/**
* Return the smallest rectangle containing the intersection of this rectangle
* and the given rectangle. Note that the region of intersection may consist
* of two disjoint rectangles, in which case a single rectangle spanning both
* of them is returned.
*#/
* public S2LatLngRect intersection(S2LatLngRect other) {
* R1Interval intersectLat = lat.intersection(other.lat);
* S1Interval intersectLng = lng.intersection(other.lng);
* if (intersectLat.isEmpty() || intersectLng.isEmpty()) {
* // The lat/lng ranges must either be both empty or both non-empty.
* return empty();
* }
* return new S2LatLngRect(intersectLat, intersectLng);
* }
*
* /**
* Return a rectangle that contains the convolution of this rectangle with a
* cap of the given angle. This expands the rectangle by a fixed distance (as
* opposed to growing the rectangle in latitude-longitude space). The returned
* rectangle includes all points whose minimum distance to the original
* rectangle is at most the given angle.
*#/
* public S2LatLngRect convolveWithCap(S1Angle angle) {
* // The most straightforward approach is to build a cap centered on each
* // vertex and take the union of all the bounding rectangles (including the
* // original rectangle; this is necessary for very large rectangles).
*
* // Optimization: convert the angle to a height exactly once.
* S2Cap cap = S2Cap.fromAxisAngle(new S2Point(1, 0, 0), angle);
*
* S2LatLngRect r = this;
* for (int k = 0; k < 4; ++k) {
* S2Cap vertexCap = S2Cap.fromAxisHeight(getVertex(k).toPoint(), cap
* .height());
* r = r.union(vertexCap.getRectBound());
* }
* return r;
* }
*
* /** Return the surface area of this rectangle on the unit sphere. *#/
* public double area() {
* if (isEmpty()) {
* return 0;
* }
*
* // This is the size difference of the two spherical caps, multiplied by
* // the longitude ratio.
* return lng().getLength() * Math.abs(Math.sin(latHi().radians()) - Math.sin(latLo().radians()));
* }
*
* /** Return true if two rectangles contains the same set of points. *#/
* @Override
* public boolean equals(Object that) {
* if (!(that instanceof S2LatLngRect)) {
* return false;
* }
* S2LatLngRect otherRect = (S2LatLngRect) that;
* return lat().equals(otherRect.lat()) && lng().equals(otherRect.lng());
* }
*
* /**
* Return true if the latitude and longitude intervals of the two rectangles
* are the same up to the given tolerance (see r1interval.h and s1interval.h
* for details).
*#/
* public boolean approxEquals(S2LatLngRect other, double maxError) {
* return (lat.approxEquals(other.lat, maxError) && lng.approxEquals(
* other.lng, maxError));
* }
*
* public boolean approxEquals(S2LatLngRect other) {
* return approxEquals(other, 1e-15);
* }
*
* @Override
* public int hashCode() {
* int value = 17;
* value = 37 * value + lat.hashCode();
* return (37 * value + lng.hashCode());
* }
*
* // //////////////////////////////////////////////////////////////////////
* // S2Region interface (see {@code S2Region} for details):
*
* @Override
* public S2Region clone() {
* return new S2LatLngRect(this.lo(), this.hi());
* }
*/
public function getCapBound()
{
// We consider two possible bounding caps, one whose axis passes
// through the center of the lat-long rectangle and one whose axis
// is the north or south pole. We return the smaller of the two caps.
if ($this->isEmpty()) {
echo __METHOD__ . " empty\n";
return S2Cap::sempty();
}
$poleZ = null;
$poleAngle = null;
if ($this->lat->lo() + $this->lat->hi() < 0) {
// South pole axis yields smaller cap.
$poleZ = -1;
$poleAngle = S2::M_PI_2 + $this->lat->hi();
} else {
$poleZ = 1;
$poleAngle = S2::M_PI_2 - $this->lat->lo();
}
$poleCap = S2Cap::fromAxisAngle(new S2Point(0, 0, $poleZ), S1Angle::sradians($poleAngle));
// For bounding rectangles that span 180 degrees or less in longitude, the
// maximum cap size is achieved at one of the rectangle vertices. For
// rectangles that are larger than 180 degrees, we punt and always return a
// bounding cap centered at one of the two poles.
$lngSpan = $this->lng->hi() - $this->lng->lo();
if (S2::IEEEremainder($lngSpan, 2 * S2::M_PI) >= 0) {
if ($lngSpan < 2 * S2::M_PI) {
$midCap = S2Cap::fromAxisAngle($this->getCenter()->toPoint(), S1Angle::sradians(0));
for ($k = 0; $k < 4; ++$k) {
$midCap = $midCap->addPoint($this->getVertex($k)->toPoint());
}
if ($midCap->height() < $poleCap->height()) {
return $midCap;
}
}
}
return $poleCap;
}
/**
* Return the average area for cells at the given level.
*#/
* public static double averageArea(int level) {
* return S2Projections.AVG_AREA.getValue(level);
* }
*
* /**
* Return the average area of cells at this level. This is accurate to within
* a factor of 1.7 (for S2_QUADRATIC_PROJECTION) and is extremely cheap to
* compute.
*#/
* public double averageArea() {
* return averageArea(level);
* }
*
* /**
* Return the approximate area of this cell. This method is accurate to within
* 3% percent for all cell sizes and accurate to within 0.1% for cells at
* level 5 or higher (i.e. 300km square or smaller). It is moderately cheap to
* compute.
*#/
* public double approxArea() {
*
* // All cells at the first two levels have the same area.
* if (level < 2) {
* return averageArea(level);
* }
*
* // First, compute the approximate area of the cell when projected
* // perpendicular to its normal. The cross product of its diagonals gives
* // the normal, and the length of the normal is twice the projected area.
* double flatArea = 0.5 * S2Point.crossProd(
* S2Point.sub(getVertex(2), getVertex(0)), S2Point.sub(getVertex(3), getVertex(1))).norm();
*
* // Now, compensate for the curvature of the cell surface by pretending
* // that the cell is shaped like a spherical cap. The ratio of the
* // area of a spherical cap to the area of its projected disc turns out
* // to be 2 / (1 + sqrt(1 - r*r)) where "r" is the radius of the disc.
* // For example, when r=0 the ratio is 1, and when r=1 the ratio is 2.
* // Here we set Pi*r*r == flat_area to find the equivalent disc.
* return flatArea * 2 / (1 + Math.sqrt(1 - Math.min(S2.M_1_PI * flatArea, 1.0)));
* }
*
* /**
* Return the area of this cell as accurately as possible. This method is more
* expensive but it is accurate to 6 digits of precision even for leaf cells
* (whose area is approximately 1e-18).
*#/
* public double exactArea() {
* S2Point v0 = getVertex(0);
* S2Point v1 = getVertex(1);
* S2Point v2 = getVertex(2);
* S2Point v3 = getVertex(3);
* return S2.area(v0, v1, v2) + S2.area(v0, v2, v3);
* }
*
* // //////////////////////////////////////////////////////////////////////
* // S2Region interface (see {@code S2Region} for details):
*
* @Override
* public S2Region clone() {
* S2Cell clone = new S2Cell();
* clone.face = this.face;
* clone.level = this.level;
* clone.orientation = this.orientation;
* clone.uv = this.uv.clone();
*
* return clone;
* }
*/
public function getCapBound()
{
// Use the cell center in (u,v)-space as the cap axis. This vector is
// very close to GetCenter() and faster to compute. Neither one of these
// vectors yields the bounding cap with minimal surface area, but they
// are both pretty close.
//
// It's possible to show that the two vertices that are furthest from
// the (u,v)-origin never determine the maximum cap size (this is a
// possible future optimization).
$u = 0.5 * ($this->uv[0][0] + $this->uv[0][1]);
$v = 0.5 * ($this->uv[1][0] + $this->uv[1][1]);
$cap = S2Cap::fromAxisHeight(S2Point::normalize(S2Projections::faceUvToXyz($this->face, $u, $v)), 0);
for ($k = 0; $k < 4; ++$k) {
$cap = $cap->addPoint($this->getVertex($k));
}
return $cap;
}