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/**
* Copyright 2010 JogAmp Community. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without modification, are
* permitted provided that the following conditions are met:
*
* 1. Redistributions of source code must retain the above copyright notice, this list of
* conditions and the following disclaimer.
*
* 2. Redistributions in binary form must reproduce the above copyright notice, this list
* of conditions and the following disclaimer in the documentation and/or other materials
* provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY JogAmp Community ``AS IS'' AND ANY EXPRESS OR IMPLIED
* WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND
* FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL JogAmp Community OR
* CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
* CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
* SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON
* ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
* NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF
* ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*
* The views and conclusions contained in the software and documentation are those of the
* authors and should not be interpreted as representing official policies, either expressed
* or implied, of JogAmp Community.
*/
package com.jogamp.opengl.math.geom;
import jogamp.graph.geom.plane.AffineTransform;
import com.jogamp.opengl.math.FloatUtil;
import com.jogamp.opengl.math.Quaternion;
import com.jogamp.opengl.math.Ray;
import com.jogamp.opengl.math.VectorUtil;
/**
* Axis Aligned Bounding Box. Defined by two 3D coordinates (low and high)
* The low being the the lower left corner of the box, and the high being the upper
* right corner of the box.
*
* A few references for collision detection, intersections:
*
* http://www.realtimerendering.com/intersections.html
* http://www.codercorner.com/RayAABB.cpp
* http://www.siggraph.org/education/materials/HyperGraph/raytrace/rtinter0.htm
* http://realtimecollisiondetection.net/files/levine_swept_sat.txt
*
*
*
*/
public class AABBox {
private static final boolean DEBUG = FloatUtil.DEBUG;
private final float[] low = new float[3];
private final float[] high = new float[3];
private final float[] center = new float[3];
/**
* Create an Axis Aligned bounding box (AABBox)
* where the low and and high MAX float Values.
*/
public AABBox() {
reset();
}
/**
* Create an AABBox copying all values from the given one
* @param src the box value to be used for the new instance
*/
public AABBox(final AABBox src) {
copy(src);
}
/**
* Create an AABBox specifying the coordinates
* of the low and high
* @param lx min x-coordinate
* @param ly min y-coordnate
* @param lz min z-coordinate
* @param hx max x-coordinate
* @param hy max y-coordinate
* @param hz max z-coordinate
*/
public AABBox(final float lx, final float ly, final float lz,
final float hx, final float hy, final float hz) {
setSize(lx, ly, lz, hx, hy, hz);
}
/**
* Create a AABBox defining the low and high
* @param low min xyz-coordinates
* @param high max xyz-coordinates
*/
public AABBox(final float[] low, final float[] high) {
setSize(low, high);
}
/**
* resets this box to the inverse low/high, allowing the next {@link #resize(float, float, float)} command to hit.
* @return this AABBox for chaining
*/
public final AABBox reset() {
setLow(Float.MAX_VALUE,Float.MAX_VALUE,Float.MAX_VALUE);
setHigh(-1*Float.MAX_VALUE,-1*Float.MAX_VALUE,-1*Float.MAX_VALUE);
center[0] = 0f;
center[1] = 0f;
center[2] = 0f;
return this;
}
/** Get the max xyz-coordinates
* @return a float array containing the max xyz coordinates
*/
public final float[] getHigh() {
return high;
}
private final void setHigh(final float hx, final float hy, final float hz) {
this.high[0] = hx;
this.high[1] = hy;
this.high[2] = hz;
}
/** Get the min xyz-coordinates
* @return a float array containing the min xyz coordinates
*/
public final float[] getLow() {
return low;
}
private final void setLow(final float lx, final float ly, final float lz) {
this.low[0] = lx;
this.low[1] = ly;
this.low[2] = lz;
}
private final void computeCenter() {
center[0] = (high[0] + low[0])/2f;
center[1] = (high[1] + low[1])/2f;
center[2] = (high[2] + low[2])/2f;
}
/**
* Copy given AABBox 'src' values to this AABBox.
*
* @param src source AABBox
* @return this AABBox for chaining
*/
public final AABBox copy(final AABBox src) {
System.arraycopy(src.low, 0, low, 0, 3);
System.arraycopy(src.high, 0, high, 0, 3);
System.arraycopy(src.center, 0, center, 0, 3);
return this;
}
/**
* Set size of the AABBox specifying the coordinates
* of the low and high.
*
* @param low min xyz-coordinates
* @param high max xyz-coordinates
* @return this AABBox for chaining
*/
public final AABBox setSize(final float[] low, final float[] high) {
return setSize(low[0],low[1],low[2], high[0],high[1],high[2]);
}
/**
* Set size of the AABBox specifying the coordinates
* of the low and high.
*
* @param lx min x-coordinate
* @param ly min y-coordnate
* @param lz min z-coordinate
* @param hx max x-coordinate
* @param hy max y-coordinate
* @param hz max z-coordinate
* @return this AABBox for chaining
*/
public final AABBox setSize(final float lx, final float ly, final float lz,
final float hx, final float hy, final float hz) {
this.low[0] = lx;
this.low[1] = ly;
this.low[2] = lz;
this.high[0] = hx;
this.high[1] = hy;
this.high[2] = hz;
computeCenter();
return this;
}
/**
* Resize the AABBox to encapsulate another AABox
* @param newBox AABBox to be encapsulated in
* @return this AABBox for chaining
*/
public final AABBox resize(final AABBox newBox) {
final float[] newLow = newBox.getLow();
final float[] newHigh = newBox.getHigh();
/** test low */
if (newLow[0] < low[0])
low[0] = newLow[0];
if (newLow[1] < low[1])
low[1] = newLow[1];
if (newLow[2] < low[2])
low[2] = newLow[2];
/** test high */
if (newHigh[0] > high[0])
high[0] = newHigh[0];
if (newHigh[1] > high[1])
high[1] = newHigh[1];
if (newHigh[2] > high[2])
high[2] = newHigh[2];
computeCenter();
return this;
}
/**
* Resize the AABBox to encapsulate another AABox, which will be transformed on the fly first.
* @param newBox AABBox to be encapsulated in
* @param t the {@link AffineTransform} applied on newBox on the fly
* @param tmpV3 temp float[3] storage
* @return this AABBox for chaining
*/
public final AABBox resize(final AABBox newBox, final AffineTransform t, final float[] tmpV3) {
/** test low */
{
final float[] newBoxLow = newBox.getLow();
t.transform(newBoxLow, tmpV3);
tmpV3[2] = newBoxLow[2];
if (tmpV3[0] < low[0])
low[0] = tmpV3[0];
if (tmpV3[1] < low[1])
low[1] = tmpV3[1];
if (tmpV3[2] < low[2])
low[2] = tmpV3[2];
}
/** test high */
{
final float[] newBoxHigh = newBox.getHigh();
t.transform(newBoxHigh, tmpV3);
tmpV3[2] = newBoxHigh[2];
if (tmpV3[0] > high[0])
high[0] = tmpV3[0];
if (tmpV3[1] > high[1])
high[1] = tmpV3[1];
if (tmpV3[2] > high[2])
high[2] = tmpV3[2];
}
computeCenter();
return this;
}
/**
* Resize the AABBox to encapsulate the passed
* xyz-coordinates.
* @param x x-axis coordinate value
* @param y y-axis coordinate value
* @param z z-axis coordinate value
* @return this AABBox for chaining
*/
public final AABBox resize(final float x, final float y, final float z) {
/** test low */
if (x < low[0]) {
low[0] = x;
}
if (y < low[1]) {
low[1] = y;
}
if (z < low[2]) {
low[2] = z;
}
/** test high */
if (x > high[0]) {
high[0] = x;
}
if (y > high[1]) {
high[1] = y;
}
if (z > high[2]) {
high[2] = z;
}
computeCenter();
return this;
}
/**
* Resize the AABBox to encapsulate the passed
* xyz-coordinates.
* @param xyz xyz-axis coordinate values
* @param offset of the array
* @return this AABBox for chaining
*/
public final AABBox resize(final float[] xyz, final int offset) {
return resize(xyz[0+offset], xyz[1+offset], xyz[2+offset]);
}
/**
* Resize the AABBox to encapsulate the passed
* xyz-coordinates.
* @param xyz xyz-axis coordinate values
* @return this AABBox for chaining
*/
public final AABBox resize(final float[] xyz) {
return resize(xyz[0], xyz[1], xyz[2]);
}
/**
* Check if the x & y coordinates are bounded/contained
* by this AABBox
* @param x x-axis coordinate value
* @param y y-axis coordinate value
* @return true if x belong to (low.x, high.x) and
* y belong to (low.y, high.y)
*/
public final boolean contains(final float x, final float y) {
if(xhigh[0]){
return false;
}
if(yhigh[1]){
return false;
}
return true;
}
/**
* Check if the xyz coordinates are bounded/contained
* by this AABBox.
* @param x x-axis coordinate value
* @param y y-axis coordinate value
* @param z z-axis coordinate value
* @return true if x belong to (low.x, high.x) and
* y belong to (low.y, high.y) and z belong to (low.z, high.z)
*/
public final boolean contains(final float x, final float y, final float z) {
if(xhigh[0]){
return false;
}
if(yhigh[1]){
return false;
}
if(zhigh[2]){
return false;
}
return true;
}
/**
* Check if there is a common region between this AABBox and the passed
* 2D region irrespective of z range
* @param x lower left x-coord
* @param y lower left y-coord
* @param w width
* @param h hight
* @return true if this AABBox might have a common region with this 2D region
*/
public final boolean intersects2DRegion(final float x, final float y, final float w, final float h) {
if (w <= 0 || h <= 0) {
return false;
}
final float _w = getWidth();
final float _h = getHeight();
if (_w <= 0 || _h <= 0) {
return false;
}
final float x0 = getMinX();
final float y0 = getMinY();
return (x + w > x0 &&
y + h > y0 &&
x < x0 + _w &&
y < y0 + _h);
}
/**
* Check if {@link Ray} intersects this bounding box.
*
* Versions uses the SAT[1], testing 6 axes.
* Original code for OBBs from MAGIC.
* Rewritten for AABBs and reorganized for early exits[2].
*
*
* [1] SAT = Separating Axis Theorem
* [2] http://www.codercorner.com/RayAABB.cpp
*
* @param ray
* @return
*/
public final boolean intersectsRay(final Ray ray) {
// diff[XYZ] -> VectorUtil.subVec3(diff, ray.orig, center);
// ext[XYZ] -> extend VectorUtil.subVec3(ext, high, center);
final float dirX = ray.dir[0];
final float diffX = ray.orig[0] - center[0];
final float extX = high[0] - center[0];
if( Math.abs(diffX) > extX && diffX*dirX >= 0f ) return false;
final float dirY = ray.dir[1];
final float diffY = ray.orig[1] - center[1];
final float extY = high[1] - center[1];
if( Math.abs(diffY) > extY && diffY*dirY >= 0f ) return false;
final float dirZ = ray.dir[2];
final float diffZ = ray.orig[2] - center[2];
final float extZ = high[2] - center[2];
if( Math.abs(diffZ) > extZ && diffZ*dirZ >= 0f ) return false;
final float absDirY = Math.abs(dirY);
final float absDirZ = Math.abs(dirZ);
float f = dirY * diffZ - dirZ * diffY;
if( Math.abs(f) > extY*absDirZ + extZ*absDirY ) return false;
final float absDirX = Math.abs(dirX);
f = dirZ * diffX - dirX * diffZ;
if( Math.abs(f) > extX*absDirZ + extZ*absDirX ) return false;
f = dirX * diffY - dirY * diffX;
if( Math.abs(f) > extX*absDirY + extY*absDirX ) return false;
return true;
}
/**
* Return intersection of a {@link Ray} with this bounding box,
* or null if none exist.
*
*
* - Original code by Andrew Woo, from "Graphics Gems", Academic Press, 1990 [2]
* - Optimized code by Pierre Terdiman, 2000 (~20-30% faster on my Celeron 500)
* - Epsilon value added by Klaus Hartmann.
*
*
*
* Method is based on the requirements:
*
* - the integer representation of 0.0f is 0x00000000
* - the sign bit of the float is the most significant one
*
*
*
* Report bugs: [email protected] (original author)
*
*
* [1] http://www.codercorner.com/RayAABB.cpp
* [2] http://tog.acm.org/resources/GraphicsGems/gems/RayBox.c
*
* @param result vec3
* @param ray
* @param epsilon
* @param assumeIntersection if true, method assumes an intersection, i.e. by pre-checking via {@link #intersectsRay(Ray)}.
* In this case method will not validate a possible non-intersection and just computes
* coordinates.
* @param tmp1V3 temp vec3
* @param tmp2V3 temp vec3
* @param tmp3V3 temp vec3
* @return float[3] result of intersection coordinates, or null if none exists
*/
public final float[] getRayIntersection(final float[] result, final Ray ray, final float epsilon,
final boolean assumeIntersection,
final float[] tmp1V3, final float[] tmp2V3, final float[] tmp3V3) {
final float[] maxT = { -1f, -1f, -1f };
final float[] origin = ray.orig;
final float[] dir = ray.dir;
boolean inside = true;
// Find candidate planes.
for(int i=0; i<3; i++) {
if(origin[i] < low[i]) {
result[i] = low[i];
inside = false;
// Calculate T distances to candidate planes
if( 0 != Float.floatToIntBits(dir[i]) ) {
maxT[i] = (low[i] - origin[i]) / dir[i];
}
} else if(origin[i] > high[i]) {
result[i] = high[i];
inside = false;
// Calculate T distances to candidate planes
if( 0 != Float.floatToIntBits(dir[i]) ) {
maxT[i] = (high[i] - origin[i]) / dir[i];
}
}
}
// Ray origin inside bounding box
if(inside) {
System.arraycopy(origin, 0, result, 0, 3);
return result;
}
// Get largest of the maxT's for final choice of intersection
int whichPlane = 0;
if(maxT[1] > maxT[whichPlane]) { whichPlane = 1; }
if(maxT[2] > maxT[whichPlane]) { whichPlane = 2; }
if( !assumeIntersection ) {
// Check final candidate actually inside box
if( 0 != ( Float.floatToIntBits(maxT[whichPlane]) & 0x80000000 ) ) {
return null;
}
/** Use unrolled version below ..
for(int i=0; i<3; i++) {
if( i!=whichPlane ) {
result[i] = origin[i] + maxT[whichPlane] * dir[i];
if(result[i] < minB[i] - epsilon || result[i] > maxB[i] + epsilon) { return null; }
// if(result[i] < minB[i] || result[i] > maxB[i] ) { return null; }
}
} */
switch( whichPlane ) {
case 0:
result[1] = origin[1] + maxT[whichPlane] * dir[1];
if(result[1] < low[1] - epsilon || result[1] > high[1] + epsilon) { return null; }
result[2] = origin[2] + maxT[whichPlane] * dir[2];
if(result[2] < low[2] - epsilon || result[2] > high[2] + epsilon) { return null; }
break;
case 1:
result[0] = origin[0] + maxT[whichPlane] * dir[0];
if(result[0] < low[0] - epsilon || result[0] > high[0] + epsilon) { return null; }
result[2] = origin[2] + maxT[whichPlane] * dir[2];
if(result[2] < low[2] - epsilon || result[2] > high[2] + epsilon) { return null; }
break;
case 2:
result[0] = origin[0] + maxT[whichPlane] * dir[0];
if(result[0] < low[0] - epsilon || result[0] > high[0] + epsilon) { return null; }
result[1] = origin[1] + maxT[whichPlane] * dir[1];
if(result[1] < low[1] - epsilon || result[1] > high[1] + epsilon) { return null; }
break;
default:
throw new InternalError("XXX");
}
} else {
switch( whichPlane ) {
case 0:
result[1] = origin[1] + maxT[whichPlane] * dir[1];
result[2] = origin[2] + maxT[whichPlane] * dir[2];
break;
case 1:
result[0] = origin[0] + maxT[whichPlane] * dir[0];
result[2] = origin[2] + maxT[whichPlane] * dir[2];
break;
case 2:
result[0] = origin[0] + maxT[whichPlane] * dir[0];
result[1] = origin[1] + maxT[whichPlane] * dir[1];
break;
default:
throw new InternalError("XXX");
}
}
return result; // ray hits box
}
/**
* Get the size of this AABBox where the size is represented by the
* length of the vector between low and high.
* @return a float representing the size of the AABBox
*/
public final float getSize() {
return VectorUtil.distVec3(low, high);
}
/**
* Get the Center of this AABBox
* @return the xyz-coordinates of the center of the AABBox
*/
public final float[] getCenter() {
return center;
}
/**
* Scale this AABBox by a constant
* @param size a constant float value
* @param tmpV3 caller provided temporary 3-component vector
* @return this AABBox for chaining
*/
public final AABBox scale(final float size, final float[] tmpV3) {
tmpV3[0] = high[0] - center[0];
tmpV3[1] = high[1] - center[1];
tmpV3[2] = high[2] - center[2];
VectorUtil.scaleVec3(tmpV3, tmpV3, size); // in-place scale
VectorUtil.addVec3(high, center, tmpV3);
tmpV3[0] = low[0] - center[0];
tmpV3[1] = low[1] - center[1];
tmpV3[2] = low[2] - center[2];
VectorUtil.scaleVec3(tmpV3, tmpV3, size); // in-place scale
VectorUtil.addVec3(low, center, tmpV3);
return this;
}
/**
* Translate this AABBox by a float[3] vector
* @param t the float[3] translation vector
* @return this AABBox for chaining
*/
public final AABBox translate(final float[] t) {
VectorUtil.addVec3(low, low, t); // in-place translate
VectorUtil.addVec3(high, high, t); // in-place translate
computeCenter();
return this;
}
/**
* Rotate this AABBox by a float[3] vector
* @param quat the {@link Quaternion} used for rotation
* @return this AABBox for chaining
*/
public final AABBox rotate(final Quaternion quat) {
quat.rotateVector(low, 0, low, 0);
quat.rotateVector(high, 0, high, 0);
computeCenter();
return this;
}
public final float getMinX() {
return low[0];
}
public final float getMinY() {
return low[1];
}
public final float getMinZ() {
return low[2];
}
public final float getMaxX() {
return high[0];
}
public final float getMaxY() {
return high[1];
}
public final float getMaxZ() {
return high[2];
}
public final float getWidth(){
return high[0] - low[0];
}
public final float getHeight() {
return high[1] - low[1];
}
public final float getDepth() {
return high[2] - low[2];
}
@Override
public final boolean equals(final Object obj) {
if( obj == this ) {
return true;
}
if( null == obj || !(obj instanceof AABBox) ) {
return false;
}
final AABBox other = (AABBox) obj;
return VectorUtil.isVec2Equal(low, 0, other.low, 0, FloatUtil.EPSILON) &&
VectorUtil.isVec3Equal(high, 0, other.high, 0, FloatUtil.EPSILON) ;
}
@Override
public final int hashCode() {
throw new InternalError("hashCode not designed");
}
/**
* Assume this bounding box as being in object space and
* compute the window bounding box.
*
* If useCenterZ is true,
* only 4 {@link FloatUtil#mapObjToWinCoords(float, float, float, float[], int[], int, float[], int, float[], float[]) mapObjToWinCoords}
* operations are made on points [1..4] using {@link #getCenter()}'s z-value.
* Otherwise 8 {@link FloatUtil#mapObjToWinCoords(float, float, float, float[], int[], int, float[], int, float[], float[]) mapObjToWinCoords}
* operation on all 8 points are performed.
*
*
* [2] ------ [4]
* | |
* | |
* [1] ------ [3]
*
* @param mat4PMv P x Mv matrix
* @param view
* @param useCenterZ
* @param vec3Tmp0 3 component vector for temp storage
* @param vec4Tmp1 4 component vector for temp storage
* @param vec4Tmp2 4 component vector for temp storage
* @return
*/
public AABBox mapToWindow(final AABBox result, final float[/*16*/] mat4PMv, final int[] view, final boolean useCenterZ,
final float[] vec3Tmp0, final float[] vec4Tmp1, final float[] vec4Tmp2) {
{
// System.err.printf("AABBox.mapToWindow.0: view[%d, %d, %d, %d], this %s%n", view[0], view[1], view[2], view[3], toString());
final float objZ = useCenterZ ? center[2] : getMinZ();
FloatUtil.mapObjToWinCoords(getMinX(), getMinY(), objZ, mat4PMv, view, 0, vec3Tmp0, 0, vec4Tmp1, vec4Tmp2);
// System.err.printf("AABBox.mapToWindow.p1: %f, %f, %f -> %f, %f, %f%n", getMinX(), getMinY(), objZ, vec3Tmp0[0], vec3Tmp0[1], vec3Tmp0[2]);
// System.err.println("AABBox.mapToWindow.p1:");
// System.err.println(FloatUtil.matrixToString(null, " mat4PMv", "%10.5f", mat4PMv, 0, 4, 4, false /* rowMajorOrder */));
result.reset();
result.resize(vec3Tmp0, 0);
FloatUtil.mapObjToWinCoords(getMinX(), getMaxY(), objZ, mat4PMv, view, 0, vec3Tmp0, 0, vec4Tmp1, vec4Tmp2);
// System.err.printf("AABBox.mapToWindow.p2: %f, %f, %f -> %f, %f, %f%n", getMinX(), getMaxY(), objZ, vec3Tmp0[0], vec3Tmp0[1], vec3Tmp0[2]);
result.resize(vec3Tmp0, 0);
FloatUtil.mapObjToWinCoords(getMaxX(), getMinY(), objZ, mat4PMv, view, 0, vec3Tmp0, 0, vec4Tmp1, vec4Tmp2);
// System.err.printf("AABBox.mapToWindow.p3: %f, %f, %f -> %f, %f, %f%n", getMaxX(), getMinY(), objZ, vec3Tmp0[0], vec3Tmp0[1], vec3Tmp0[2]);
result.resize(vec3Tmp0, 0);
FloatUtil.mapObjToWinCoords(getMaxX(), getMaxY(), objZ, mat4PMv, view, 0, vec3Tmp0, 0, vec4Tmp1, vec4Tmp2);
// System.err.printf("AABBox.mapToWindow.p4: %f, %f, %f -> %f, %f, %f%n", getMaxX(), getMaxY(), objZ, vec3Tmp0[0], vec3Tmp0[1], vec3Tmp0[2]);
result.resize(vec3Tmp0, 0);
}
if( !useCenterZ ) {
final float objZ = getMaxZ();
FloatUtil.mapObjToWinCoords(getMinX(), getMinY(), objZ, mat4PMv, view, 0, vec3Tmp0, 0, vec4Tmp1, vec4Tmp2);
result.resize(vec3Tmp0, 0);
FloatUtil.mapObjToWinCoords(getMinX(), getMaxY(), objZ, mat4PMv, view, 0, vec3Tmp0, 0, vec4Tmp1, vec4Tmp2);
result.resize(vec3Tmp0, 0);
FloatUtil.mapObjToWinCoords(getMaxX(), getMinY(), objZ, mat4PMv, view, 0, vec3Tmp0, 0, vec4Tmp1, vec4Tmp2);
result.resize(vec3Tmp0, 0);
FloatUtil.mapObjToWinCoords(getMaxX(), getMaxY(), objZ, mat4PMv, view, 0, vec3Tmp0, 0, vec4Tmp1, vec4Tmp2);
result.resize(vec3Tmp0, 0);
}
if( DEBUG ) {
System.err.printf("AABBox.mapToWindow: view[%d, %d], this %s -> %s%n", view[0], view[1], toString(), result.toString());
}
return result;
}
@Override
public final String toString() {
return "[ dim "+getWidth()+" x "+getHeight()+" x "+getDepth()+
", box "+low[0]+" / "+low[1]+" / "+low[2]+" .. "+high[0]+" / "+high[1]+" / "+high[2]+
", ctr "+center[0]+" / "+center[1]+" / "+center[2]+" ]";
}
}
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