedFace returns the (u,v) coordinates for the portion of the edge AB that +// intersects the given face, but rather than clipping to the square [-1,1]x[-1,1] +// in (u,v) space, this method clips to [-R,R]x[-R,R] where R=(1+padding). +// Padding must be non-negative. +func ClipToPaddedFace(a, b Point, f int, padding float64) (aUV, bUV r2.Point, intersects bool) { + // Fast path: both endpoints are on the given face. + if face(a.Vector) == f && face(b.Vector) == f { + au, av := validFaceXYZToUV(f, a.Vector) + bu, bv := validFaceXYZToUV(f, b.Vector) + return r2.Point{au, av}, r2.Point{bu, bv}, true + } + + // Convert everything into the (u,v,w) coordinates of the given face. Note + // that the cross product *must* be computed in the original (x,y,z) + // coordinate system because PointCross (unlike the mathematical cross + // product) can produce different results in different coordinate systems + // when one argument is a linear multiple of the other, due to the use of + // symbolic perturbations. + normUVW := pointUVW(faceXYZtoUVW(f, a.PointCross(b))) + aUVW := pointUVW(faceXYZtoUVW(f, a)) + bUVW := pointUVW(faceXYZtoUVW(f, b)) + + // Padding is handled by scaling the u- and v-components of the normal. + // Letting R=1+padding, this means that when we compute the dot product of + // the normal with a cube face vertex (such as (-1,-1,1)), we will actually + // compute the dot product with the scaled vertex (-R,-R,1). This allows + // methods such as intersectsFace, exitAxis, etc, to handle padding + // with no further modifications. + scaleUV := 1 + padding + scaledN := pointUVW{r3.Vector{X: scaleUV * normUVW.X, Y: scaleUV * normUVW.Y, Z: normUVW.Z}} + if !scaledN.intersectsFace() { + return aUV, bUV, false + } + + // TODO(roberts): This is a workaround for extremely small vectors where some + // loss of precision can occur in Normalize causing underflow. When PointCross + // is updated to work around this, this can be removed. + if math.Max(math.Abs(normUVW.X), math.Max(math.Abs(normUVW.Y), math.Abs(normUVW.Z))) < math.Ldexp(1, -511) { + normUVW = pointUVW{normUVW.Mul(math.Ldexp(1, 563))} + } + + normUVW = pointUVW{normUVW.Normalize()} + + aTan := pointUVW{normUVW.Cross(aUVW.Vector)} + bTan := pointUVW{bUVW.Cross(normUVW.Vector)} + + // As described in clipDestination, if the sum of the scores from clipping the two + // endpoints is 3 or more, then the segment does not intersect this face. + aUV, aScore := clipDestination(bUVW, aUVW, pointUVW{scaledN.Mul(-1)}, bTan, aTan, scaleUV) + bUV, bScore := clipDestination(aUVW, bUVW, scaledN, aTan, bTan, scaleUV) + + return aUV, bUV, aScore+bScore < 3 +} + +// ClipEdge returns the portion of the edge defined by AB that is contained by the +// given rectangle. If there is no intersection, false is returned and aClip and bClip +// are undefined. +func ClipEdge(a, b r2.Point, clip r2.Rect) (aClip, bClip r2.Point, intersects bool) { + // Compute the bounding rectangle of AB, clip it, and then extract the new + // endpoints from the clipped bound. + bound := r2.RectFromPoints(a, b) + if bound, intersects = clipEdgeBound(a, b, clip, bound); !intersects { + return aClip, bClip, false + } + ai := 0 + if a.X > b.X { + ai = 1 + } + aj := 0 + if a.Y > b.Y { + aj = 1 + } + + return bound.VertexIJ(ai, aj), bound.VertexIJ(1-ai, 1-aj), true +} + +// The three functions below (sumEqual, intersectsFace, intersectsOppositeEdges) +// all compare a sum (u + v) to a third value w. They are implemented in such a +// way that they produce an exact result even though all calculations are done +// with ordinary floating-point operations. Here are the principles on which these +// functions are based: +// +// A. If u + v < w in floating-point, then u + v < w in exact arithmetic. +// +// B. If u + v < w in exact arithmetic, then at least one of the following +// expressions is true in floating-point: +// u + v < w +// u < w - v +// v < w - u +// +// Proof: By rearranging terms and substituting ">" for "<", we can assume +// that all values are non-negative. Now clearly "w" is not the smallest +// value, so assume WLOG that "u" is the smallest. We want to show that +// u < w - v in floating-point. If v >= w/2, the calculation of w - v is +// exact since the result is smaller in magnitude than either input value, +// so the result holds. Otherwise we have u <= v < w/2 and w - v >= w/2 +// (even in floating point), so the result also holds. + +// sumEqual reports whether u + v == w exactly. +func sumEqual(u, v, w float64) bool { + return (u+v == w) && (u == w-v) && (v == w-u) +} + +// pointUVW represents a Point in (u,v,w) coordinate space of a cube face. +type pointUVW Point + +// intersectsFace reports whether a given directed line L intersects the cube face F. +// The line L is defined by its normal N in the (u,v,w) coordinates of F. +func (p pointUVW) intersectsFace() bool { + // L intersects the [-1,1]x[-1,1] square in (u,v) if and only if the dot + // products of N with the four corner vertices (-1,-1,1), (1,-1,1), (1,1,1), + // and (-1,1,1) do not all have the same sign. This is true exactly when + // |Nu| + |Nv| >= |Nw|. The code below evaluates this expression exactly. + u := math.Abs(p.X) + v := math.Abs(p.Y) + w := math.Abs(p.Z) + + // We only need to consider the cases where u or v is the smallest value, + // since if w is the smallest then both expressions below will have a + // positive LHS and a negative RHS. + return (v >= w-u) && (u >= w-v) +} + +// intersectsOppositeEdges reports whether a directed line L intersects two +// opposite edges of a cube face F. This includs the case where L passes +// exactly through a corner vertex of F. The directed line L is defined +// by its normal N in the (u,v,w) coordinates of F. +func (p pointUVW) intersectsOppositeEdges() bool { + // The line L intersects opposite edges of the [-1,1]x[-1,1] (u,v) square if + // and only exactly two of the corner vertices lie on each side of L. This + // is true exactly when ||Nu| - |Nv|| >= |Nw|. The code below evaluates this + // expression exactly. + u := math.Abs(p.X) + v := math.Abs(p.Y) + w := math.Abs(p.Z) + + // If w is the smallest, the following line returns an exact result. + if math.Abs(u-v) != w { + return math.Abs(u-v) >= w + } + + // Otherwise u - v = w exactly, or w is not the smallest value. In either + // case the following returns the correct result. + if u >= v { + return u-w >= v + } + return v-w >= u +} + +// axis represents the possible results of exitAxis. +type axis int + +const ( + axisU axis = iota + axisV +) + +// exitAxis reports which axis the directed line L exits the cube face F on. +// The directed line L is represented by its CCW normal N in the (u,v,w) coordinates +// of F. It returns axisU if L exits through the u=-1 or u=+1 edge, and axisV if L exits +// through the v=-1 or v=+1 edge. Either result is acceptable if L exits exactly +// through a corner vertex of the cube face. +func (p pointUVW) exitAxis() axis { + if p.intersectsOppositeEdges() { + // The line passes through through opposite edges of the face. + // It exits through the v=+1 or v=-1 edge if the u-component of N has a + // larger absolute magnitude than the v-component. + if math.Abs(p.X) >= math.Abs(p.Y) { + return axisV + } + return axisU + } + + // The line passes through through two adjacent edges of the face. + // It exits the v=+1 or v=-1 edge if an even number of the components of N + // are negative. We test this using signbit() rather than multiplication + // to avoid the possibility of underflow. + var x, y, z int + if math.Signbit(p.X) { + x = 1 + } + if math.Signbit(p.Y) { + y = 1 + } + if math.Signbit(p.Z) { + z = 1 + } + + if x^y^z == 0 { + return axisV + } + return axisU +} + +// exitPoint returns the UV coordinates of the point where a directed line L (represented +// by the CCW normal of this point), exits the cube face this point is derived from along +// the given axis. +func (p pointUVW) exitPoint(a axis) r2.Point { + if a == axisU { + u := -1.0 + if p.Y > 0 { + u = 1.0 + } + return r2.Point{u, (-u*p.X - p.Z) / p.Y} + } + + v := -1.0 + if p.X < 0 { + v = 1.0 + } + return r2.Point{(-v*p.Y - p.Z) / p.X, v} +} + +// clipDestination returns a score which is used to indicate if the clipped edge AB +// on the given face intersects the face at all. This function returns the score for +// the given endpoint, which is an integer ranging from 0 to 3. If the sum of the scores +// from both of the endpoints is 3 or more, then edge AB does not intersect this face. +// +// First, it clips the line segment AB to find the clipped destination B' on a given +// face. (The face is specified implicitly by expressing *all arguments* in the (u,v,w) +// coordinates of that face.) Second, it partially computes whether the segment AB +// intersects this face at all. The actual condition is fairly complicated, but it +// turns out that it can be expressed as a "score" that can be computed independently +// when clipping the two endpoints A and B. +func clipDestination(a, b, scaledN, aTan, bTan pointUVW, scaleUV float64) (r2.Point, int) { + var uv r2.Point + + // Optimization: if B is within the safe region of the face, use it. + maxSafeUVCoord := 1 - faceClipErrorUVCoord + if b.Z > 0 { + uv = r2.Point{b.X / b.Z, b.Y / b.Z} + if math.Max(math.Abs(uv.X), math.Abs(uv.Y)) <= maxSafeUVCoord { + return uv, 0 + } + } + + // Otherwise find the point B' where the line AB exits the face. + uv = scaledN.exitPoint(scaledN.exitAxis()).Mul(scaleUV) + + p := pointUVW(Point{r3.Vector{uv.X, uv.Y, 1.0}}) + + // Determine if the exit point B' is contained within the segment. We do this + // by computing the dot products with two inward-facing tangent vectors at A + // and B. If either dot product is negative, we say that B' is on the "wrong + // side" of that point. As the point B' moves around the great circle AB past + // the segment endpoint B, it is initially on the wrong side of B only; as it + // moves further it is on the wrong side of both endpoints; and then it is on + // the wrong side of A only. If the exit point B' is on the wrong side of + // either endpoint, we can't use it; instead the segment is clipped at the + // original endpoint B. + // + // We reject the segment if the sum of the scores of the two endpoints is 3 + // or more. Here is what that rule encodes: + // - If B' is on the wrong side of A, then the other clipped endpoint A' + // must be in the interior of AB (otherwise AB' would go the wrong way + // around the circle). There is a similar rule for A'. + // - If B' is on the wrong side of either endpoint (and therefore we must + // use the original endpoint B instead), then it must be possible to + // project B onto this face (i.e., its w-coordinate must be positive). + // This rule is only necessary to handle certain zero-length edges (A=B). + score := 0 + if p.Sub(a.Vector).Dot(aTan.Vector) < 0 { + score = 2 // B' is on wrong side of A. + } else if p.Sub(b.Vector).Dot(bTan.Vector) < 0 { + score = 1 // B' is on wrong side of B. + } + + if score > 0 { // B' is not in the interior of AB. + if b.Z <= 0 { + score = 3 // B cannot be projected onto this face. + } else { + uv = r2.Point{b.X / b.Z, b.Y / b.Z} + } + } + + return uv, score +} + +// updateEndpoint returns the interval with the specified endpoint updated to +// the given value. If the value lies beyond the opposite endpoint, nothing is +// changed and false is returned. +func updateEndpoint(bound r1.Interval, highEndpoint bool, value float64) (r1.Interval, bool) { + if !highEndpoint { + if bound.Hi < value { + return bound, false + } + if bound.Lo < value { + bound.Lo = value + } + return bound, true + } + + if bound.Lo > value { + return bound, false + } + if bound.Hi > value { + bound.Hi = value + } + return bound, true +} + +// clipBoundAxis returns the clipped versions of the bounding intervals for the given +// axes for the line segment from (a0,a1) to (b0,b1) so that neither extends beyond the +// given clip interval. negSlope is a precomputed helper variable that indicates which +// diagonal of the bounding box is spanned by AB; it is false if AB has positive slope, +// and true if AB has negative slope. If the clipping interval doesn't overlap the bounds, +// false is returned. +func clipBoundAxis(a0, b0 float64, bound0 r1.Interval, a1, b1 float64, bound1 r1.Interval, + negSlope bool, clip r1.Interval) (bound0c, bound1c r1.Interval, updated bool) { + + if bound0.Lo < clip.Lo { + // If the upper bound is below the clips lower bound, there is nothing to do. + if bound0.Hi < clip.Lo { + return bound0, bound1, false + } + // narrow the intervals lower bound to the clip bound. + bound0.Lo = clip.Lo + if bound1, updated = updateEndpoint(bound1, negSlope, interpolateFloat64(clip.Lo, a0, b0, a1, b1)); !updated { + return bound0, bound1, false + } + } + + if bound0.Hi > clip.Hi { + // If the lower bound is above the clips upper bound, there is nothing to do. + if bound0.Lo > clip.Hi { + return bound0, bound1, false + } + // narrow the intervals upper bound to the clip bound. + bound0.Hi = clip.Hi + if bound1, updated = updateEndpoint(bound1, !negSlope, interpolateFloat64(clip.Hi, a0, b0, a1, b1)); !updated { + return bound0, bound1, false + } + } + return bound0, bound1, true +} + +// edgeIntersectsRect reports whether the edge defined by AB intersects the +// given closed rectangle to within the error bound. +func edgeIntersectsRect(a, b r2.Point, r r2.Rect) bool { + // First check whether the bounds of a Rect around AB intersects the given rect. + if !r.Intersects(r2.RectFromPoints(a, b)) { + return false + } + + // Otherwise AB intersects the rect if and only if all four vertices of rect + // do not lie on the same side of the extended line AB. We test this by finding + // the two vertices of rect with minimum and maximum projections onto the normal + // of AB, and computing their dot products with the edge normal. + n := b.Sub(a).Ortho() + + i := 0 + if n.X >= 0 { + i = 1 + } + j := 0 + if n.Y >= 0 { + j = 1 + } + + max := n.Dot(r.VertexIJ(i, j).Sub(a)) + min := n.Dot(r.VertexIJ(1-i, 1-j).Sub(a)) + + return (max >= 0) && (min <= 0) +} + +// clippedEdgeBound returns the bounding rectangle of the portion of the edge defined +// by AB intersected by clip. The resulting bound may be empty. This is a convenience +// function built on top of clipEdgeBound. +func clippedEdgeBound(a, b r2.Point, clip r2.Rect) r2.Rect { + bound := r2.RectFromPoints(a, b) + if b1, intersects := clipEdgeBound(a, b, clip, bound); intersects { + return b1 + } + return r2.EmptyRect() +} + +// clipEdgeBound clips an edge AB to sequence of rectangles efficiently. +// It represents the clipped edges by their bounding boxes rather than as a pair of +// endpoints. Specifically, let A'B' be some portion of an edge AB, and let bound be +// a tight bound of A'B'. This function returns the bound that is a tight bound +// of A'B' intersected with a given rectangle. If A'B' does not intersect clip, +// it returns false and the original bound. +func clipEdgeBound(a, b r2.Point, clip, bound r2.Rect) (r2.Rect, bool) { + // negSlope indicates which diagonal of the bounding box is spanned by AB: it + // is false if AB has positive slope, and true if AB has negative slope. This is + // used to determine which interval endpoints need to be updated each time + // the edge is clipped. + negSlope := (a.X > b.X) != (a.Y > b.Y) + + b0x, b0y, up1 := clipBoundAxis(a.X, b.X, bound.X, a.Y, b.Y, bound.Y, negSlope, clip.X) + if !up1 { + return bound, false + } + b1y, b1x, up2 := clipBoundAxis(a.Y, b.Y, b0y, a.X, b.X, b0x, negSlope, clip.Y) + if !up2 { + return r2.Rect{b0x, b0y}, false + } + return r2.Rect{X: b1x, Y: b1y}, true +} + +// interpolateFloat64 returns a value with the same combination of a1 and b1 as the +// given value x is of a and b. This function makes the following guarantees: +// - If x == a, then x1 = a1 (exactly). +// - If x == b, then x1 = b1 (exactly). +// - If a <= x <= b, then a1 <= x1 <= b1 (even if a1 == b1). +// This requires a != b. +func interpolateFloat64(x, a, b, a1, b1 float64) float64 { + // To get results that are accurate near both A and B, we interpolate + // starting from the closer of the two points. + if math.Abs(a-x) <= math.Abs(b-x) { + return a1 + (b1-a1)*(x-a)/(b-a) + } + return b1 + (a1-b1)*(x-b)/(a-b) +} + +// FaceSegment represents an edge AB clipped to an S2 cube face. It is +// represented by a face index and a pair of (u,v) coordinates. +type FaceSegment struct { + face int + a, b r2.Point +} + +// FaceSegments subdivides the given edge AB at every point where it crosses the +// boundary between two S2 cube faces and returns the corresponding FaceSegments. +// The segments are returned in order from A toward B. The input points must be +// unit length. +// +// This function guarantees that the returned segments form a continuous path +// from A to B, and that all vertices are within faceClipErrorUVDist of the +// line AB. All vertices lie within the [-1,1]x[-1,1] cube face rectangles. +// The results are consistent with Sign, i.e. the edge is well-defined even its +// endpoints are antipodal. +// TODO(roberts): Extend the implementation of PointCross so that this is true. +func FaceSegments(a, b Point) []FaceSegment { + var segment FaceSegment + + // Fast path: both endpoints are on the same face. + var aFace, bFace int + aFace, segment.a.X, segment.a.Y = xyzToFaceUV(a.Vector) + bFace, segment.b.X, segment.b.Y = xyzToFaceUV(b.Vector) + if aFace == bFace { + segment.face = aFace + return []FaceSegment{segment} + } + + // Starting at A, we follow AB from face to face until we reach the face + // containing B. The following code is designed to ensure that we always + // reach B, even in the presence of numerical errors. + // + // First we compute the normal to the plane containing A and B. This normal + // becomes the ultimate definition of the line AB; it is used to resolve all + // questions regarding where exactly the line goes. Unfortunately due to + // numerical errors, the line may not quite intersect the faces containing + // the original endpoints. We handle this by moving A and/or B slightly if + // necessary so that they are on faces intersected by the line AB. + ab := a.PointCross(b) + + aFace, segment.a = moveOriginToValidFace(aFace, a, ab, segment.a) + bFace, segment.b = moveOriginToValidFace(bFace, b, Point{ab.Mul(-1)}, segment.b) + + // Now we simply follow AB from face to face until we reach B. + var segments []FaceSegment + segment.face = aFace + bSaved := segment.b + + for face := aFace; face != bFace; { + // Complete the current segment by finding the point where AB + // exits the current face. + z := faceXYZtoUVW(face, ab) + n := pointUVW{z.Vector} + + exitAxis := n.exitAxis() + segment.b = n.exitPoint(exitAxis) + segments = append(segments, segment) + + // Compute the next face intersected by AB, and translate the exit + // point of the current segment into the (u,v) coordinates of the + // next face. This becomes the first point of the next segment. + exitXyz := faceUVToXYZ(face, segment.b.X, segment.b.Y) + face = nextFace(face, segment.b, exitAxis, n, bFace) + exitUvw := faceXYZtoUVW(face, Point{exitXyz}) + segment.face = face + segment.a = r2.Point{exitUvw.X, exitUvw.Y} + } + // Finish the last segment. + segment.b = bSaved + return append(segments, segment) +} + +// moveOriginToValidFace updates the origin point to a valid face if necessary. +// Given a line segment AB whose origin A has been projected onto a given cube +// face, determine whether it is necessary to project A onto a different face +// instead. This can happen because the normal of the line AB is not computed +// exactly, so that the line AB (defined as the set of points perpendicular to +// the normal) may not intersect the cube face containing A. Even if it does +// intersect the face, the exit point of the line from that face may be on +// the wrong side of A (i.e., in the direction away from B). If this happens, +// we reproject A onto the adjacent face where the line AB approaches A most +// closely. This moves the origin by a small amount, but never more than the +// error tolerances. +func moveOriginToValidFace(face int, a, ab Point, aUV r2.Point) (int, r2.Point) { + // Fast path: if the origin is sufficiently far inside the face, it is + // always safe to use it. + const maxSafeUVCoord = 1 - faceClipErrorUVCoord + if math.Max(math.Abs((aUV).X), math.Abs((aUV).Y)) <= maxSafeUVCoord { + return face, aUV + } + + // Otherwise check whether the normal AB even intersects this face. + z := faceXYZtoUVW(face, ab) + n := pointUVW{z.Vector} + if n.intersectsFace() { + // Check whether the point where the line AB exits this face is on the + // wrong side of A (by more than the acceptable error tolerance). + uv := n.exitPoint(n.exitAxis()) + exit := faceUVToXYZ(face, uv.X, uv.Y) + aTangent := ab.Normalize().Cross(a.Vector) + + // We can use the given face. + if exit.Sub(a.Vector).Dot(aTangent) >= -faceClipErrorRadians { + return face, aUV + } + } + + // Otherwise we reproject A to the nearest adjacent face. (If line AB does + // not pass through a given face, it must pass through all adjacent faces.) + var dir int + if math.Abs((aUV).X) >= math.Abs((aUV).Y) { + // U-axis + if aUV.X > 0 { + dir = 1 + } + face = uvwFace(face, 0, dir) + } else { + // V-axis + if aUV.Y > 0 { + dir = 1 + } + face = uvwFace(face, 1, dir) + } + + aUV.X, aUV.Y = validFaceXYZToUV(face, a.Vector) + aUV.X = math.Max(-1.0, math.Min(1.0, aUV.X)) + aUV.Y = math.Max(-1.0, math.Min(1.0, aUV.Y)) + + return face, aUV +} + +// nextFace returns the next face that should be visited by FaceSegments, given that +// we have just visited face and we are following the line AB (represented +// by its normal N in the (u,v,w) coordinates of that face). The other +// arguments include the point where AB exits face, the corresponding +// exit axis, and the target face containing the destination point B. +func nextFace(face int, exit r2.Point, axis axis, n pointUVW, targetFace int) int { + // this bit is to work around C++ cleverly casting bools to ints for you. + exitA := exit.X + exit1MinusA := exit.Y + + if axis == axisV { + exitA = exit.Y + exit1MinusA = exit.X + } + exitAPos := 0 + if exitA > 0 { + exitAPos = 1 + } + exit1MinusAPos := 0 + if exit1MinusA > 0 { + exit1MinusAPos = 1 + } + + // We return the face that is adjacent to the exit point along the given + // axis. If line AB exits *exactly* through a corner of the face, there are + // two possible next faces. If one is the target face containing B, then + // we guarantee that we advance to that face directly. + // + // The three conditions below check that (1) AB exits approximately through + // a corner, (2) the adjacent face along the non-exit axis is the target + // face, and (3) AB exits *exactly* through the corner. (The sumEqual + // code checks whether the dot product of (u,v,1) and n is exactly zero.) + if math.Abs(exit1MinusA) == 1 && + uvwFace(face, int(1-axis), exit1MinusAPos) == targetFace && + sumEqual(exit.X*n.X, exit.Y*n.Y, -n.Z) { + return targetFace + } + + // Otherwise return the face that is adjacent to the exit point in the + // direction of the exit axis. + return uvwFace(face, int(axis), exitAPos) +} |