static char help[] = "Tests for cell geometry\n\n";

#include <petscdmplex.h>

typedef enum {
  RUN_REFERENCE,
  RUN_HEX_CURVED,
  RUN_FILE,
  RUN_DISPLAY
} RunType;

typedef struct {
  DM        dm;
  RunType   runType;   /* Type of mesh to use */
  PetscBool transform; /* Use random coordinate transformations */
  /* Data for input meshes */
  PetscReal *v0, *J, *invJ, *detJ;    /* FEM data */
  PetscReal *centroid, *normal, *vol; /* FVM data */
} AppCtx;

static PetscErrorCode ReadMesh(MPI_Comm comm, AppCtx *user, DM *dm)
{
  PetscFunctionBegin;
  PetscCall(DMCreate(comm, dm));
  PetscCall(DMSetType(*dm, DMPLEX));
  PetscCall(DMSetFromOptions(*dm));
  PetscCall(DMSetApplicationContext(*dm, user));
  PetscCall(PetscObjectSetName((PetscObject)*dm, "Input Mesh"));
  PetscCall(DMViewFromOptions(*dm, NULL, "-dm_view"));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode ProcessOptions(MPI_Comm comm, AppCtx *options)
{
  const char *runTypes[4] = {"reference", "hex_curved", "file", "display"};
  PetscInt    run;

  PetscFunctionBeginUser;
  options->runType   = RUN_REFERENCE;
  options->transform = PETSC_FALSE;

  PetscOptionsBegin(comm, "", "Geometry Test Options", "DMPLEX");
  run = options->runType;
  PetscCall(PetscOptionsEList("-run_type", "The run type", "ex8.c", runTypes, 3, runTypes[options->runType], &run, NULL));
  options->runType = (RunType)run;
  PetscCall(PetscOptionsBool("-transform", "Use random transforms", "ex8.c", options->transform, &options->transform, NULL));

  if (options->runType == RUN_FILE) {
    PetscInt  dim, cStart, cEnd, numCells, n;
    PetscBool flg, feFlg;

    PetscCall(ReadMesh(PETSC_COMM_WORLD, options, &options->dm));
    PetscCall(DMGetDimension(options->dm, &dim));
    PetscCall(DMPlexGetHeightStratum(options->dm, 0, &cStart, &cEnd));
    numCells = cEnd - cStart;
    PetscCall(PetscMalloc4(numCells * dim, &options->v0, numCells * dim * dim, &options->J, numCells * dim * dim, &options->invJ, numCells, &options->detJ));
    PetscCall(PetscMalloc1(numCells * dim, &options->centroid));
    PetscCall(PetscMalloc1(numCells * dim, &options->normal));
    PetscCall(PetscMalloc1(numCells, &options->vol));
    n = numCells * dim;
    PetscCall(PetscOptionsRealArray("-v0", "Input v0 for each cell", "ex8.c", options->v0, &n, &feFlg));
    PetscCheck(!feFlg || n == numCells * dim, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Invalid size of v0 %" PetscInt_FMT " should be %" PetscInt_FMT, n, numCells * dim);
    n = numCells * dim * dim;
    PetscCall(PetscOptionsRealArray("-J", "Input Jacobian for each cell", "ex8.c", options->J, &n, &flg));
    PetscCheck(!flg || n == numCells * dim * dim, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Invalid size of J %" PetscInt_FMT " should be %" PetscInt_FMT, n, numCells * dim * dim);
    n = numCells * dim * dim;
    PetscCall(PetscOptionsRealArray("-invJ", "Input inverse Jacobian for each cell", "ex8.c", options->invJ, &n, &flg));
    PetscCheck(!flg || n == numCells * dim * dim, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Invalid size of invJ %" PetscInt_FMT " should be %" PetscInt_FMT, n, numCells * dim * dim);
    n = numCells;
    PetscCall(PetscOptionsRealArray("-detJ", "Input Jacobian determinant for each cell", "ex8.c", options->detJ, &n, &flg));
    PetscCheck(!flg || n == numCells, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Invalid size of detJ %" PetscInt_FMT " should be %" PetscInt_FMT, n, numCells);
    n = numCells * dim;
    if (!feFlg) {
      PetscCall(PetscFree4(options->v0, options->J, options->invJ, options->detJ));
      options->v0 = options->J = options->invJ = options->detJ = NULL;
    }
    PetscCall(PetscOptionsRealArray("-centroid", "Input centroid for each cell", "ex8.c", options->centroid, &n, &flg));
    PetscCheck(!flg || n == numCells * dim, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Invalid size of centroid %" PetscInt_FMT " should be %" PetscInt_FMT, n, numCells * dim);
    if (!flg) {
      PetscCall(PetscFree(options->centroid));
      options->centroid = NULL;
    }
    n = numCells * dim;
    PetscCall(PetscOptionsRealArray("-normal", "Input normal for each cell", "ex8.c", options->normal, &n, &flg));
    PetscCheck(!flg || n == numCells * dim, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Invalid size of normal %" PetscInt_FMT " should be %" PetscInt_FMT, n, numCells * dim);
    if (!flg) {
      PetscCall(PetscFree(options->normal));
      options->normal = NULL;
    }
    n = numCells;
    PetscCall(PetscOptionsRealArray("-vol", "Input volume for each cell", "ex8.c", options->vol, &n, &flg));
    PetscCheck(!flg || n == numCells, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Invalid size of vol %" PetscInt_FMT " should be %" PetscInt_FMT, n, numCells);
    if (!flg) {
      PetscCall(PetscFree(options->vol));
      options->vol = NULL;
    }
  } else if (options->runType == RUN_DISPLAY) {
    PetscCall(ReadMesh(PETSC_COMM_WORLD, options, &options->dm));
  }
  PetscOptionsEnd();

  if (options->transform) PetscCall(PetscPrintf(comm, "Using random transforms\n"));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode ChangeCoordinates(DM dm, PetscInt spaceDim, PetscScalar vertexCoords[])
{
  PetscSection coordSection;
  Vec          coordinates;
  PetscScalar *coords;
  PetscInt     vStart, vEnd, v, d, coordSize;

  PetscFunctionBegin;
  PetscCall(DMPlexGetDepthStratum(dm, 0, &vStart, &vEnd));
  PetscCall(DMGetCoordinateSection(dm, &coordSection));
  PetscCall(PetscSectionSetNumFields(coordSection, 1));
  PetscCall(PetscSectionSetFieldComponents(coordSection, 0, spaceDim));
  PetscCall(PetscSectionSetChart(coordSection, vStart, vEnd));
  for (v = vStart; v < vEnd; ++v) {
    PetscCall(PetscSectionSetDof(coordSection, v, spaceDim));
    PetscCall(PetscSectionSetFieldDof(coordSection, v, 0, spaceDim));
  }
  PetscCall(PetscSectionSetUp(coordSection));
  PetscCall(PetscSectionGetStorageSize(coordSection, &coordSize));
  PetscCall(VecCreate(PETSC_COMM_SELF, &coordinates));
  PetscCall(PetscObjectSetName((PetscObject)coordinates, "coordinates"));
  PetscCall(VecSetSizes(coordinates, coordSize, PETSC_DETERMINE));
  PetscCall(VecSetFromOptions(coordinates));
  PetscCall(VecGetArray(coordinates, &coords));
  for (v = vStart; v < vEnd; ++v) {
    PetscInt off;

    PetscCall(PetscSectionGetOffset(coordSection, v, &off));
    for (d = 0; d < spaceDim; ++d) coords[off + d] = vertexCoords[(v - vStart) * spaceDim + d];
  }
  PetscCall(VecRestoreArray(coordinates, &coords));
  PetscCall(DMSetCoordinateDim(dm, spaceDim));
  PetscCall(DMSetCoordinatesLocal(dm, coordinates));
  PetscCall(VecDestroy(&coordinates));
  PetscCall(DMViewFromOptions(dm, NULL, "-dm_view"));
  PetscFunctionReturn(PETSC_SUCCESS);
}

#define RelativeError(a, b) PetscAbs(a - b) / (1.0 + PetscMax(PetscAbs(a), PetscAbs(b)))

static PetscErrorCode CheckFEMGeometry(DM dm, PetscInt cell, PetscInt spaceDim, PetscReal v0Ex[], PetscReal JEx[], PetscReal invJEx[], PetscReal detJEx)
{
  PetscReal v0[3], J[9], invJ[9], detJ;
  PetscInt  d, i, j;

  PetscFunctionBegin;
  PetscCall(DMPlexComputeCellGeometryFEM(dm, cell, NULL, v0, J, invJ, &detJ));
  for (d = 0; d < spaceDim; ++d) {
    if (v0[d] != v0Ex[d]) {
      switch (spaceDim) {
      case 2:
        SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "Invalid v0 (%g, %g) != (%g, %g)", (double)v0[0], (double)v0[1], (double)v0Ex[0], (double)v0Ex[1]);
      case 3:
        SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "Invalid v0 (%g, %g, %g) != (%g, %g, %g)", (double)v0[0], (double)v0[1], (double)v0[2], (double)v0Ex[0], (double)v0Ex[1], (double)v0Ex[2]);
      default:
        SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "Invalid space dimension %" PetscInt_FMT, spaceDim);
      }
    }
  }
  for (i = 0; i < spaceDim; ++i) {
    for (j = 0; j < spaceDim; ++j) {
      PetscCheck(RelativeError(J[i * spaceDim + j], JEx[i * spaceDim + j]) < 10 * PETSC_SMALL, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Invalid J[%" PetscInt_FMT ",%" PetscInt_FMT "]: %g != %g", i, j, (double)J[i * spaceDim + j], (double)JEx[i * spaceDim + j]);
      PetscCheck(RelativeError(invJ[i * spaceDim + j], invJEx[i * spaceDim + j]) < 10 * PETSC_SMALL, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Invalid invJ[%" PetscInt_FMT ",%" PetscInt_FMT "]: %g != %g", i, j, (double)invJ[i * spaceDim + j], (double)invJEx[i * spaceDim + j]);
    }
  }
  PetscCheck(RelativeError(detJ, detJEx) < 10 * PETSC_SMALL, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Invalid |J| = %g != %g diff %g", (double)detJ, (double)detJEx, (double)(detJ - detJEx));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode CheckFVMGeometry(DM dm, PetscInt cell, PetscInt spaceDim, PetscReal centroidEx[], PetscReal normalEx[], PetscReal volEx)
{
  PetscReal tol = PetscMax(10 * PETSC_SMALL, 1e-10);
  PetscReal centroid[3], normal[3], vol;
  PetscInt  d;

  PetscFunctionBegin;
  PetscCall(DMPlexComputeCellGeometryFVM(dm, cell, volEx ? &vol : NULL, centroidEx ? centroid : NULL, normalEx ? normal : NULL));
  for (d = 0; d < spaceDim; ++d) {
    if (centroidEx)
      PetscCheck(RelativeError(centroid[d], centroidEx[d]) < tol, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Cell %" PetscInt_FMT ", Invalid centroid[%" PetscInt_FMT "]: %g != %g diff %g", cell, d, (double)centroid[d], (double)centroidEx[d], (double)(centroid[d] - centroidEx[d]));
    if (normalEx) PetscCheck(RelativeError(normal[d], normalEx[d]) < tol, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Cell %" PetscInt_FMT ", Invalid normal[%" PetscInt_FMT "]: %g != %g", cell, d, (double)normal[d], (double)normalEx[d]);
  }
  if (volEx) PetscCheck(RelativeError(volEx, vol) < tol, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Cell %" PetscInt_FMT ", Invalid volume = %g != %g diff %g", cell, (double)vol, (double)volEx, (double)(vol - volEx));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode CheckGaussLaw(DM dm, PetscInt cell)
{
  DMPolytopeType  ct;
  PetscReal       tol = PetscMax(10 * PETSC_SMALL, 1e-10);
  PetscReal       normal[3], integral[3] = {0., 0., 0.}, area;
  const PetscInt *cone, *ornt;
  PetscInt        coneSize, f, dim, cdim, d;

  PetscFunctionBegin;
  PetscCall(DMGetDimension(dm, &dim));
  PetscCall(DMGetCoordinateDim(dm, &cdim));
  if (dim != cdim) PetscFunctionReturn(PETSC_SUCCESS);
  PetscCall(DMPlexGetCellType(dm, cell, &ct));
  if (ct == DM_POLYTOPE_TRI_PRISM_TENSOR) PetscFunctionReturn(PETSC_SUCCESS);
  PetscCall(DMPlexGetConeSize(dm, cell, &coneSize));
  PetscCall(DMPlexGetCone(dm, cell, &cone));
  PetscCall(DMPlexGetConeOrientation(dm, cell, &ornt));
  for (f = 0; f < coneSize; ++f) {
    const PetscInt sgn = dim == 1 ? (f == 0 ? -1 : 1) : (ornt[f] < 0 ? -1 : 1);

    PetscCall(DMPlexComputeCellGeometryFVM(dm, cone[f], &area, NULL, normal));
    for (d = 0; d < cdim; ++d) integral[d] += sgn * area * normal[d];
  }
  for (d = 0; d < cdim; ++d)
    PetscCheck(PetscAbsReal(integral[d]) < tol, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Cell %" PetscInt_FMT " Surface integral for component %" PetscInt_FMT ": %g != 0. as it should be for a constant field", cell, d, (double)integral[d]);
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode CheckCell(DM dm, PetscInt cell, PetscBool transform, PetscReal v0Ex[], PetscReal JEx[], PetscReal invJEx[], PetscReal detJEx, PetscReal centroidEx[], PetscReal normalEx[], PetscReal volEx, PetscReal faceCentroidEx[], PetscReal faceNormalEx[], PetscReal faceVolEx[])
{
  const PetscInt *cone;
  PetscInt        coneSize, c;
  PetscInt        dim, depth, cdim;

  PetscFunctionBegin;
  PetscCall(DMPlexGetDepth(dm, &depth));
  PetscCall(DMGetDimension(dm, &dim));
  PetscCall(DMGetCoordinateDim(dm, &cdim));
  if (v0Ex) PetscCall(CheckFEMGeometry(dm, cell, cdim, v0Ex, JEx, invJEx, detJEx));
  if (dim == depth && centroidEx) {
    PetscCall(CheckFVMGeometry(dm, cell, cdim, centroidEx, normalEx, volEx));
    PetscCall(CheckGaussLaw(dm, cell));
    if (faceCentroidEx) {
      PetscCall(DMPlexGetConeSize(dm, cell, &coneSize));
      PetscCall(DMPlexGetCone(dm, cell, &cone));
      for (c = 0; c < coneSize; ++c) PetscCall(CheckFVMGeometry(dm, cone[c], dim, &faceCentroidEx[c * dim], &faceNormalEx[c * dim], faceVolEx[c]));
    }
  }
  if (transform) {
    Vec          coordinates;
    PetscSection coordSection;
    PetscScalar *coords = NULL, *origCoords, *newCoords;
    PetscReal   *v0ExT, *JExT, *invJExT, detJExT = 0, *centroidExT, *normalExT, volExT = 0;
    PetscReal   *faceCentroidExT, *faceNormalExT, faceVolExT;
    PetscRandom  r, ang, ang2;
    PetscInt     coordSize, numCorners, t;

    PetscCall(DMGetCoordinatesLocal(dm, &coordinates));
    PetscCall(DMGetCoordinateSection(dm, &coordSection));
    PetscCall(DMPlexVecGetClosure(dm, coordSection, coordinates, cell, &coordSize, &coords));
    PetscCall(PetscMalloc2(coordSize, &origCoords, coordSize, &newCoords));
    PetscCall(PetscMalloc5(cdim, &v0ExT, cdim * cdim, &JExT, cdim * cdim, &invJExT, cdim, &centroidExT, cdim, &normalExT));
    PetscCall(PetscMalloc2(cdim, &faceCentroidExT, cdim, &faceNormalExT));
    for (c = 0; c < coordSize; ++c) origCoords[c] = coords[c];
    PetscCall(DMPlexVecRestoreClosure(dm, coordSection, coordinates, cell, &coordSize, &coords));
    numCorners = coordSize / cdim;

    PetscCall(PetscRandomCreate(PETSC_COMM_SELF, &r));
    PetscCall(PetscRandomSetFromOptions(r));
    PetscCall(PetscRandomSetInterval(r, 0.0, 10.0));
    PetscCall(PetscRandomCreate(PETSC_COMM_SELF, &ang));
    PetscCall(PetscRandomSetFromOptions(ang));
    PetscCall(PetscRandomSetInterval(ang, 0.0, 2 * PETSC_PI));
    PetscCall(PetscRandomCreate(PETSC_COMM_SELF, &ang2));
    PetscCall(PetscRandomSetFromOptions(ang2));
    PetscCall(PetscRandomSetInterval(ang2, 0.0, PETSC_PI));
    for (t = 0; t < 1; ++t) {
      PetscScalar trans[3];
      PetscReal   R[9], rot[3], rotM[9];
      PetscReal   scale, phi, theta, psi = 0.0, norm;
      PetscInt    d, e, f, p;

      for (c = 0; c < coordSize; ++c) newCoords[c] = origCoords[c];
      PetscCall(PetscRandomGetValueReal(r, &scale));
      PetscCall(PetscRandomGetValueReal(ang, &phi));
      PetscCall(PetscRandomGetValueReal(ang2, &theta));
      for (d = 0; d < cdim; ++d) PetscCall(PetscRandomGetValue(r, &trans[d]));
      switch (cdim) {
      case 2:
        R[0] = PetscCosReal(phi);
        R[1] = -PetscSinReal(phi);
        R[2] = PetscSinReal(phi);
        R[3] = PetscCosReal(phi);
        break;
      case 3: {
        const PetscReal ct = PetscCosReal(theta), st = PetscSinReal(theta);
        const PetscReal cp = PetscCosReal(phi), sp = PetscSinReal(phi);
        const PetscReal cs = PetscCosReal(psi), ss = PetscSinReal(psi);
        R[0] = ct * cs;
        R[1] = sp * st * cs - cp * ss;
        R[2] = sp * ss + cp * st * cs;
        R[3] = ct * ss;
        R[4] = cp * cs + sp * st * ss;
        R[5] = cp * st * ss - sp * cs;
        R[6] = -st;
        R[7] = sp * ct;
        R[8] = cp * ct;
        break;
      }
      default:
        SETERRQ(PetscObjectComm((PetscObject)dm), PETSC_ERR_ARG_WRONG, "Invalid coordinate dimension %" PetscInt_FMT, cdim);
      }
      if (v0Ex) {
        detJExT = detJEx;
        for (d = 0; d < cdim; ++d) {
          v0ExT[d] = v0Ex[d];
          for (e = 0; e < cdim; ++e) {
            JExT[d * cdim + e]    = JEx[d * cdim + e];
            invJExT[d * cdim + e] = invJEx[d * cdim + e];
          }
        }
        for (d = 0; d < cdim; ++d) {
          v0ExT[d] *= scale;
          v0ExT[d] += PetscRealPart(trans[d]);
          /* Only scale dimensions in the manifold */
          for (e = 0; e < dim; ++e) {
            JExT[d * cdim + e] *= scale;
            invJExT[d * cdim + e] /= scale;
          }
          if (d < dim) detJExT *= scale;
        }
        /* Do scaling and translation before rotation, so that we can leave out the normal dimension for lower dimensional manifolds */
        for (d = 0; d < cdim; ++d) {
          for (e = 0, rot[d] = 0.0; e < cdim; ++e) rot[d] += R[d * cdim + e] * v0ExT[e];
        }
        for (d = 0; d < cdim; ++d) v0ExT[d] = rot[d];
        for (d = 0; d < cdim; ++d) {
          for (e = 0; e < cdim; ++e) {
            for (f = 0, rotM[d * cdim + e] = 0.0; f < cdim; ++f) rotM[d * cdim + e] += R[d * cdim + f] * JExT[f * cdim + e];
          }
        }
        for (d = 0; d < cdim; ++d) {
          for (e = 0; e < cdim; ++e) JExT[d * cdim + e] = rotM[d * cdim + e];
        }
        for (d = 0; d < cdim; ++d) {
          for (e = 0; e < cdim; ++e) {
            for (f = 0, rotM[d * cdim + e] = 0.0; f < cdim; ++f) rotM[d * cdim + e] += invJExT[d * cdim + f] * R[e * cdim + f];
          }
        }
        for (d = 0; d < cdim; ++d) {
          for (e = 0; e < cdim; ++e) invJExT[d * cdim + e] = rotM[d * cdim + e];
        }
      }
      if (centroidEx) {
        volExT = volEx;
        for (d = 0; d < cdim; ++d) {
          centroidExT[d] = centroidEx[d];
          normalExT[d]   = normalEx[d];
        }
        for (d = 0; d < cdim; ++d) {
          centroidExT[d] *= scale;
          centroidExT[d] += PetscRealPart(trans[d]);
          normalExT[d] /= scale;
          /* Only scale dimensions in the manifold */
          if (d < dim) volExT *= scale;
        }
        /* Do scaling and translation before rotation, so that we can leave out the normal dimension for lower dimensional manifolds */
        for (d = 0; d < cdim; ++d) {
          for (e = 0, rot[d] = 0.0; e < cdim; ++e) rot[d] += R[d * cdim + e] * centroidExT[e];
        }
        for (d = 0; d < cdim; ++d) centroidExT[d] = rot[d];
        for (d = 0; d < cdim; ++d) {
          for (e = 0, rot[d] = 0.0; e < cdim; ++e) rot[d] += R[d * cdim + e] * normalExT[e];
        }
        for (d = 0; d < cdim; ++d) normalExT[d] = rot[d];
        for (d = 0, norm = 0.0; d < cdim; ++d) norm += PetscSqr(normalExT[d]);
        norm = PetscSqrtReal(norm);
        if (norm != 0.)
          for (d = 0; d < cdim; ++d) normalExT[d] /= norm;
      }
      for (d = 0; d < cdim; ++d) {
        for (p = 0; p < numCorners; ++p) {
          newCoords[p * cdim + d] *= scale;
          newCoords[p * cdim + d] += trans[d];
        }
      }
      for (p = 0; p < numCorners; ++p) {
        for (d = 0; d < cdim; ++d) {
          for (e = 0, rot[d] = 0.0; e < cdim; ++e) rot[d] += R[d * cdim + e] * PetscRealPart(newCoords[p * cdim + e]);
        }
        for (d = 0; d < cdim; ++d) newCoords[p * cdim + d] = rot[d];
      }

      PetscCall(ChangeCoordinates(dm, cdim, newCoords));
      if (v0Ex) PetscCall(CheckFEMGeometry(dm, 0, cdim, v0ExT, JExT, invJExT, detJExT));
      if (dim == depth && centroidEx) {
        PetscCall(CheckFVMGeometry(dm, cell, cdim, centroidExT, normalExT, volExT));
        PetscCall(CheckGaussLaw(dm, cell));
        if (faceCentroidEx) {
          PetscCall(DMPlexGetConeSize(dm, cell, &coneSize));
          PetscCall(DMPlexGetCone(dm, cell, &cone));
          for (c = 0; c < coneSize; ++c) {
            PetscInt off = c * cdim;

            faceVolExT = faceVolEx[c];
            for (d = 0; d < cdim; ++d) {
              faceCentroidExT[d] = faceCentroidEx[off + d];
              faceNormalExT[d]   = faceNormalEx[off + d];
            }
            for (d = 0; d < cdim; ++d) {
              faceCentroidExT[d] *= scale;
              faceCentroidExT[d] += PetscRealPart(trans[d]);
              faceNormalExT[d] /= scale;
              /* Only scale dimensions in the manifold */
              if (d < dim - 1) faceVolExT *= scale;
            }
            for (d = 0; d < cdim; ++d) {
              for (e = 0, rot[d] = 0.0; e < cdim; ++e) rot[d] += R[d * cdim + e] * faceCentroidExT[e];
            }
            for (d = 0; d < cdim; ++d) faceCentroidExT[d] = rot[d];
            for (d = 0; d < cdim; ++d) {
              for (e = 0, rot[d] = 0.0; e < cdim; ++e) rot[d] += R[d * cdim + e] * faceNormalExT[e];
            }
            for (d = 0; d < cdim; ++d) faceNormalExT[d] = rot[d];
            for (d = 0, norm = 0.0; d < cdim; ++d) norm += PetscSqr(faceNormalExT[d]);
            norm = PetscSqrtReal(norm);
            for (d = 0; d < cdim; ++d) faceNormalExT[d] /= norm;

            PetscCall(CheckFVMGeometry(dm, cone[c], cdim, faceCentroidExT, faceNormalExT, faceVolExT));
          }
        }
      }
    }
    PetscCall(PetscRandomDestroy(&r));
    PetscCall(PetscRandomDestroy(&ang));
    PetscCall(PetscRandomDestroy(&ang2));
    PetscCall(PetscFree2(origCoords, newCoords));
    PetscCall(PetscFree5(v0ExT, JExT, invJExT, centroidExT, normalExT));
    PetscCall(PetscFree2(faceCentroidExT, faceNormalExT));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TestTriangle(MPI_Comm comm, PetscBool transform)
{
  DM dm;

  PetscFunctionBegin;
  PetscCall(DMPlexCreateReferenceCell(comm, DM_POLYTOPE_TRIANGLE, &dm));
  PetscCall(DMViewFromOptions(dm, NULL, "-dm_view"));
  /* Check reference geometry: determinant is scaled by reference volume (2.0) */
  {
    PetscReal v0Ex[2]       = {-1.0, -1.0};
    PetscReal JEx[4]        = {1.0, 0.0, 0.0, 1.0};
    PetscReal invJEx[4]     = {1.0, 0.0, 0.0, 1.0};
    PetscReal detJEx        = 1.0;
    PetscReal centroidEx[2] = {-((PetscReal)1.) / ((PetscReal)3.), -((PetscReal)1.) / ((PetscReal)3.)};
    PetscReal normalEx[2]   = {0.0, 0.0};
    PetscReal volEx         = 2.0;

    PetscCall(CheckCell(dm, 0, transform, v0Ex, JEx, invJEx, detJEx, centroidEx, normalEx, volEx, NULL, NULL, NULL));
  }
  /* Move to 3D: Check reference geometry: determinant is scaled by reference volume (2.0) */
  {
    PetscScalar vertexCoords[9] = {-1.0, -1.0, 0.0, 1.0, -1.0, 0.0, -1.0, 1.0, 0.0};
    PetscReal   v0Ex[3]         = {-1.0, -1.0, 0.0};
    PetscReal   JEx[9]          = {1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 1.0};
    PetscReal   invJEx[9]       = {1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 1.0};
    PetscReal   detJEx          = 1.0;
    PetscReal   centroidEx[3]   = {-((PetscReal)1.) / ((PetscReal)3.), -((PetscReal)1.) / ((PetscReal)3.), 0.0};
    PetscReal   normalEx[3]     = {0.0, 0.0, 1.0};
    PetscReal   volEx           = 2.0;

    PetscCall(ChangeCoordinates(dm, 3, vertexCoords));
    PetscCall(CheckCell(dm, 0, transform, v0Ex, JEx, invJEx, detJEx, centroidEx, normalEx, volEx, NULL, NULL, NULL));
  }
  /* Cleanup */
  PetscCall(DMDestroy(&dm));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TestQuadrilateral(MPI_Comm comm, PetscBool transform)
{
  DM dm;

  PetscFunctionBegin;
  PetscCall(DMPlexCreateReferenceCell(comm, DM_POLYTOPE_QUADRILATERAL, &dm));
  PetscCall(DMViewFromOptions(dm, NULL, "-dm_view"));
  /* Check reference geometry: determinant is scaled by reference volume (2.0) */
  {
    PetscReal v0Ex[2]       = {-1.0, -1.0};
    PetscReal JEx[4]        = {1.0, 0.0, 0.0, 1.0};
    PetscReal invJEx[4]     = {1.0, 0.0, 0.0, 1.0};
    PetscReal detJEx        = 1.0;
    PetscReal centroidEx[2] = {0.0, 0.0};
    PetscReal normalEx[2]   = {0.0, 0.0};
    PetscReal volEx         = 4.0;

    PetscCall(CheckCell(dm, 0, transform, v0Ex, JEx, invJEx, detJEx, centroidEx, normalEx, volEx, NULL, NULL, NULL));
  }
  /* Move to 3D: Check reference geometry: determinant is scaled by reference volume (4.0) */
  {
    PetscScalar vertexCoords[12] = {-1.0, -1.0, 0.0, 1.0, -1.0, 0.0, 1.0, 1.0, 0.0, -1.0, 1.0, 0.0};
    PetscReal   v0Ex[3]          = {-1.0, -1.0, 0.0};
    PetscReal   JEx[9]           = {1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 1.0};
    PetscReal   invJEx[9]        = {1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 1.0};
    PetscReal   detJEx           = 1.0;
    PetscReal   centroidEx[3]    = {0.0, 0.0, 0.0};
    PetscReal   normalEx[3]      = {0.0, 0.0, 1.0};
    PetscReal   volEx            = 4.0;

    PetscCall(ChangeCoordinates(dm, 3, vertexCoords));
    PetscCall(CheckCell(dm, 0, transform, v0Ex, JEx, invJEx, detJEx, centroidEx, normalEx, volEx, NULL, NULL, NULL));
  }
  /* Cleanup */
  PetscCall(DMDestroy(&dm));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TestTetrahedron(MPI_Comm comm, PetscBool transform)
{
  DM dm;

  PetscFunctionBegin;
  PetscCall(DMPlexCreateReferenceCell(comm, DM_POLYTOPE_TETRAHEDRON, &dm));
  PetscCall(DMViewFromOptions(dm, NULL, "-dm_view"));
  /* Check reference geometry: determinant is scaled by reference volume (4/3) */
  {
    PetscReal v0Ex[3]       = {-1.0, -1.0, -1.0};
    PetscReal JEx[9]        = {1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 1.0};
    PetscReal invJEx[9]     = {1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 1.0};
    PetscReal detJEx        = 1.0;
    PetscReal centroidEx[3] = {-0.5, -0.5, -0.5};
    PetscReal normalEx[3]   = {0.0, 0.0, 0.0};
    PetscReal volEx         = (PetscReal)4.0 / (PetscReal)3.0;

    PetscCall(CheckCell(dm, 0, transform, v0Ex, JEx, invJEx, detJEx, centroidEx, normalEx, volEx, NULL, NULL, NULL));
  }
  /* Cleanup */
  PetscCall(DMDestroy(&dm));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TestHexahedron(MPI_Comm comm, PetscBool transform)
{
  DM dm;

  PetscFunctionBegin;
  PetscCall(DMPlexCreateReferenceCell(comm, DM_POLYTOPE_HEXAHEDRON, &dm));
  PetscCall(DMViewFromOptions(dm, NULL, "-dm_view"));
  /* Check reference geometry: determinant is scaled by reference volume 8.0 */
  {
    PetscReal v0Ex[3]       = {-1.0, -1.0, -1.0};
    PetscReal JEx[9]        = {1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 1.0};
    PetscReal invJEx[9]     = {1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 1.0};
    PetscReal detJEx        = 1.0;
    PetscReal centroidEx[3] = {0.0, 0.0, 0.0};
    PetscReal normalEx[3]   = {0.0, 0.0, 0.0};
    PetscReal volEx         = 8.0;

    PetscCall(CheckCell(dm, 0, transform, v0Ex, JEx, invJEx, detJEx, centroidEx, normalEx, volEx, NULL, NULL, NULL));
  }
  /* Cleanup */
  PetscCall(DMDestroy(&dm));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode TestHexahedronCurved(MPI_Comm comm)
{
  DM          dm;
  PetscScalar coords[24] = {-1.0, -1.0, -1.0, -1.0, 1.0, -1.0, 1.0, 1.0, -1.0, 1.0, -1.0, -1.0, -1.0, -1.0, 1.1, 1.0, -1.0, 1.0, 1.0, 1.0, 1.1, -1.0, 1.0, 1.0};

  PetscFunctionBegin;
  PetscCall(DMPlexCreateReferenceCell(comm, DM_POLYTOPE_HEXAHEDRON, &dm));
  PetscCall(ChangeCoordinates(dm, 3, coords));
  PetscCall(DMViewFromOptions(dm, NULL, "-dm_view"));
  {
    PetscReal centroidEx[3] = {0.0, 0.0, 0.016803278688524603};
    PetscReal normalEx[3]   = {0.0, 0.0, 0.0};
    PetscReal volEx         = 8.1333333333333346;

    PetscCall(CheckCell(dm, 0, PETSC_FALSE, NULL, NULL, NULL, 0.0, centroidEx, normalEx, volEx, NULL, NULL, NULL));
  }
  PetscCall(DMDestroy(&dm));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* This wedge is a tensor product cell, rather than a normal wedge */
static PetscErrorCode TestWedge(MPI_Comm comm, PetscBool transform)
{
  DM dm;

  PetscFunctionBegin;
  PetscCall(DMPlexCreateReferenceCell(comm, DM_POLYTOPE_TRI_PRISM_TENSOR, &dm));
  PetscCall(DMViewFromOptions(dm, NULL, "-dm_view"));
  /* Check reference geometry: determinant is scaled by reference volume 4.0 */
  {
#if 0
    /* FEM geometry is not functional for wedges */
    PetscReal v0Ex[3]   = {-1.0, -1.0, -1.0};
    PetscReal JEx[9]    = {1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 1.0};
    PetscReal invJEx[9] = {1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 1.0};
    PetscReal detJEx    = 1.0;
#endif

    {
      PetscReal centroidEx[3]      = {-((PetscReal)1.) / ((PetscReal)3.), -((PetscReal)1.) / ((PetscReal)3.), 0.0};
      PetscReal normalEx[3]        = {0.0, 0.0, 0.0};
      PetscReal volEx              = 4.0;
      PetscReal faceVolEx[5]       = {2.0, 2.0, 4.0, PETSC_SQRT2 * 4.0, 4.0};
      PetscReal faceNormalEx[15]   = {0.0, 0.0, 1.0, 0.0, 0.0, 1.0, 0.0, -1.0, 0.0, PETSC_SQRT2 / 2.0, PETSC_SQRT2 / 2.0, 0.0, -1.0, 0.0, 0.0};
      PetscReal faceCentroidEx[15] = {-((PetscReal)1.) / ((PetscReal)3.), -((PetscReal)1.) / ((PetscReal)3.), -1.0, -((PetscReal)1.) / ((PetscReal)3.), -((PetscReal)1.) / ((PetscReal)3.), 1.0, 0.0, -1.0, 0.0, 0.0, 0.0, 0.0, -1.0, 0.0, 0.0};

      PetscCall(CheckCell(dm, 0, transform, NULL, NULL, NULL, 0.0, centroidEx, normalEx, volEx, faceCentroidEx, faceNormalEx, faceVolEx));
    }
  }
  /* Cleanup */
  PetscCall(DMDestroy(&dm));
  PetscFunctionReturn(PETSC_SUCCESS);
}

int main(int argc, char **argv)
{
  AppCtx user;

  PetscFunctionBeginUser;
  PetscCall(PetscInitialize(&argc, &argv, NULL, help));
  PetscCall(ProcessOptions(PETSC_COMM_WORLD, &user));
  if (user.runType == RUN_REFERENCE) {
    PetscCall(TestTriangle(PETSC_COMM_SELF, user.transform));
    PetscCall(TestQuadrilateral(PETSC_COMM_SELF, user.transform));
    PetscCall(TestTetrahedron(PETSC_COMM_SELF, user.transform));
    PetscCall(TestHexahedron(PETSC_COMM_SELF, user.transform));
    PetscCall(TestWedge(PETSC_COMM_SELF, user.transform));
  } else if (user.runType == RUN_HEX_CURVED) {
    PetscCall(TestHexahedronCurved(PETSC_COMM_SELF));
  } else if (user.runType == RUN_FILE) {
    PetscInt dim, cStart, cEnd, c;

    PetscCall(DMGetDimension(user.dm, &dim));
    PetscCall(DMPlexGetHeightStratum(user.dm, 0, &cStart, &cEnd));
    for (c = 0; c < cEnd - cStart; ++c) {
      PetscReal *v0       = PetscSafePointerPlusOffset(user.v0, c * dim);
      PetscReal *J        = PetscSafePointerPlusOffset(user.J, c * dim * dim);
      PetscReal *invJ     = PetscSafePointerPlusOffset(user.invJ, c * dim * dim);
      PetscReal  detJ     = user.detJ ? user.detJ[c] : 0.0;
      PetscReal *centroid = PetscSafePointerPlusOffset(user.centroid, c * dim);
      PetscReal *normal   = PetscSafePointerPlusOffset(user.normal, c * dim);
      PetscReal  vol      = user.vol ? user.vol[c] : 0.0;

      PetscCall(CheckCell(user.dm, c + cStart, PETSC_FALSE, v0, J, invJ, detJ, centroid, normal, vol, NULL, NULL, NULL));
    }
    PetscCall(PetscFree4(user.v0, user.J, user.invJ, user.detJ));
    PetscCall(PetscFree(user.centroid));
    PetscCall(PetscFree(user.normal));
    PetscCall(PetscFree(user.vol));
    PetscCall(DMDestroy(&user.dm));
  } else if (user.runType == RUN_DISPLAY) {
    DM                 gdm, dmCell;
    Vec                cellgeom, facegeom;
    const PetscScalar *cgeom;
    PetscInt           dim, d, cStart, cEnd, cEndInterior, c;

    PetscCall(DMGetCoordinateDim(user.dm, &dim));
    PetscCall(DMPlexConstructGhostCells(user.dm, NULL, NULL, &gdm));
    if (gdm) {
      PetscCall(DMDestroy(&user.dm));
      user.dm = gdm;
    }
    PetscCall(DMPlexComputeGeometryFVM(user.dm, &cellgeom, &facegeom));
    PetscCall(DMPlexGetHeightStratum(user.dm, 0, &cStart, &cEnd));
    PetscCall(DMPlexGetCellTypeStratum(user.dm, DM_POLYTOPE_FV_GHOST, &cEndInterior, NULL));
    if (cEndInterior >= 0) cEnd = cEndInterior;
    PetscCall(VecGetDM(cellgeom, &dmCell));
    PetscCall(VecGetArrayRead(cellgeom, &cgeom));
    for (c = 0; c < cEnd - cStart; ++c) {
      PetscFVCellGeom *cg;

      PetscCall(DMPlexPointLocalRead(dmCell, c, cgeom, &cg));
      PetscCall(PetscPrintf(PETSC_COMM_SELF, "Cell %4" PetscInt_FMT ": Centroid (", c));
      for (d = 0; d < dim; ++d) {
        if (d > 0) PetscCall(PetscPrintf(PETSC_COMM_SELF, ", "));
        PetscCall(PetscPrintf(PETSC_COMM_SELF, "%12.2g", (double)cg->centroid[d]));
      }
      PetscCall(PetscPrintf(PETSC_COMM_SELF, ") Vol %12.2g\n", (double)cg->volume));
    }
    PetscCall(VecRestoreArrayRead(cellgeom, &cgeom));
    PetscCall(VecDestroy(&cellgeom));
    PetscCall(VecDestroy(&facegeom));
    PetscCall(DMDestroy(&user.dm));
  }
  PetscCall(PetscFinalize());
  return 0;
}

/*TEST

  test:
    suffix: 1
    args: -dm_view ascii::ascii_info_detail
  test:
    suffix: 2
    args: -run_type hex_curved
    output_file: output/empty.out
  test:
    suffix: 3
    args: -transform
  test:
    suffix: 4
    requires: exodusii
    args: -run_type file -dm_plex_filename ${wPETSC_DIR}/share/petsc/datafiles/meshes/simpleblock-100.exo -dm_view ascii::ascii_info_detail -v0 -1.5,-0.5,0.5,-0.5,-0.5,0.5,0.5,-0.5,0.5 -J 0.0,0.0,0.5,0.0,0.5,0.0,-0.5,0.0,0.0,0.0,0.0,0.5,0.0,0.5,0.0,-0.5,0.0,0.0,0.0,0.0,0.5,0.0,0.5,0.0,-0.5,0.0,0.0 -invJ 0.0,0.0,-2.0,0.0,2.0,0.0,2.0,0.0,0.0,0.0,0.0,-2.0,0.0,2.0,0.0,2.0,0.0,0.0,0.0,0.0,-2.0,0.0,2.0,0.0,2.0,0.0,0.0 -detJ 0.125,0.125,0.125 -centroid -1.0,0.0,0.0,0.0,0.0,0.0,1.0,0.0,0.0 -normal 0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0 -vol 1.0,1.0,1.0
  test:
    suffix: 5
    args: -run_type file -dm_plex_dim 3 -dm_plex_simplex 0 -dm_plex_box_faces 3,1,1 -dm_plex_box_lower -1.5,-0.5,-0.5 -dm_plex_box_upper 1.5,0.5,0.5 -dm_view ascii::ascii_info_detail -centroid -1.0,0.0,0.0,0.0,0.0,0.0,1.0,0.0,0.0 -normal 0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0 -vol 1.0,1.0,1.0
  test:
    suffix: 6
    args: -run_type file -dm_plex_dim 1 -dm_plex_simplex 0 -dm_plex_box_faces 3 -dm_plex_box_lower -1.5 -dm_plex_box_upper 1.5 -dm_view ascii::ascii_info_detail -centroid -1.0,0.0,1.0 -vol 1.0,1.0,1.0
TEST*/