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MapsUtil.h
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30 //
31 /// @file MapsUtil.h
32 
33 #ifndef OPENVDB_UTIL_MAPSUTIL_HAS_BEEN_INCLUDED
34 #define OPENVDB_UTIL_MAPSUTIL_HAS_BEEN_INCLUDED
35 
36 #include <openvdb/math/Maps.h>
37 
38 
39 namespace openvdb {
41 namespace OPENVDB_VERSION_NAME {
42 namespace util {
43 
44 // Utility methods for calculating bounding boxes
45 
46 /// @brief Calculate an axis-aligned bounding box in the given map's domain
47 /// (e.g., index space) from an axis-aligned bounding box in its range
48 /// (e.g., world space)
49 template<typename MapType>
50 inline void
51 calculateBounds(const MapType& map, const BBoxd& in, BBoxd& out)
52 {
53  const Vec3d& min = in.min();
54  const Vec3d& max = in.max();
55 
56  // the pre-image of the 8 corners of the box
57  Vec3d corners[8];
58  corners[0] = in.min();;
59  corners[1] = Vec3d(min(0), min(1), min(2));
60  corners[2] = Vec3d(max(0), max(1), min(2));
61  corners[3] = Vec3d(min(0), max(1), min(2));
62  corners[4] = Vec3d(min(0), min(1), max(2));
63  corners[5] = Vec3d(max(0), min(1), max(2));
64  corners[6] = max;
65  corners[7] = Vec3d(min(0), max(1), max(2));
66 
67  Vec3d pre_image;
68  Vec3d& out_min = out.min();
69  Vec3d& out_max = out.max();
70  out_min = map.applyInverseMap(corners[0]);
71  out_max = min;
72  for (int i = 1; i < 8; ++i) {
73  pre_image = map.applyInverseMap(corners[i]);
74  for (int j = 0; j < 3; ++j) {
75  out_min(j) = std::min( out_min(j), pre_image(j));
76  out_max(j) = std::max( out_max(j), pre_image(j));
77  }
78  }
79 }
80 
81 
82 /// @brief Calculate an axis-aligned bounding box in the given map's domain
83 /// from a spherical bounding box in its range.
84 template<typename MapType>
85 inline void
86 calculateBounds(const MapType& map, const Vec3d& center, const Real radius, BBoxd& out)
87 {
88  // On return, out gives a bounding box in continuous index space
89  // that encloses the sphere.
90  //
91  // the image of a sphere under the inverse of the linearMap will be an ellipsoid.
92 
94  // I want to find extrema for three functions f(x', y', z') = x', or = y', or = z'
95  // with the constraint that g = (x-xo)^2 + (y-yo)^2 + (z-zo)^2 = r^2.
96  // Where the point x,y,z is the image of x',y',z'
97  // Solve: \lambda Grad(g) = Grad(f) and g = r^2.
98  // Note: here (x,y,z) is the image of (x',y',z'), and the gradient
99  // is w.r.t the (') space.
100  //
101  // This can be solved exactly: e_a^T (x' -xo') =\pm r\sqrt(e_a^T J^(-1)J^(-T)e_a)
102  // where e_a is one of the three unit vectors. - djh.
103 
104  /// find the image of the center of the sphere
105  Vec3d center_pre_image = map.applyInverseMap(center);
106 
107  std::vector<Vec3d> coordinate_units;
108  coordinate_units.push_back(Vec3d(1,0,0));
109  coordinate_units.push_back(Vec3d(0,1,0));
110  coordinate_units.push_back(Vec3d(0,0,1));
111 
112  Vec3d& out_min = out.min();
113  Vec3d& out_max = out.max();
114  for (int direction = 0; direction < 3; ++direction) {
115  Vec3d temp = map.applyIJT(coordinate_units[direction]);
116  double offset =
117  radius * sqrt(temp.x()*temp.x() + temp.y()*temp.y() + temp.z()*temp.z());
118  out_min(direction) = center_pre_image(direction) - offset;
119  out_max(direction) = center_pre_image(direction) + offset;
120  }
121 
122  } else {
123  // This is some unknown map type. In this case, we form an axis-aligned
124  // bounding box for the sphere in world space and find the pre-images of
125  // the corners in index space. From these corners we compute an axis-aligned
126  // bounding box in index space.
127  BBoxd bounding_box(center - radius*Vec3d(1,1,1), center + radius*Vec3d(1,1,1));
128  calculateBounds<MapType>(map, bounding_box, out);
129  }
130 }
131 
132 
133 namespace { // anonymous namespace for this helper function
134 
135 /// @brief Find the intersection of a line passing through the point
136 /// \f$ (x=0, z=-1/g)\f$ with the circle \f$ (x-xo)^2 + (z-zo)^2 = r^2 \f$
137 /// at a point tangent to the circle.
138 /// @return 0 if the focal point (0, -1/g) is inside the circle,
139 /// 1 if the focal point touches the circle, or 2 when both points are found.
140 inline int
141 findTangentPoints(const double g, const double xo, const double zo,
142  const double r, double& xp, double& zp, double& xm, double& zm)
143 {
144  double x2 = xo * xo;
145  double r2 = r * r;
146  double xd = g * xo;
147  double xd2 = xd*xd;
148  double zd = g * zo + 1.;
149  double zd2 = zd*zd;
150  double rd2 = r2*g*g;
151 
152  double distA = xd2 + zd2;
153  double distB = distA - rd2;
154 
155  if (distB > 0) {
156  double discriminate = sqrt(distB);
157 
158  xp = xo - xo*rd2/distA + r * zd *discriminate / distA;
159  xm = xo - xo*rd2/distA - r * zd *discriminate / distA;
160 
161  zp = (zo*zd2 + zd*g*(x2 - r2) - xo*xo*g - r*xd*discriminate) / distA;
162  zm = (zo*zd2 + zd*g*(x2 - r2) - xo*xo*g + r*xd*discriminate) / distA;
163 
164  return 2;
165 
166  } if (0 >= distB && distB >= -1e-9) {
167  // the circle touches the focal point (x=0, z = -1/g)
168  xp = 0; xm = 0;
169  zp = -1/g; zm = -1/g;
170 
171  return 1;
172  }
173 
174  return 0;
175 }
176 
177 } // end anonymous namespace
178 
179 
180 /// @brief Calculate an axis-aligned bounding box in index space
181 /// from a spherical bounding box in world space.
182 /// @note This specialization is optimized for a frustum map
183 template<>
184 inline void
185 calculateBounds<math::NonlinearFrustumMap>(const math::NonlinearFrustumMap& frustum,
186  const Vec3d& center, const Real radius, BBoxd& out)
187 {
188  // The frustum is a nonlinear map followed by a uniform scale, rotation, translation.
189  // First we invert the translation, rotation and scale to find the spherical pre-image
190  // of the sphere in "local" coordinates where the frustum is aligned with the near plane
191  // on the z=0 plane and the "camera" is located at (x=0, y=0, z=-1/g).
192 
193  // check that the internal map has no shear.
194  const math::AffineMap& secondMap = frustum.secondMap();
195  // test if the linear part has shear or non-uniform scaling
196  if (!frustum.hasSimpleAffine()) {
197 
198  // In this case, we form an axis-aligned bounding box for sphere in world space
199  // and find the pre_images of the corners in voxel space. From these corners we
200  // compute an axis-algined bounding box in voxel-spae
201  BBoxd bounding_box(center - radius*Vec3d(1,1,1), center + radius*Vec3d(1,1,1));
202  calculateBounds<math::NonlinearFrustumMap>(frustum, bounding_box, out);
203  return;
204  }
205 
206  // for convenience
207  Vec3d& out_min = out.min();
208  Vec3d& out_max = out.max();
209 
210  Vec3d centerLS = secondMap.applyInverseMap(center);
211  Vec3d voxelSize = secondMap.voxelSize();
212 
213  // all the voxels have the same size since we know this is a simple affine map
214  double radiusLS = radius / voxelSize(0);
215 
216  double gamma = frustum.getGamma();
217  double xp;
218  double zp;
219  double xm;
220  double zm;
221  int soln_number;
222 
223  // the bounding box in index space for the points in the frustum
224  const BBoxd& bbox = frustum.getBBox();
225  // initialize min and max
226  const double x_min = bbox.min().x();
227  const double y_min = bbox.min().y();
228  const double z_min = bbox.min().z();
229 
230  const double x_max = bbox.max().x();
231  const double y_max = bbox.max().y();
232  const double z_max = bbox.max().z();
233 
234  out_min.x() = x_min;
235  out_max.x() = x_max;
236  out_min.y() = y_min;
237  out_max.y() = y_max;
238 
239  Vec3d extreme;
240  Vec3d extreme2;
241  Vec3d pre_image;
242  // find the x-range
243  soln_number = findTangentPoints(gamma, centerLS.x(), centerLS.z(), radiusLS, xp, zp, xm, zm);
244  if (soln_number == 2) {
245  extreme.x() = xp;
246  extreme.y() = centerLS.y();
247  extreme.z() = zp;
248 
249  // location in world space of the tangent point
250  extreme2 = secondMap.applyMap(extreme);
251  // convert back to voxel space
252  pre_image = frustum.applyInverseMap(extreme2);
253  out_max.x() = std::max(x_min, std::min(x_max, pre_image.x()));
254 
255  extreme.x() = xm;
256  extreme.y() = centerLS.y();
257  extreme.z() = zm;
258  // location in world space of the tangent point
259  extreme2 = secondMap.applyMap(extreme);
260 
261  // convert back to voxel space
262  pre_image = frustum.applyInverseMap(extreme2);
263  out_min.x() = std::max(x_min, std::min(x_max, pre_image.x()));
264 
265  } else if (soln_number == 1) {
266  // the circle was tangent at the focal point
267  } else if (soln_number == 0) {
268  // the focal point was inside the circle
269  }
270 
271  // find the y-range
272  soln_number = findTangentPoints(gamma, centerLS.y(), centerLS.z(), radiusLS, xp, zp, xm, zm);
273  if (soln_number == 2) {
274  extreme.x() = centerLS.x();
275  extreme.y() = xp;
276  extreme.z() = zp;
277 
278  // location in world space of the tangent point
279  extreme2 = secondMap.applyMap(extreme);
280  // convert back to voxel space
281  pre_image = frustum.applyInverseMap(extreme2);
282  out_max.y() = std::max(y_min, std::min(y_max, pre_image.y()));
283 
284  extreme.x() = centerLS.x();
285  extreme.y() = xm;
286  extreme.z() = zm;
287  extreme2 = secondMap.applyMap(extreme);
288 
289  // convert back to voxel space
290  pre_image = frustum.applyInverseMap(extreme2);
291  out_min.y() = std::max(y_min, std::min(y_max, pre_image.y()));
292 
293  } else if (soln_number == 1) {
294  // the circle was tangent at the focal point
295  } else if (soln_number == 0) {
296  // the focal point was inside the circle
297  }
298 
299  // the near and far
300  // the closest point. The front of the frustum is at 0 in index space
301  double near_dist = std::max(centerLS.z() - radiusLS, 0.);
302  // the farthest point. The back of the frustum is at mDepth in index space
303  double far_dist = std::min(centerLS.z() + radiusLS, frustum.getDepth() );
304 
305  Vec3d near_point(0.f, 0.f, near_dist);
306  Vec3d far_point(0.f, 0.f, far_dist);
307 
308  out_min.z() = std::max(z_min, frustum.applyInverseMap(secondMap.applyMap(near_point)).z());
309  out_max.z() = std::min(z_max, frustum.applyInverseMap(secondMap.applyMap(far_point)).z());
310 
311 }
312 
313 } // namespace util
314 } // namespace OPENVDB_VERSION_NAME
315 } // namespace openvdb
316 
317 #endif // OPENVDB_UTIL_MAPSUTIL_HAS_BEEN_INCLUDED
318 
319 // Copyright (c) 2012-2017 DreamWorks Animation LLC
320 // All rights reserved. This software is distributed under the
321 // Mozilla Public License 2.0 ( http://www.mozilla.org/MPL/2.0/ )
Vec3d voxelSize() const override
Return the lengths of the images of the segments (0,0,0)-(1,0,0), (0,0,0)-(0,1,0) and (0...
Definition: Maps.h:490
const hboost::disable_if_c< VecTraits< T >::IsVec, T >::type & min(const T &a, const T &b)
Definition: Composite.h:128
IMF_EXPORT IMATH_NAMESPACE::V3f direction(const IMATH_NAMESPACE::Box2i &dataWindow, const IMATH_NAMESPACE::V2f &pixelPosition)
GLdouble GLdouble GLdouble z
Definition: glcorearb.h:847
GLboolean GLboolean g
Definition: glcorearb.h:1221
This map is composed of three steps. First it will take a box of size (Lx X Ly X Lz) defined by a mem...
Definition: Maps.h:1902
png_uint_32 i
Definition: png.h:2877
const hboost::disable_if_c< VecTraits< T >::IsVec, T >::type & max(const T &a, const T &b)
Definition: Composite.h:132
GLfloat f
Definition: glcorearb.h:1925
const Vec3T & min() const
Return a const reference to the minimum point of the BBox.
Definition: BBox.h:84
#define OPENVDB_VERSION_NAME
Definition: version.h:43
A general linear transform using homogeneous coordinates to perform rotation, scaling, shear and translation.
Definition: Maps.h:324
GLintptr offset
Definition: glcorearb.h:664
Vec3d applyInverseMap(const Vec3d &in) const override
Return the pre-image of in under the map.
Definition: Maps.h:442
const Vec3T & max() const
Return a const reference to the maximum point of the BBox.
Definition: BBox.h:87
Vec3d applyMap(const Vec3d &in) const override
Return the image of in under the map.
Definition: Maps.h:440
void calculateBounds(const MapType &map, const BBoxd &in, BBoxd &out)
Calculate an axis-aligned bounding box in the given map's domain (e.g., index space) from an axis-ali...
Definition: MapsUtil.h:51
T & x()
Reference to the component, e.g. v.x() = 4.5f;.
Definition: Vec3.h:109
GLboolean r
Definition: glcorearb.h:1221
#define OPENVDB_USE_VERSION_NAMESPACE
Definition: version.h:71