OpenSubdiv/opensubdiv/osd/glslPatchCommon.glsl

//
//   Copyright 2013 Pixar
//
//   Licensed under the Apache License, Version 2.0 (the "Apache License")
//   with the following modification; you may not use this file except in
//   compliance with the Apache License and the following modification to it:
//   Section 6. Trademarks. is deleted and replaced with:
//
//   6. Trademarks. This License does not grant permission to use the trade
//      names, trademarks, service marks, or product names of the Licensor
//      and its affiliates, except as required to comply with Section 4(c) of
//      the License and to reproduce the content of the NOTICE file.
//
//   You may obtain a copy of the Apache License at
//
//       http://www.apache.org/licenses/LICENSE-2.0
//
//   Unless required by applicable law or agreed to in writing, software
//   distributed under the Apache License with the above modification is
//   distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
//   KIND, either express or implied. See the Apache License for the specific
//   language governing permissions and limitations under the Apache License.
//

//
// typical shader composition ordering (see glDrawRegistry:_CompileShader)
//
//
// - glsl version string  (#version 430)
//
// - common defines       (#define OSD_ENABLE_PATCH_CULL, ...)
// - source defines       (#define VERTEX_SHADER, ...)
//
// - osd headers          (glslPatchCommon: varying structs,
//                         glslPtexCommon: ptex functions)
// - client header        (Osd*Matrix(), displacement callback, ...)
//
// - osd shader source    (glslPatchBSpline, glslPatchGregory, ...)
//     or
//   client shader source (vertex/geometry/fragment shader)
//

//----------------------------------------------------------
// Patches.Common
//----------------------------------------------------------

// XXXdyu all handling of varying data can be managed by client code
#ifndef OSD_USER_VARYING_DECLARE
#define OSD_USER_VARYING_DECLARE
// type var;
#endif

#ifndef OSD_USER_VARYING_ATTRIBUTE_DECLARE
#define OSD_USER_VARYING_ATTRIBUTE_DECLARE
// layout(location = loc) in type var;
#endif

#ifndef OSD_USER_VARYING_PER_VERTEX
#define OSD_USER_VARYING_PER_VERTEX()
// output.var = var;
#endif

#ifndef OSD_USER_VARYING_PER_CONTROL_POINT
#define OSD_USER_VARYING_PER_CONTROL_POINT(ID_OUT, ID_IN)
// output[ID_OUT].var = input[ID_IN].var
#endif

#ifndef OSD_USER_VARYING_PER_EVAL_POINT
#define OSD_USER_VARYING_PER_EVAL_POINT(UV, a, b, c, d)
// output.var =
//     mix(mix(input[a].var, input[b].var, UV.x),
//         mix(input[c].var, input[d].var, UV.x), UV.y)
#endif

// For now, fractional spacing is supported only with screen space tessellation
#ifndef OSD_ENABLE_SCREENSPACE_TESSELLATION
#undef OSD_FRACTIONAL_EVEN_SPACING
#undef OSD_FRACTIONAL_ODD_SPACING
#endif

#if defined OSD_FRACTIONAL_EVEN_SPACING
  #define OSD_SPACING fractional_even_spacing
#elif defined OSD_FRACTIONAL_ODD_SPACING
  #define OSD_SPACING fractional_odd_spacing
#else
  #define OSD_SPACING equal_spacing
#endif

#if __VERSION__ < 420
    #define centroid
#endif

struct ControlVertex {
    vec4 position;
#ifdef OSD_ENABLE_PATCH_CULL
    ivec3 clipFlag;
#endif
};

// XXXdyu all downstream data can be handled by client code
struct OutputVertex {
    vec4 position;
    vec3 normal;
    vec3 tangent;
    vec3 bitangent;
    centroid vec4 patchCoord; // u, v, faceLevel, faceId
    centroid vec2 tessCoord; // tesscoord.st
#if defined OSD_COMPUTE_NORMAL_DERIVATIVES
    vec3 Nu;
    vec3 Nv;
#endif
};

// osd shaders need following functions defined
mat4 OsdModelViewMatrix();
mat4 OsdProjectionMatrix();
mat4 OsdModelViewProjectionMatrix();
float OsdTessLevel();
int OsdGregoryQuadOffsetBase();
int OsdPrimitiveIdBase();
int OsdBaseVertex();

#ifndef OSD_DISPLACEMENT_CALLBACK
#define OSD_DISPLACEMENT_CALLBACK
#endif

// ----------------------------------------------------------------------------
// Patch Parameters
// ----------------------------------------------------------------------------

//
// Each patch has a corresponding patchParam. This is a set of three values
// specifying additional information about the patch:
//
//    faceId    -- topological face identifier (e.g. Ptex FaceId)
//    bitfield  -- refinement-level, non-quad, boundary, transition, uv-offset
//    sharpness -- crease sharpness for single-crease patches
//
// These are stored in OsdPatchParamBuffer indexed by the value returned
// from OsdGetPatchIndex() which is a function of the current PrimitiveID
// along with an optional client provided offset.
//

uniform isamplerBuffer OsdPatchParamBuffer;

int OsdGetPatchIndex(int primitiveId)
{
    return (primitiveId + OsdPrimitiveIdBase());
}

ivec3 OsdGetPatchParam(int patchIndex)
{
    return texelFetch(OsdPatchParamBuffer, patchIndex).xyz;
}

int OsdGetPatchFaceId(ivec3 patchParam)
{
    return (patchParam.x & 0xfffffff);
}

int OsdGetPatchFaceLevel(ivec3 patchParam)
{
    return (1 << ((patchParam.y & 0xf) - ((patchParam.y >> 4) & 1)));
}

int OsdGetPatchRefinementLevel(ivec3 patchParam)
{
    return (patchParam.y & 0xf);
}

int OsdGetPatchBoundaryMask(ivec3 patchParam)
{
    return ((patchParam.y >> 7) & 0x1f);
}

int OsdGetPatchTransitionMask(ivec3 patchParam)
{
    return ((patchParam.x >> 28) & 0xf);
}

ivec2 OsdGetPatchFaceUV(ivec3 patchParam)
{
    int u = (patchParam.y >> 22) & 0x3ff;
    int v = (patchParam.y >> 12) & 0x3ff;
    return ivec2(u,v);
}

bool OsdGetPatchIsRegular(ivec3 patchParam)
{
    return ((patchParam.y >> 5) & 0x1) != 0;
}

float OsdGetPatchSharpness(ivec3 patchParam)
{
    return intBitsToFloat(patchParam.z);
}

float OsdGetPatchSingleCreaseSegmentParameter(ivec3 patchParam, vec2 uv)
{
    int boundaryMask = OsdGetPatchBoundaryMask(patchParam);
    float s = 0;
    if ((boundaryMask & 1) != 0) {
        s = 1 - uv.y;
    } else if ((boundaryMask & 2) != 0) {
        s = uv.x;
    } else if ((boundaryMask & 4) != 0) {
        s = uv.y;
    } else if ((boundaryMask & 8) != 0) {
        s = 1 - uv.x;
    }
    return s;
}

ivec4 OsdGetPatchCoord(ivec3 patchParam)
{
    int faceId = OsdGetPatchFaceId(patchParam);
    int faceLevel = OsdGetPatchFaceLevel(patchParam);
    ivec2 faceUV = OsdGetPatchFaceUV(patchParam);
    return ivec4(faceUV.x, faceUV.y, faceLevel, faceId);
}

vec4 OsdInterpolatePatchCoord(vec2 localUV, ivec3 patchParam)
{
    ivec4 perPrimPatchCoord = OsdGetPatchCoord(patchParam);
    int faceId = perPrimPatchCoord.w;
    int faceLevel = perPrimPatchCoord.z;
    vec2 faceUV = vec2(perPrimPatchCoord.x, perPrimPatchCoord.y);
    vec2 uv = localUV/faceLevel + faceUV/faceLevel;
    // add 0.5 to integer values for more robust interpolation
    return vec4(uv.x, uv.y, faceLevel+0.5f, faceId+0.5f);
}

// ----------------------------------------------------------------------------
// patch culling
// ----------------------------------------------------------------------------

#ifdef OSD_ENABLE_PATCH_CULL

#define OSD_PATCH_CULL_COMPUTE_CLIPFLAGS(P)                     \
    vec4 clipPos = OsdModelViewProjectionMatrix() * P;          \
    bvec3 clip0 = lessThan(clipPos.xyz, vec3(clipPos.w));       \
    bvec3 clip1 = greaterThan(clipPos.xyz, -vec3(clipPos.w));   \
    outpt.v.clipFlag = ivec3(clip0) + 2*ivec3(clip1);           \

#define OSD_PATCH_CULL(N)                            \
    ivec3 clipFlag = ivec3(0);                       \
    for(int i = 0; i < N; ++i) {                     \
        clipFlag |= inpt[i].v.clipFlag;              \
    }                                                \
    if (clipFlag != ivec3(3) ) {                     \
        gl_TessLevelInner[0] = 0;                    \
        gl_TessLevelInner[1] = 0;                    \
        gl_TessLevelOuter[0] = 0;                    \
        gl_TessLevelOuter[1] = 0;                    \
        gl_TessLevelOuter[2] = 0;                    \
        gl_TessLevelOuter[3] = 0;                    \
        return;                                      \
    }

#else
#define OSD_PATCH_CULL_COMPUTE_CLIPFLAGS(P)
#define OSD_PATCH_CULL(N)
#endif

// ----------------------------------------------------------------------------

void
OsdUnivar4x4(in float u, out float B[4], out float D[4])
{
    float t = u;
    float s = 1.0f - u;

    float A0 = s * s;
    float A1 = 2 * s * t;
    float A2 = t * t;

    B[0] = s * A0;
    B[1] = t * A0 + s * A1;
    B[2] = t * A1 + s * A2;
    B[3] = t * A2;

    D[0] =    - A0;
    D[1] = A0 - A1;
    D[2] = A1 - A2;
    D[3] = A2;
}

void
OsdUnivar4x4(in float u, out float B[4], out float D[4], out float C[4])
{
    float t = u;
    float s = 1.0f - u;

    float A0 = s * s;
    float A1 = 2 * s * t;
    float A2 = t * t;

    B[0] = s * A0;
    B[1] = t * A0 + s * A1;
    B[2] = t * A1 + s * A2;
    B[3] = t * A2;

    D[0] =    - A0;
    D[1] = A0 - A1;
    D[2] = A1 - A2;
    D[3] = A2;

    A0 =   - s;
    A1 = s - t;
    A2 = t;

    C[0] =    - A0;
    C[1] = A0 - A1;
    C[2] = A1 - A2;
    C[3] = A2;
}

// ----------------------------------------------------------------------------

struct OsdPerPatchVertexBezier {
    ivec3 patchParam;
    vec3 P;
#if defined OSD_PATCH_ENABLE_SINGLE_CREASE
    vec3 P1;
    vec3 P2;
    vec2 vSegments;
#endif
};

vec3
OsdEvalBezier(vec3 cp[16], vec2 uv)
{
    vec3 BUCP[4] = vec3[4](vec3(0), vec3(0), vec3(0), vec3(0));

    float B[4], D[4];

    OsdUnivar4x4(uv.x, B, D);
    for (int i=0; i<4; ++i) {
        for (int j=0; j<4; ++j) {
            vec3 A = cp[4*i + j];
            BUCP[i] += A * B[j];
        }
    }

    vec3 P = vec3(0);

    OsdUnivar4x4(uv.y, B, D);
    for (int k=0; k<4; ++k) {
        P += B[k] * BUCP[k];
    }

    return P;
}

// When OSD_PATCH_ENABLE_SINGLE_CREASE is defined,
// this function evaluates single-crease patch, which is segmented into
// 3 parts in the v-direction.
//
//  v=0             vSegment.x        vSegment.y              v=1
//   +------------------+-------------------+------------------+
//   |       cp 0       |     cp 1          |      cp 2        |
//   | (infinite sharp) | (floor sharpness) | (ceil sharpness) |
//   +------------------+-------------------+------------------+
//
vec3
OsdEvalBezier(OsdPerPatchVertexBezier cp[16], ivec3 patchParam, vec2 uv)
{
    vec3 BUCP[4] = vec3[4](vec3(0), vec3(0), vec3(0), vec3(0));

    float B[4], D[4];
    float s = OsdGetPatchSingleCreaseSegmentParameter(patchParam, uv);

    OsdUnivar4x4(uv.x, B, D);
#if defined OSD_PATCH_ENABLE_SINGLE_CREASE
    vec2 vSegments = cp[0].vSegments;
    if (s <= vSegments.x) {
        for (int i=0; i<4; ++i) {
            for (int j=0; j<4; ++j) {
                vec3 A = cp[4*i + j].P;
                BUCP[i] += A * B[j];
            }
        }
    } else if (s <= vSegments.y) {
        for (int i=0; i<4; ++i) {
            for (int j=0; j<4; ++j) {
                vec3 A = cp[4*i + j].P1;
                BUCP[i] += A * B[j];
            }
        }
    } else {
        for (int i=0; i<4; ++i) {
            for (int j=0; j<4; ++j) {
                vec3 A = cp[4*i + j].P2;
                BUCP[i] += A * B[j];
            }
        }
    }
#else
    for (int i=0; i<4; ++i) {
        for (int j=0; j<4; ++j) {
            vec3 A = cp[4*i + j].P;
            BUCP[i] += A * B[j];
        }
    }
#endif

    vec3 P = vec3(0);

    OsdUnivar4x4(uv.y, B, D);
    for (int k=0; k<4; ++k) {
        P += B[k] * BUCP[k];
    }

    return P;
}

// ----------------------------------------------------------------------------
// Boundary Interpolation
// ----------------------------------------------------------------------------

void
OsdComputeBSplineBoundaryPoints(inout vec3 cpt[16], ivec3 patchParam)
{
    int boundaryMask = OsdGetPatchBoundaryMask(patchParam);

    if ((boundaryMask & 1) != 0) {
        cpt[0] = 2*cpt[4] - cpt[8];
        cpt[1] = 2*cpt[5] - cpt[9];
        cpt[2] = 2*cpt[6] - cpt[10];
        cpt[3] = 2*cpt[7] - cpt[11];
    }
    if ((boundaryMask & 2) != 0) {
        cpt[3] = 2*cpt[2] - cpt[1];
        cpt[7] = 2*cpt[6] - cpt[5];
        cpt[11] = 2*cpt[10] - cpt[9];
        cpt[15] = 2*cpt[14] - cpt[13];
    }
    if ((boundaryMask & 4) != 0) {
        cpt[12] = 2*cpt[8] - cpt[4];
        cpt[13] = 2*cpt[9] - cpt[5];
        cpt[14] = 2*cpt[10] - cpt[6];
        cpt[15] = 2*cpt[11] - cpt[7];
    }
    if ((boundaryMask & 8) != 0) {
        cpt[0] = 2*cpt[1] - cpt[2];
        cpt[4] = 2*cpt[5] - cpt[6];
        cpt[8] = 2*cpt[9] - cpt[10];
        cpt[12] = 2*cpt[13] - cpt[14];
    }
}

// ----------------------------------------------------------------------------
// Tessellation
// ----------------------------------------------------------------------------

//
// Organization of B-spline and Bezier control points.
//
// Each patch is defined by 16 control points (labeled 0-15).
//
// The patch will be evaluated across the domain from (0,0) at
// the lower-left to (1,1) at the upper-right. When computing
// adaptive tessellation metrics, we consider refined vertex-vertex
// and edge-vertex points along the transition edges of the patch
// (labeled vv* and ev* respectively).
//
// The two segments of each transition edge are labeled Lo and Hi,
// with the Lo segment occurring before the Hi segment along the
// transition edge's domain parameterization. These Lo and Hi segment
// tessellation levels determine how domain evaluation coordinates
// are remapped along transition edges. The Hi segment value will
// be zero for a non-transition edge.
//
// (0,1)                                         (1,1)
//
//   vv3                  ev23                   vv2
//        |       Lo3       |       Hi3       |
//      --O-----------O-----+-----O-----------O--
//        | 12        | 13     14 |        15 |
//        |           |           |           |
//        |           |           |           |
//    Hi0 |           |           |           | Hi2
//        |           |           |           |
//        O-----------O-----------O-----------O
//        | 8         | 9      10 |        11 |
//        |           |           |           |
// ev03 --+           |           |           +-- ev12
//        |           |           |           |
//        | 4         | 5       6 |         7 |
//        O-----------O-----------O-----------O
//        |           |           |           |
//    Lo0 |           |           |           | Lo2
//        |           |           |           |
//        |           |           |           |
//        | 0         | 1       2 |         3 |
//      --O-----------O-----+-----O-----------O--
//        |       Lo1       |       Hi1       |
//   vv0                  ev01                   vv1
//
// (0,0)                                         (1,0)
//

#define OSD_MAX_TESS_LEVEL gl_MaxTessGenLevel

float OsdComputePostProjectionSphereExtent(vec3 center, float diameter)
{
    vec4 p = OsdProjectionMatrix() * vec4(center, 1.0);
    return abs(diameter * OsdProjectionMatrix()[1][1] / p.w);
}

float OsdComputeTessLevel(vec3 p0, vec3 p1)
{
    // Adaptive factor can be any computation that depends only on arg values.
    // Project the diameter of the edge's bounding sphere instead of using the
    // length of the projected edge itself to avoid problems near silhouettes.
    p0 = (OsdModelViewMatrix() * vec4(p0, 1.0)).xyz;
    p1 = (OsdModelViewMatrix() * vec4(p1, 1.0)).xyz;
    vec3 center = (p0 + p1) / 2.0;
    float diameter = distance(p0, p1);
    float projLength = OsdComputePostProjectionSphereExtent(center, diameter);
    float tessLevel = max(1.0, OsdTessLevel() * projLength);

    // We restrict adaptive tessellation levels to half of the device
    // supported maximum because transition edges are split into two
    // halves and the sum of the two corresponding levels must not exceed
    // the device maximum. We impose this limit even for non-transition
    // edges because a non-transition edge must be able to match up with
    // one half of the transition edge of an adjacent transition patch.
    return min(tessLevel, OSD_MAX_TESS_LEVEL / 2);
}

void
OsdGetTessLevelsUniform(ivec3 patchParam,
                        out vec4 tessOuterLo, out vec4 tessOuterHi)
{
    // Uniform factors are simple powers of two for each level.
    // The maximum here can be increased if we know the maximum
    // refinement level of the mesh:
    //     min(OSD_MAX_TESS_LEVEL, pow(2, MaximumRefinementLevel-1)
    int refinementLevel = OsdGetPatchRefinementLevel(patchParam);
    float tessLevel = min(OsdTessLevel(), OSD_MAX_TESS_LEVEL) /
                        pow(2, refinementLevel-1);

    // tessLevels of transition edge should be clamped to 2.
    int transitionMask = OsdGetPatchTransitionMask(patchParam);
    vec4 tessLevelMin = vec4(1) + vec4(((transitionMask & 8) >> 3),
                                       ((transitionMask & 1) >> 0),
                                       ((transitionMask & 2) >> 1),
                                       ((transitionMask & 4) >> 2));

    tessOuterLo = max(vec4(tessLevel), tessLevelMin);
    tessOuterHi = vec4(0);
}

void
OsdGetTessLevelsRefinedPoints(vec3 cp[16], ivec3 patchParam,
                              out vec4 tessOuterLo, out vec4 tessOuterHi)
{
    // Each edge of a transition patch is adjacent to one or two patches
    // at the next refined level of subdivision. We compute the corresponding
    // vertex-vertex and edge-vertex refined points along the edges of the
    // patch using Catmull-Clark subdivision stencil weights.
    // For simplicity, we let the optimizer discard unused computation.

    vec3 vv0 = (cp[0] + cp[2] + cp[8] + cp[10]) * 0.015625 +
               (cp[1] + cp[4] + cp[6] + cp[9]) * 0.09375 + cp[5] * 0.5625;
    vec3 ev01 = (cp[1] + cp[2] + cp[9] + cp[10]) * 0.0625 +
                (cp[5] + cp[6]) * 0.375;

    vec3 vv1 = (cp[1] + cp[3] + cp[9] + cp[11]) * 0.015625 +
               (cp[2] + cp[5] + cp[7] + cp[10]) * 0.09375 + cp[6] * 0.5625;
    vec3 ev12 = (cp[5] + cp[7] + cp[9] + cp[11]) * 0.0625 +
                (cp[6] + cp[10]) * 0.375;

    vec3 vv2 = (cp[5] + cp[7] + cp[13] + cp[15]) * 0.015625 +
               (cp[6] + cp[9] + cp[11] + cp[14]) * 0.09375 + cp[10] * 0.5625;
    vec3 ev23 = (cp[5] + cp[6] + cp[13] + cp[14]) * 0.0625 +
                (cp[9] + cp[10]) * 0.375;

    vec3 vv3 = (cp[4] + cp[6] + cp[12] + cp[14]) * 0.015625 +
               (cp[5] + cp[8] + cp[10] + cp[13]) * 0.09375 + cp[9] * 0.5625;
    vec3 ev03 = (cp[4] + cp[6] + cp[8] + cp[10]) * 0.0625 +
                (cp[5] + cp[9]) * 0.375;

    tessOuterLo = vec4(0);
    tessOuterHi = vec4(0);

    int transitionMask = OsdGetPatchTransitionMask(patchParam);

    if ((transitionMask & 8) != 0) {
        tessOuterLo[0] = OsdComputeTessLevel(vv0, ev03);
        tessOuterHi[0] = OsdComputeTessLevel(vv3, ev03);
    } else {
        tessOuterLo[0] = OsdComputeTessLevel(cp[5], cp[9]);
    }
    if ((transitionMask & 1) != 0) {
        tessOuterLo[1] = OsdComputeTessLevel(vv0, ev01);
        tessOuterHi[1] = OsdComputeTessLevel(vv1, ev01);
    } else {
        tessOuterLo[1] = OsdComputeTessLevel(cp[5], cp[6]);
    }
    if ((transitionMask & 2) != 0) {
        tessOuterLo[2] = OsdComputeTessLevel(vv1, ev12);
        tessOuterHi[2] = OsdComputeTessLevel(vv2, ev12);
    } else {
        tessOuterLo[2] = OsdComputeTessLevel(cp[6], cp[10]);
    }
    if ((transitionMask & 4) != 0) {
        tessOuterLo[3] = OsdComputeTessLevel(vv3, ev23);
        tessOuterHi[3] = OsdComputeTessLevel(vv2, ev23);
    } else {
        tessOuterLo[3] = OsdComputeTessLevel(cp[9], cp[10]);
    }
}

void
OsdGetTessLevelsLimitPoints(OsdPerPatchVertexBezier cpBezier[16],
                 ivec3 patchParam, out vec4 tessOuterLo, out vec4 tessOuterHi)
{
    // Each edge of a transition patch is adjacent to one or two patches
    // at the next refined level of subdivision. When the patch control
    // points have been converted to the Bezier basis, the control points
    // at the four corners are on the limit surface (since a Bezier patch
    // interpolates its corner control points). We can compute an adaptive
    // tessellation level for transition edges on the limit surface by
    // evaluating a limit position at the mid point of each transition edge.

    tessOuterLo = vec4(0);
    tessOuterHi = vec4(0);

    int transitionMask = OsdGetPatchTransitionMask(patchParam);

#if defined OSD_PATCH_ENABLE_SINGLE_CREASE
    // PERFOMANCE: we just need to pick the correct corner points from P, P1, P2
    vec3 p0 = OsdEvalBezier(cpBezier, patchParam, vec2(0.0, 0.0));
    vec3 p3 = OsdEvalBezier(cpBezier, patchParam, vec2(1.0, 0.0));
    vec3 p12 = OsdEvalBezier(cpBezier, patchParam, vec2(0.0, 1.0));
    vec3 p15 = OsdEvalBezier(cpBezier, patchParam, vec2(1.0, 1.0));
    if ((transitionMask & 8) != 0) {
        vec3 ev03 = OsdEvalBezier(cpBezier, patchParam, vec2(0.0, 0.5));
        tessOuterLo[0] = OsdComputeTessLevel(p0, ev03);
        tessOuterHi[0] = OsdComputeTessLevel(p12, ev03);
    } else {
        tessOuterLo[0] = OsdComputeTessLevel(p0, p12);
    }
    if ((transitionMask & 1) != 0) {
        vec3 ev01 = OsdEvalBezier(cpBezier, patchParam, vec2(0.5, 0.0));
        tessOuterLo[1] = OsdComputeTessLevel(p0, ev01);
        tessOuterHi[1] = OsdComputeTessLevel(p3, ev01);
    } else {
        tessOuterLo[1] = OsdComputeTessLevel(p0, p3);
    }
    if ((transitionMask & 2) != 0) {
        vec3 ev12 = OsdEvalBezier(cpBezier, patchParam, vec2(1.0, 0.5));
        tessOuterLo[2] = OsdComputeTessLevel(p3, ev12);
        tessOuterHi[2] = OsdComputeTessLevel(p15, ev12);
    } else {
        tessOuterLo[2] = OsdComputeTessLevel(p3, p15);
    }
    if ((transitionMask & 4) != 0) {
        vec3 ev23 = OsdEvalBezier(cpBezier, patchParam, vec2(0.5, 1.0));
        tessOuterLo[3] = OsdComputeTessLevel(p12, ev23);
        tessOuterHi[3] = OsdComputeTessLevel(p15, ev23);
    } else {
        tessOuterLo[3] = OsdComputeTessLevel(p12, p15);
    }
#else

    if ((transitionMask & 8) != 0) {
        vec3 ev03 = OsdEvalBezier(cpBezier, patchParam, vec2(0.0, 0.5));
        tessOuterLo[0] = OsdComputeTessLevel(cpBezier[0].P, ev03);
        tessOuterHi[0] = OsdComputeTessLevel(cpBezier[12].P, ev03);
    } else {
        tessOuterLo[0] = OsdComputeTessLevel(cpBezier[0].P, cpBezier[12].P);
    }
    if ((transitionMask & 1) != 0) {
        vec3 ev01 = OsdEvalBezier(cpBezier, patchParam, vec2(0.5, 0.0));
        tessOuterLo[1] = OsdComputeTessLevel(cpBezier[0].P, ev01);
        tessOuterHi[1] = OsdComputeTessLevel(cpBezier[3].P, ev01);
    } else {
        tessOuterLo[1] = OsdComputeTessLevel(cpBezier[0].P, cpBezier[3].P);
    }
    if ((transitionMask & 2) != 0) {
        vec3 ev12 = OsdEvalBezier(cpBezier, patchParam, vec2(1.0, 0.5));
        tessOuterLo[2] = OsdComputeTessLevel(cpBezier[3].P, ev12);
        tessOuterHi[2] = OsdComputeTessLevel(cpBezier[15].P, ev12);
    } else {
        tessOuterLo[2] = OsdComputeTessLevel(cpBezier[3].P, cpBezier[15].P);
    }
    if ((transitionMask & 4) != 0) {
        vec3 ev23 = OsdEvalBezier(cpBezier, patchParam, vec2(0.5, 1.0));
        tessOuterLo[3] = OsdComputeTessLevel(cpBezier[12].P, ev23);
        tessOuterHi[3] = OsdComputeTessLevel(cpBezier[15].P, ev23);
    } else {
        tessOuterLo[3] = OsdComputeTessLevel(cpBezier[12].P, cpBezier[15].P);
    }
#endif
}

// Round up to the nearest even integer
float OsdRoundUpEven(float x) {
    return 2*ceil(x/2);
}

// Round up to the nearest odd integer
float OsdRoundUpOdd(float x) {
    return 2*ceil((x+1)/2)-1;
}

// Compute outer and inner tessellation levels taking into account the
// current tessellation spacing mode.
void
OsdComputeTessLevels(inout vec4 tessOuterLo, inout vec4 tessOuterHi,
                     out vec4 tessLevelOuter, out vec2 tessLevelInner)
{
    // Outer levels are the sum of the Lo and Hi segments where the Hi
    // segments will have lengths of zero for non-transition edges.

#if defined OSD_FRACTIONAL_EVEN_SPACING
    // Combine fractional outer transition edge levels before rounding.
    vec4 combinedOuter = tessOuterLo + tessOuterHi;

    // Round the segments of transition edges separately. We will recover the
    // fractional parameterization of transition edges after tessellation.

    tessLevelOuter = combinedOuter;
    if (tessOuterHi[0] > 0) {
        tessLevelOuter[0] =
            OsdRoundUpEven(tessOuterLo[0]) + OsdRoundUpEven(tessOuterHi[0]);
    }
    if (tessOuterHi[1] > 0) {
        tessLevelOuter[1] =
            OsdRoundUpEven(tessOuterLo[1]) + OsdRoundUpEven(tessOuterHi[1]);
    }
    if (tessOuterHi[2] > 0) {
        tessLevelOuter[2] =
            OsdRoundUpEven(tessOuterLo[2]) + OsdRoundUpEven(tessOuterHi[2]);
    }
    if (tessOuterHi[3] > 0) {
        tessLevelOuter[3] =
            OsdRoundUpEven(tessOuterLo[3]) + OsdRoundUpEven(tessOuterHi[3]);
    }
#elif defined OSD_FRACTIONAL_ODD_SPACING
    // Combine fractional outer transition edge levels before rounding.
    vec4 combinedOuter = tessOuterLo + tessOuterHi;

    // Round the segments of transition edges separately. We will recover the
    // fractional parameterization of transition edges after tessellation.
    //
    // The sum of the two outer odd segment lengths will be an even number
    // which the tessellator will increase by +1 so that there will be a
    // total odd number of segments. We clamp the combinedOuter tess levels
    // (used to compute the inner tess levels) so that the outer transition
    // edges will be sampled without degenerate triangles.

    tessLevelOuter = combinedOuter;
    if (tessOuterHi[0] > 0) {
        tessLevelOuter[0] =
            OsdRoundUpOdd(tessOuterLo[0]) + OsdRoundUpOdd(tessOuterHi[0]);
        combinedOuter = max(vec4(3), combinedOuter);
    }
    if (tessOuterHi[1] > 0) {
        tessLevelOuter[1] =
            OsdRoundUpOdd(tessOuterLo[1]) + OsdRoundUpOdd(tessOuterHi[1]);
        combinedOuter = max(vec4(3), combinedOuter);
    }
    if (tessOuterHi[2] > 0) {
        tessLevelOuter[2] =
            OsdRoundUpOdd(tessOuterLo[2]) + OsdRoundUpOdd(tessOuterHi[2]);
        combinedOuter = max(vec4(3), combinedOuter);
    }
    if (tessOuterHi[3] > 0) {
        tessLevelOuter[3] =
            OsdRoundUpOdd(tessOuterLo[3]) + OsdRoundUpOdd(tessOuterHi[3]);
        combinedOuter = max(vec4(3), combinedOuter);
    }
#else
    // Round equally spaced transition edge levels before combining.
    tessOuterLo = round(tessOuterLo);
    tessOuterHi = round(tessOuterHi);

    vec4 combinedOuter = tessOuterLo + tessOuterHi;
    tessLevelOuter = combinedOuter;
#endif

    // Inner levels are the averages the corresponding outer levels.
    tessLevelInner[0] = (combinedOuter[1] + combinedOuter[3]) * 0.5;
    tessLevelInner[1] = (combinedOuter[0] + combinedOuter[2]) * 0.5;
}

void
OsdGetTessLevelsUniform(ivec3 patchParam,
                 out vec4 tessLevelOuter, out vec2 tessLevelInner,
                 out vec4 tessOuterLo, out vec4 tessOuterHi)
{
    // uniform tessellation
    OsdGetTessLevelsUniform(patchParam, tessOuterLo, tessOuterHi);

    OsdComputeTessLevels(tessOuterLo, tessOuterHi,
                         tessLevelOuter, tessLevelInner);
}

void
OsdGetTessLevelsAdaptiveRefinedPoints(vec3 cpRefined[16], ivec3 patchParam,
                        out vec4 tessLevelOuter, out vec2 tessLevelInner,
                        out vec4 tessOuterLo, out vec4 tessOuterHi)
{
    OsdGetTessLevelsRefinedPoints(cpRefined, patchParam,
                                  tessOuterLo, tessOuterHi);

    OsdComputeTessLevels(tessOuterLo, tessOuterHi,
                         tessLevelOuter, tessLevelInner);
}

void
OsdGetTessLevelsAdaptiveLimitPoints(OsdPerPatchVertexBezier cpBezier[16],
                 ivec3 patchParam,
                 out vec4 tessLevelOuter, out vec2 tessLevelInner,
                 out vec4 tessOuterLo, out vec4 tessOuterHi)
{
    OsdGetTessLevelsLimitPoints(cpBezier, patchParam,
                                tessOuterLo, tessOuterHi);

    OsdComputeTessLevels(tessOuterLo, tessOuterHi,
                         tessLevelOuter, tessLevelInner);
}

void
OsdGetTessLevels(vec3 cp0, vec3 cp1, vec3 cp2, vec3 cp3,
                 ivec3 patchParam,
                 out vec4 tessLevelOuter, out vec2 tessLevelInner)
{
    vec4 tessOuterLo = vec4(0);
    vec4 tessOuterHi = vec4(0);

#if defined OSD_ENABLE_SCREENSPACE_TESSELLATION
    tessOuterLo[0] = OsdComputeTessLevel(cp0, cp1);
    tessOuterLo[1] = OsdComputeTessLevel(cp0, cp3);
    tessOuterLo[2] = OsdComputeTessLevel(cp2, cp3);
    tessOuterLo[3] = OsdComputeTessLevel(cp1, cp2);
    tessOuterHi = vec4(0);
#else
    OsdGetTessLevelsUniform(patchParam, tessOuterLo, tessOuterHi);
#endif

    OsdComputeTessLevels(tessOuterLo, tessOuterHi,
                         tessLevelOuter, tessLevelInner);
}

#if defined OSD_FRACTIONAL_EVEN_SPACING || defined OSD_FRACTIONAL_ODD_SPACING
float
OsdGetTessFractionalSplit(float t, float level, float levelUp)
{
    // Fractional tessellation of an edge will produce n segments where n
    // is the tessellation level of the edge (level) rounded up to the
    // nearest even or odd integer (levelUp). There will be n-2 segments of
    // equal length (dx1) and two additional segments of equal length (dx0)
    // that are typically shorter than the other segments. The two additional
    // segments should be placed symmetrically on opposite sides of the
    // edge (offset).

#if defined OSD_FRACTIONAL_EVEN_SPACING
    if (level <= 2) return t;

    float base = pow(2.0,floor(log2(levelUp)));
    float offset = 1.0/(int(2*base-levelUp)/2 & int(base/2-1));

#elif defined OSD_FRACTIONAL_ODD_SPACING
    if (level <= 1) return t;

    float base = pow(2.0,floor(log2(levelUp)));
    float offset = 1.0/(((int(2*base-levelUp)/2+1) & int(base/2-1))+1);
#endif

    float dx0 = (1.0 - (levelUp-level)/2) / levelUp;
    float dx1 = (1.0 - 2.0*dx0) / (levelUp - 2.0*ceil(dx0));

    if (t < 0.5) {
        float x = levelUp/2 - round(t*levelUp);
        return 0.5 - (x*dx1 + int(x*offset > 1) * (dx0 - dx1));
    } else if (t > 0.5) {
        float x = round(t*levelUp) - levelUp/2;
        return 0.5 + (x*dx1 + int(x*offset > 1) * (dx0 - dx1));
    } else {
        return t;
    }
}
#endif

float
OsdGetTessTransitionSplit(float t, float lo, float hi)
{
#if defined OSD_FRACTIONAL_EVEN_SPACING
    float loRoundUp = OsdRoundUpEven(lo);
    float hiRoundUp = OsdRoundUpEven(hi);

    // Convert the parametric t into a segment index along the combined edge.
    float ti = round(t * (loRoundUp + hiRoundUp));

    if (ti <= loRoundUp) {
        float t0 = ti / loRoundUp;
        return OsdGetTessFractionalSplit(t0, lo, loRoundUp) * 0.5;
    } else {
        float t1 = (ti - loRoundUp) / hiRoundUp;
        return OsdGetTessFractionalSplit(t1, hi, hiRoundUp) * 0.5 + 0.5;
    }
#elif defined OSD_FRACTIONAL_ODD_SPACING
    float loRoundUp = OsdRoundUpOdd(lo);
    float hiRoundUp = OsdRoundUpOdd(hi);

    // Convert the parametric t into a segment index along the combined edge.
    // The +1 below is to account for the extra segment produced by the
    // tessellator since the sum of two odd tess levels will be rounded
    // up by one to the next odd integer tess level.
    float ti = round(t * (loRoundUp + hiRoundUp + 1));

    if (ti <= loRoundUp) {
        float t0 = ti / loRoundUp;
        return OsdGetTessFractionalSplit(t0, lo, loRoundUp) * 0.5;
    } else if (ti > (loRoundUp+1)) {
        float t1 = (ti - (loRoundUp+1)) / hiRoundUp;
        return OsdGetTessFractionalSplit(t1, hi, hiRoundUp) * 0.5 + 0.5;
    } else {
        return 0.5;
    }
#else
    // Convert the parametric t into a segment index along the combined edge.
    float ti = round(t * (lo + hi));

    if (ti <= lo) {
        return (ti / lo) * 0.5;
    } else {
        return ((ti - lo) / hi) * 0.5 + 0.5;
    }
#endif
}

vec2
OsdGetTessParameterization(vec2 uv, vec4 tessOuterLo, vec4 tessOuterHi)
{
    vec2 UV = uv;
    if (UV.x == 0 && tessOuterHi[0] > 0) {
        UV.y = OsdGetTessTransitionSplit(UV.y, tessOuterLo[0], tessOuterHi[0]);
    } else
    if (UV.y == 0 && tessOuterHi[1] > 0) {
        UV.x = OsdGetTessTransitionSplit(UV.x, tessOuterLo[1], tessOuterHi[1]);
    } else
    if (UV.x == 1 && tessOuterHi[2] > 0) {
        UV.y = OsdGetTessTransitionSplit(UV.y, tessOuterLo[2], tessOuterHi[2]);
    } else
    if (UV.y == 1 && tessOuterHi[3] > 0) {
        UV.x = OsdGetTessTransitionSplit(UV.x, tessOuterLo[3], tessOuterHi[3]);
    }
    return UV;
}

// ----------------------------------------------------------------------------
// BSpline
// ----------------------------------------------------------------------------

// compute single-crease patch matrix
mat4
OsdComputeMs(float sharpness)
{
    float s = pow(2.0f, sharpness);
    float s2 = s*s;
    float s3 = s2*s;

    mat4 m = mat4(
        0, s + 1 + 3*s2 - s3, 7*s - 2 - 6*s2 + 2*s3, (1-s)*(s-1)*(s-1),
        0,       (1+s)*(1+s),        6*s - 2 - 2*s2,       (s-1)*(s-1),
        0,               1+s,               6*s - 2,               1-s,
        0,                 1,               6*s - 2,                 1);

    m /= (s*6.0);
    m[0][0] = 1.0/6.0;

    return m;
}

// flip matrix orientation
mat4
OsdFlipMatrix(mat4 m)
{
    return mat4(m[3][3], m[3][2], m[3][1], m[3][0],
                m[2][3], m[2][2], m[2][1], m[2][0],
                m[1][3], m[1][2], m[1][1], m[1][0],
                m[0][3], m[0][2], m[0][1], m[0][0]);
}

// Regular BSpline to Bezier
uniform mat4 Q = mat4(
    1.f/6.f, 4.f/6.f, 1.f/6.f, 0.f,
    0.f,     4.f/6.f, 2.f/6.f, 0.f,
    0.f,     2.f/6.f, 4.f/6.f, 0.f,
    0.f,     1.f/6.f, 4.f/6.f, 1.f/6.f
);

// Infinitely Sharp (boundary)
uniform mat4 Mi = mat4(
    1.f/6.f, 4.f/6.f, 1.f/6.f, 0.f,
    0.f,     4.f/6.f, 2.f/6.f, 0.f,
    0.f,     2.f/6.f, 4.f/6.f, 0.f,
    0.f,     0.f,     1.f,     0.f
);

// convert BSpline cv to Bezier cv
void
OsdComputePerPatchVertexBSpline(ivec3 patchParam, int ID, vec3 cv[16],
                                out OsdPerPatchVertexBezier result)
{
    result.patchParam = patchParam;

    int i = ID%4;
    int j = ID/4;

#if defined OSD_PATCH_ENABLE_SINGLE_CREASE

    vec3 P  = vec3(0); // 0 to 1-2^(-Sf)
    vec3 P1 = vec3(0); // 1-2^(-Sf) to 1-2^(-Sc)
    vec3 P2 = vec3(0); // 1-2^(-Sc) to 1

    float sharpness = OsdGetPatchSharpness(patchParam);
    if (sharpness > 0) {
        float Sf = floor(sharpness);
        float Sc = ceil(sharpness);
        float Sr = fract(sharpness);
        mat4 Mf = OsdComputeMs(Sf);
        mat4 Mc = OsdComputeMs(Sc);
        mat4 Mj = (1-Sr) * Mf + Sr * Mi;
        mat4 Ms = (1-Sr) * Mf + Sr * Mc;
        float s0 = 1 - pow(2, -floor(sharpness));
        float s1 = 1 - pow(2, -ceil(sharpness));
        result.vSegments = vec2(s0, s1);

        mat4 MUi = Q, MUj = Q, MUs = Q;
        mat4 MVi = Q, MVj = Q, MVs = Q;

        int boundaryMask = OsdGetPatchBoundaryMask(patchParam);
        if ((boundaryMask & 1) != 0) {
            MVi = OsdFlipMatrix(Mi);
            MVj = OsdFlipMatrix(Mj);
            MVs = OsdFlipMatrix(Ms);
        }
        if ((boundaryMask & 2) != 0) {
            MUi = Mi;
            MUj = Mj;
            MUs = Ms;
        }
        if ((boundaryMask & 4) != 0) {
            MVi = Mi;
            MVj = Mj;
            MVs = Ms;
        }
        if ((boundaryMask & 8) != 0) {
            MUi = OsdFlipMatrix(Mi);
            MUj = OsdFlipMatrix(Mj);
            MUs = OsdFlipMatrix(Ms);
        }

        vec3 Hi[4], Hj[4], Hs[4];
        for (int l=0; l<4; ++l) {
            Hi[l] = Hj[l] = Hs[l] = vec3(0);
            for (int k=0; k<4; ++k) {
                Hi[l] += MUi[i][k] * cv[l*4 + k];
                Hj[l] += MUj[i][k] * cv[l*4 + k];
                Hs[l] += MUs[i][k] * cv[l*4 + k];
            }
        }
        for (int k=0; k<4; ++k) {
            P  += MVi[j][k]*Hi[k];
            P1 += MVj[j][k]*Hj[k];
            P2 += MVs[j][k]*Hs[k];
        }

        result.P  = P;
        result.P1 = P1;
        result.P2 = P2;
    } else {
        result.vSegments = vec2(0);

        OsdComputeBSplineBoundaryPoints(cv, patchParam);

        vec3 Hi[4];
        for (int l=0; l<4; ++l) {
            Hi[l] = vec3(0);
            for (int k=0; k<4; ++k) {
                Hi[l] += Q[i][k] * cv[l*4 + k];
            }
        }
        for (int k=0; k<4; ++k) {
            P += Q[j][k]*Hi[k];
        }

        result.P  = P;
        result.P1 = P;
        result.P2 = P;
    }
#else
    OsdComputeBSplineBoundaryPoints(cv, patchParam);

    vec3 H[4];
    for (int l=0; l<4; ++l) {
        H[l] = vec3(0);
        for (int k=0; k<4; ++k) {
            H[l] += Q[i][k] * cv[l*4 + k];
        }
    }
    {
        result.P = vec3(0);
        for (int k=0; k<4; ++k) {
            result.P += Q[j][k]*H[k];
        }
    }
#endif
}

void
OsdEvalPatchBezier(ivec3 patchParam, vec2 UV,
                   OsdPerPatchVertexBezier cv[16],
                   out vec3 P, out vec3 dPu, out vec3 dPv,
                   out vec3 N, out vec3 dNu, out vec3 dNv)
{
    //
    //  Use the recursive nature of the basis functions to compute a 2x2 set
    //  of intermediate points (via repeated linear interpolation).  These
    //  points define a bilinear surface tangent to the desired surface at P
    //  and so containing dPu and dPv.  The cost of computing P, dPu and dPv
    //  this way is comparable to that of typical tensor product evaluation
    //  (if not faster).
    //
    //  If N = dPu X dPv degenerates, it often results from an edge of the
    //  2x2 bilinear hull collapsing or two adjacent edges colinear. In both
    //  cases, the expected non-planar quad degenerates into a triangle, and
    //  the tangent plane of that triangle provides the desired normal N.
    //

    //  Reduce 4x4 points to 2x4 -- two levels of linear interpolation in U
    //  and so 3 original rows contributing to each of the 2 resulting rows:
    float u    = UV.x;
    float uinv = 1.0f - u;

    float u0 = uinv * uinv;
    float u1 = u * uinv * 2.0f;
    float u2 = u * u;

    vec3 LROW[4], RROW[4];
#ifndef OSD_PATCH_ENABLE_SINGLE_CREASE
    LROW[0] = u0 * cv[ 0].P + u1 * cv[ 1].P + u2 * cv[ 2].P;
    LROW[1] = u0 * cv[ 4].P + u1 * cv[ 5].P + u2 * cv[ 6].P;
    LROW[2] = u0 * cv[ 8].P + u1 * cv[ 9].P + u2 * cv[10].P;
    LROW[3] = u0 * cv[12].P + u1 * cv[13].P + u2 * cv[14].P;

    RROW[0] = u0 * cv[ 1].P + u1 * cv[ 2].P + u2 * cv[ 3].P;
    RROW[1] = u0 * cv[ 5].P + u1 * cv[ 6].P + u2 * cv[ 7].P;
    RROW[2] = u0 * cv[ 9].P + u1 * cv[10].P + u2 * cv[11].P;
    RROW[3] = u0 * cv[13].P + u1 * cv[14].P + u2 * cv[15].P;
#else
    vec2 vSegments = cv[0].vSegments;
    float s = OsdGetPatchSingleCreaseSegmentParameter(patchParam, UV);

    for (int i = 0; i < 4; ++i) {
        int j = i*4;
        if (s <= vSegments.x) {
            LROW[i] = u0 * cv[ j ].P + u1 * cv[j+1].P + u2 * cv[j+2].P;
            RROW[i] = u0 * cv[j+1].P + u1 * cv[j+2].P + u2 * cv[j+3].P;
        } else if (s <= vSegments.y) {
            LROW[i] = u0 * cv[ j ].P1 + u1 * cv[j+1].P1 + u2 * cv[j+2].P1;
            RROW[i] = u0 * cv[j+1].P1 + u1 * cv[j+2].P1 + u2 * cv[j+3].P1;
        } else {
            LROW[i] = u0 * cv[ j ].P2 + u1 * cv[j+1].P2 + u2 * cv[j+2].P2;
            RROW[i] = u0 * cv[j+1].P2 + u1 * cv[j+2].P2 + u2 * cv[j+3].P2;
        }
    }
#endif

    //  Reduce 2x4 points to 2x2 -- two levels of linear interpolation in V
    //  and so 3 original pairs contributing to each of the 2 resulting:
    float v    = UV.y;
    float vinv = 1.0f - v;

    float v0 = vinv * vinv;
    float v1 = v * vinv * 2.0f;
    float v2 = v * v;

    vec3 LPAIR[2], RPAIR[2];
    LPAIR[0] = v0 * LROW[0] + v1 * LROW[1] + v2 * LROW[2];
    RPAIR[0] = v0 * RROW[0] + v1 * RROW[1] + v2 * RROW[2];

    LPAIR[1] = v0 * LROW[1] + v1 * LROW[2] + v2 * LROW[3];
    RPAIR[1] = v0 * RROW[1] + v1 * RROW[2] + v2 * RROW[3];

    //  Interpolate points on the edges of the 2x2 bilinear hull from which
    //  both position and partials are trivially determined:
    vec3 DU0 = vinv * LPAIR[0] + v * LPAIR[1];
    vec3 DU1 = vinv * RPAIR[0] + v * RPAIR[1];
    vec3 DV0 = uinv * LPAIR[0] + u * RPAIR[0];
    vec3 DV1 = uinv * LPAIR[1] + u * RPAIR[1];

    int level = OsdGetPatchFaceLevel(patchParam);
    dPu = (DU1 - DU0) * 3 * level;
    dPv = (DV1 - DV0) * 3 * level;

    P = u * DU1 + uinv * DU0;

    //  Compute the normal and test for degeneracy:
    //
    //  We need a geometric measure of the size of the patch for a suitable
    //  tolerance.  Magnitudes of the partials are generally proportional to
    //  that size -- the sum of the partials is readily available, cheap to
    //  compute, and has proved effective in most cases (though not perfect).
    //  The size of the bounding box of the patch, or some approximation to
    //  it, would be better but more costly to compute.
    //
    float proportionalNormalTolerance = 0.00001f;

    float nEpsilon = (length(dPu) + length(dPv)) * proportionalNormalTolerance;

    N = cross(dPu, dPv);

    float nLength = length(N);
    if (nLength > nEpsilon) {
        N = N / nLength;
    } else {
        vec3 diagCross = cross(RPAIR[1] - LPAIR[0], LPAIR[1] - RPAIR[0]);
        float diagCrossLength = length(diagCross);
        if (diagCrossLength > nEpsilon) {
            N = diagCross / diagCrossLength;
        }
    }

#ifndef OSD_COMPUTE_NORMAL_DERIVATIVES
    dNu = vec3(0);
    dNv = vec3(0);
#else
    //
    //  Compute 2nd order partials of P(u,v) in order to compute 1st order partials
    //  for the un-normalized n(u,v) = dPu X dPv, then project into the tangent
    //  plane of normalized N.  With resulting dNu and dNv we can make another
    //  attempt to resolve a still-degenerate normal.
    //
    //  We don't use the Weingarten equations here as they require N != 0 and also
    //  are a little less numerically stable/accurate in single precision.
    //
    float B0u[4], B1u[4], B2u[4];
    float B0v[4], B1v[4], B2v[4];

    OsdUnivar4x4(UV.x, B0u, B1u, B2u);
    OsdUnivar4x4(UV.y, B0v, B1v, B2v);

    vec3 dUU = vec3(0);
    vec3 dVV = vec3(0);
    vec3 dUV = vec3(0);

    for (int i=0; i<4; ++i) {
        for (int j=0; j<4; ++j) {
#ifdef OSD_PATCH_ENABLE_SINGLE_CREASE
            int k = 4*i + j;
            vec3 CV = (s <= vSegments.x) ? cv[k].P
                 :   ((s <= vSegments.y) ? cv[k].P1
                                         : cv[k].P2);
#else
            vec3 CV = cv[4*i + j].P;
#endif
            dUU += (B0v[i] * B2u[j]) * CV;
            dVV += (B2v[i] * B0u[j]) * CV;
            dUV += (B1v[i] * B1u[j]) * CV;
        }
    }

    dUU *= 6 * level;
    dVV *= 6 * level;
    dUV *= 9 * level;

    dNu = cross(dUU, dPv) + cross(dPu, dUV);
    dNv = cross(dUV, dPv) + cross(dPu, dVV);

    float nLengthInv = 1.0;
    if (nLength > nEpsilon) {
        nLengthInv = 1.0 / nLength;
    } else {
        //  N may have been resolved above if degenerate, but if N was resolved
        //  we don't have an accurate length for its un-normalized value, and that
        //  length is needed to project the un-normalized dNu and dNv into the
        //  tangent plane of N.
        //
        //  So compute N more accurately with available second derivatives, i.e.
        //  with a 1st order Taylor approximation to un-normalized N(u,v).

        float DU = (UV.x == 1.0f) ? -1.0f : 1.0f;
        float DV = (UV.y == 1.0f) ? -1.0f : 1.0f;

        N = DU * dNu + DV * dNv;

        nLength = length(N);
        if (nLength > nEpsilon) {
            nLengthInv = 1.0f / nLength;
            N = N * nLengthInv;
        }
    }

    //  Project derivatives of non-unit normals into tangent plane of N:
    dNu = (dNu - dot(dNu,N) * N) * nLengthInv;
    dNv = (dNv - dot(dNv,N) * N) * nLengthInv;
#endif
}

// ----------------------------------------------------------------------------
// Gregory Basis
// ----------------------------------------------------------------------------

struct OsdPerPatchVertexGregoryBasis {
    ivec3 patchParam;
    vec3 P;
};

void
OsdComputePerPatchVertexGregoryBasis(ivec3 patchParam, int ID, vec3 cv,
                                     out OsdPerPatchVertexGregoryBasis result)
{
    result.patchParam = patchParam;
    result.P = cv;
}

void
OsdEvalPatchGregory(ivec3 patchParam, vec2 UV, vec3 cv[20],
                    out vec3 P, out vec3 dPu, out vec3 dPv,
                    out vec3 N, out vec3 dNu, out vec3 dNv)
{
    float u = UV.x, v = UV.y;
    float U = 1-u, V = 1-v;

    //(0,1)                              (1,1)
    //   P3         e3-      e2+         P2
    //      15------17-------11-------10
    //      |        |        |        |
    //      |        |        |        |
    //      |        | f3-    | f2+    |
    //      |       19       13        |
    //  e3+ 16-----18          14-----12 e2-
    //      |     f3+          f2-     |
    //      |                          |
    //      |                          |
    //      |     f0-         f1+      |
    //  e0- 2------4            8------6 e1+
    //      |        3 f0+    9        |
    //      |        |        | f1-    |
    //      |        |        |        |
    //      |        |        |        |
    //      0--------1--------7--------5
    //    P0        e0+      e1-         P1
    //(0,0)                               (1,0)

    float d11 = u+v;
    float d12 = U+v;
    float d21 = u+V;
    float d22 = U+V;

    OsdPerPatchVertexBezier bezcv[16];

    bezcv[ 5].P = (d11 == 0.0) ? cv[3]  : (u*cv[3] + v*cv[4])/d11;
    bezcv[ 6].P = (d12 == 0.0) ? cv[8]  : (U*cv[9] + v*cv[8])/d12;
    bezcv[ 9].P = (d21 == 0.0) ? cv[18] : (u*cv[19] + V*cv[18])/d21;
    bezcv[10].P = (d22 == 0.0) ? cv[13] : (U*cv[13] + V*cv[14])/d22;

    bezcv[ 0].P = cv[0];
    bezcv[ 1].P = cv[1];
    bezcv[ 2].P = cv[7];
    bezcv[ 3].P = cv[5];
    bezcv[ 4].P = cv[2];
    bezcv[ 7].P = cv[6];
    bezcv[ 8].P = cv[16];
    bezcv[11].P = cv[12];
    bezcv[12].P = cv[15];
    bezcv[13].P = cv[17];
    bezcv[14].P = cv[11];
    bezcv[15].P = cv[10];

    OsdEvalPatchBezier(patchParam, UV, bezcv, P, dPu, dPv, N, dNu, dNv);
}