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https://github.com/PixarAnimationStudios/OpenSubdiv
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506 lines
19 KiB
C++
506 lines
19 KiB
C++
//
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// Copyright 2013 Pixar
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//
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// Licensed under the Apache License, Version 2.0 (the "Apache License")
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// with the following modification; you may not use this file except in
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// compliance with the Apache License and the following modification to it:
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// Section 6. Trademarks. is deleted and replaced with:
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//
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// 6. Trademarks. This License does not grant permission to use the trade
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// names, trademarks, service marks, or product names of the Licensor
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// and its affiliates, except as required to comply with Section 4(c) of
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// the License and to reproduce the content of the NOTICE file.
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//
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// You may obtain a copy of the Apache License at
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//
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// http://www.apache.org/licenses/LICENSE-2.0
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//
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// Unless required by applicable law or agreed to in writing, software
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// distributed under the Apache License with the above modification is
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// distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
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// KIND, either express or implied. See the Apache License for the specific
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// language governing permissions and limitations under the Apache License.
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//
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#include "../far/patchBasis.h"
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#include <cassert>
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#include <cstring>
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namespace OpenSubdiv {
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namespace OPENSUBDIV_VERSION {
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namespace Far {
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namespace internal {
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enum SplineBasis {
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BASIS_BILINEAR,
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BASIS_BEZIER,
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BASIS_BSPLINE,
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BASIS_BOX_SPLINE
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};
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template <SplineBasis BASIS>
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class Spline {
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public:
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// curve weights
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static void GetWeights(float t, float point[], float deriv[], float deriv2[]);
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// box-spline weights
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static void GetWeights(float v, float w, float point[]);
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// patch weights
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static void GetPatchWeights(PatchParam const & param,
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float s, float t, float point[], float deriv1[], float deriv2[], float deriv11[], float deriv12[], float deriv22[]);
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// adjust patch weights for boundary (and corner) edges
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static void AdjustBoundaryWeights(PatchParam const & param,
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float sWeights[4], float tWeights[4]);
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};
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template <>
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inline void Spline<BASIS_BEZIER>::GetWeights(
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float t, float point[4], float deriv[4], float deriv2[4]) {
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// The four uniform cubic Bezier basis functions (in terms of t and its
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// complement tC) evaluated at t:
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float t2 = t*t;
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float tC = 1.0f - t;
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float tC2 = tC * tC;
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assert(point);
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point[0] = tC2 * tC;
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point[1] = tC2 * t * 3.0f;
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point[2] = t2 * tC * 3.0f;
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point[3] = t2 * t;
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// Derivatives of the above four basis functions at t:
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if (deriv) {
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deriv[0] = -3.0f * tC2;
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deriv[1] = 9.0f * t2 - 12.0f * t + 3.0f;
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deriv[2] = -9.0f * t2 + 6.0f * t;
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deriv[3] = 3.0f * t2;
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}
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// Second derivatives of the basis functions at t:
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if (deriv2) {
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deriv2[0] = 6.0f * tC;
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deriv2[1] = 18.0f * t - 12.0f;
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deriv2[2] = -18.0f * t + 6.0f;
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deriv2[3] = 6.0f * t;
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}
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}
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template <>
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inline void Spline<BASIS_BSPLINE>::GetWeights(
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float t, float point[4], float deriv[4], float deriv2[4]) {
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// The four uniform cubic B-Spline basis functions evaluated at t:
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float const one6th = 1.0f / 6.0f;
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float t2 = t * t;
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float t3 = t * t2;
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assert(point);
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point[0] = one6th * (1.0f - 3.0f*(t - t2) - t3);
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point[1] = one6th * (4.0f - 6.0f*t2 + 3.0f*t3);
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point[2] = one6th * (1.0f + 3.0f*(t + t2 - t3));
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point[3] = one6th * ( t3);
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// Derivatives of the above four basis functions at t:
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if (deriv) {
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deriv[0] = -0.5f*t2 + t - 0.5f;
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deriv[1] = 1.5f*t2 - 2.0f*t;
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deriv[2] = -1.5f*t2 + t + 0.5f;
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deriv[3] = 0.5f*t2;
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}
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// Second derivatives of the basis functions at t:
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if (deriv2) {
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deriv2[0] = - t + 1.0f;
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deriv2[1] = 3.0f * t - 2.0f;
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deriv2[2] = -3.0f * t + 1.0f;
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deriv2[3] = t;
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}
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}
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template <>
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inline void Spline<BASIS_BOX_SPLINE>::GetWeights(
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float v, float w, float point[12]) {
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float u = 1.0f - v - w;
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//
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// The 12 basis functions of the quartic box spline (unscaled by their common
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// factor of 1/12 until later, and formatted to make it easy to spot any
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// typing errors):
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//
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// 15 terms for the 3 points above the triangle corners
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// 9 terms for the 3 points on faces opposite the triangle edges
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// 2 terms for the 6 points on faces opposite the triangle corners
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//
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// Powers of each variable for notational convenience:
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float u2 = u*u;
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float u3 = u*u2;
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float u4 = u*u3;
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float v2 = v*v;
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float v3 = v*v2;
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float v4 = v*v3;
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float w2 = w*w;
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float w3 = w*w2;
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float w4 = w*w3;
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// And now the basis functions:
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point[ 0] = u4 + 2.0f*u3*v;
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point[ 1] = u4 + 2.0f*u3*w;
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point[ 8] = w4 + 2.0f*w3*u;
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point[11] = w4 + 2.0f*w3*v;
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point[ 9] = v4 + 2.0f*v3*w;
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point[ 5] = v4 + 2.0f*v3*u;
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point[ 2] = u4 + 2.0f*u3*w + 6.0f*u3*v + 6.0f*u2*v*w + 12.0f*u2*v2 +
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v4 + 2.0f*v3*w + 6.0f*v3*u + 6.0f*v2*u*w;
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point[ 4] = w4 + 2.0f*w3*v + 6.0f*w3*u + 6.0f*w2*u*v + 12.0f*w2*u2 +
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u4 + 2.0f*u3*v + 6.0f*u3*w + 6.0f*u2*v*w;
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point[10] = v4 + 2.0f*v3*u + 6.0f*v3*w + 6.0f*v2*w*u + 12.0f*v2*w2 +
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w4 + 2.0f*w3*u + 6.0f*w3*v + 6.0f*w3*u*v;
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point[ 3] = v4 + 6*v3*w + 8*v3*u + 36*v2*w*u + 24*v2*u2 + 24*v*u3 +
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w4 + 6*w3*v + 8*w3*u + 36*w2*v*u + 24*w2*u2 + 24*w*u3 + 6*u4 + 60*u2*v*w + 12*v2*w2;
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point[ 6] = w4 + 6*w3*u + 8*w3*v + 36*w2*u*v + 24*w2*v2 + 24*w*v3 +
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u4 + 6*u3*w + 8*u3*v + 36*u2*v*w + 24*u2*v2 + 24*u*v3 + 6*v4 + 60*v2*w*u + 12*w2*u2;
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point[ 7] = u4 + 6*u3*v + 8*u3*w + 36*u2*v*w + 24*u2*w2 + 24*u*w3 +
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v4 + 6*v3*u + 8*v3*w + 36*v2*u*w + 24*v2*w2 + 24*v*w3 + 6*w4 + 60*w2*u*v + 12*u2*v2;
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for (int i = 0; i < 12; ++i) {
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point[i] *= 1.0f / 12.0f;
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}
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}
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template <>
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inline void Spline<BASIS_BILINEAR>::GetPatchWeights(PatchParam const & param,
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float s, float t, float point[4], float derivS[4], float derivT[4], float derivSS[4], float derivST[4], float derivTT[4]) {
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param.Normalize(s,t);
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float sC = 1.0f - s,
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tC = 1.0f - t;
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if (point) {
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point[0] = sC * tC;
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point[1] = s * tC;
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point[2] = s * t;
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point[3] = sC * t;
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}
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if (derivS and derivT) {
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float dScale = (float)(1 << param.GetDepth());
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derivS[0] = -tC * dScale;
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derivS[1] = tC * dScale;
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derivS[2] = t * dScale;
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derivS[3] = -t * dScale;
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derivT[0] = -sC * dScale;
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derivT[1] = -s * dScale;
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derivT[2] = s * dScale;
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derivT[3] = sC * dScale;
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if (derivSS and derivST and derivTT) {
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float d2Scale = dScale * dScale;
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for(int i=0;i<4;i++) {
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derivSS[i] = 0;
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derivTT[i] = 0;
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}
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derivST[0] = d2Scale;
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derivST[1] = -d2Scale;
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derivST[2] = -d2Scale;
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derivST[3] = d2Scale;
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}
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}
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}
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template <SplineBasis BASIS>
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void Spline<BASIS>::AdjustBoundaryWeights(PatchParam const & param,
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float sWeights[4], float tWeights[4]) {
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int boundary = param.GetBoundary();
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if (boundary & 1) {
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tWeights[2] -= tWeights[0];
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tWeights[1] += 2*tWeights[0];
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tWeights[0] = 0;
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}
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if (boundary & 2) {
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sWeights[1] -= sWeights[3];
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sWeights[2] += 2*sWeights[3];
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sWeights[3] = 0;
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}
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if (boundary & 4) {
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tWeights[1] -= tWeights[3];
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tWeights[2] += 2*tWeights[3];
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tWeights[3] = 0;
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}
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if (boundary & 8) {
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sWeights[2] -= sWeights[0];
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sWeights[1] += 2*sWeights[0];
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sWeights[0] = 0;
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}
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}
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template <SplineBasis BASIS>
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void Spline<BASIS>::GetPatchWeights(PatchParam const & param,
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float s, float t, float point[16], float derivS[16], float derivT[16], float derivSS[16], float derivST[16], float derivTT[16]) {
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float sWeights[4], tWeights[4], dsWeights[4], dtWeights[4], dssWeights[4], dttWeights[4];
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param.Normalize(s,t);
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Spline<BASIS>::GetWeights(s, point ? sWeights : 0, derivS ? dsWeights : 0, derivSS ? dssWeights : 0);
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Spline<BASIS>::GetWeights(t, point ? tWeights : 0, derivT ? dtWeights : 0, derivTT ? dttWeights : 0);
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if (point) {
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// Compute the tensor product weight of the (s,t) basis function
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// corresponding to each control vertex:
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AdjustBoundaryWeights(param, sWeights, tWeights);
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for (int i = 0; i < 4; ++i) {
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for (int j = 0; j < 4; ++j) {
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point[4*i+j] = sWeights[j] * tWeights[i];
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}
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}
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}
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if (derivS and derivT) {
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// Compute the tensor product weight of the differentiated (s,t) basis
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// function corresponding to each control vertex (scaled accordingly):
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float dScale = (float)(1 << param.GetDepth());
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AdjustBoundaryWeights(param, dsWeights, dtWeights);
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for (int i = 0; i < 4; ++i) {
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for (int j = 0; j < 4; ++j) {
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derivS[4*i+j] = dsWeights[j] * tWeights[i] * dScale;
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derivT[4*i+j] = sWeights[j] * dtWeights[i] * dScale;
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}
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}
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if (derivSS and derivST and derivTT) {
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// Compute the tensor product weight of appropriate differentiated
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// (s,t) basis functions for each control vertex (scaled accordingly):
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float d2Scale = dScale * dScale;
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AdjustBoundaryWeights(param, dssWeights, dttWeights);
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for (int i = 0; i < 4; ++i) {
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for (int j = 0; j < 4; ++j) {
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derivSS[4*i+j] = dssWeights[j] * tWeights[i] * d2Scale;
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derivST[4*i+j] = dsWeights[j] * dtWeights[i] * d2Scale;
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derivTT[4*i+j] = sWeights[j] * dttWeights[i] * d2Scale;
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}
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}
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}
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}
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}
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void GetBilinearWeights(PatchParam const & param,
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float s, float t, float point[4], float deriv1[4], float deriv2[4], float deriv11[4], float deriv12[4], float deriv22[4]) {
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Spline<BASIS_BILINEAR>::GetPatchWeights(param, s, t, point, deriv1, deriv2, deriv11, deriv12, deriv22);
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}
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void GetBezierWeights(PatchParam const param,
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float s, float t, float point[16], float deriv1[16], float deriv2[16], float deriv11[16], float deriv12[16], float deriv22[16]) {
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Spline<BASIS_BEZIER>::GetPatchWeights(param, s, t, point, deriv1, deriv2, deriv11, deriv12, deriv22);
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}
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void GetBSplineWeights(PatchParam const & param,
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float s, float t, float point[16], float deriv1[16], float deriv2[16], float deriv11[16], float deriv12[16], float deriv22[16]) {
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Spline<BASIS_BSPLINE>::GetPatchWeights(param, s, t, point, deriv1, deriv2, deriv11, deriv12, deriv22);
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}
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void GetGregoryWeights(PatchParam const & param,
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float s, float t, float point[20], float deriv1[20], float deriv2[20], float deriv11[20], float deriv12[20], float deriv22[20]) {
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//
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// P3 e3- e2+ P2
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// 15------17-------11--------10
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// | | | |
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// | | | |
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// | | f3- | f2+ |
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// | 19 13 |
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// e3+ 16-----18 14-----12 e2-
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// | f3+ f2- |
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// | |
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// | |
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// | f0- f1+ |
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// e0- 2------4 8------6 e1+
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// | 3 9 |
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// | | f0+ | f1- |
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// | | | |
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// | | | |
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// O--------1--------7--------5
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// P0 e0+ e1- P1
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//
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// Indices of boundary and interior points and their corresponding Bezier points
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// (this can be reduced with more direct indexing and unrolling of loops):
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//
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static int const boundaryGregory[12] = { 0, 1, 7, 5, 2, 6, 16, 12, 15, 17, 11, 10 };
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static int const boundaryBezSCol[12] = { 0, 1, 2, 3, 0, 3, 0, 3, 0, 1, 2, 3 };
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static int const boundaryBezTRow[12] = { 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 3 };
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static int const interiorGregory[8] = { 3, 4, 8, 9, 13, 14, 18, 19 };
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static int const interiorBezSCol[8] = { 1, 1, 2, 2, 2, 2, 1, 1 };
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static int const interiorBezTRow[8] = { 1, 1, 1, 1, 2, 2, 2, 2 };
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//
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// Bezier basis functions are denoted with B while the rational multipliers for the
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// interior points will be denoted G -- so we have B(s), B(t) and G(s,t):
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//
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// Directional Bezier basis functions B at s and t:
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float Bs[4], Bds[4], Bdss[4];
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float Bt[4], Bdt[4], Bdtt[4];
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param.Normalize(s,t);
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Spline<BASIS_BEZIER>::GetWeights(s, Bs, deriv1 ? Bds : 0, deriv11 ? Bdss : 0);
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Spline<BASIS_BEZIER>::GetWeights(t, Bt, deriv2 ? Bdt : 0, deriv22 ? Bdtt : 0);
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// Rational multipliers G at s and t:
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float sC = 1.0f - s;
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float tC = 1.0f - t;
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// Use <= here to avoid compiler warnings -- the sums should always be non-negative:
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float df0 = s + t; df0 = (df0 <= 0.0f) ? 1.0f : (1.0f / df0);
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float df1 = sC + t; df1 = (df1 <= 0.0f) ? 1.0f : (1.0f / df1);
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float df2 = sC + tC; df2 = (df2 <= 0.0f) ? 1.0f : (1.0f / df2);
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float df3 = s + tC; df3 = (df3 <= 0.0f) ? 1.0f : (1.0f / df3);
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float G[8] = { s*df0, t*df0, t*df1, sC*df1, sC*df2, tC*df2, tC*df3, s*df3 };
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// Combined weights for boundary and interior points:
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for (int i = 0; i < 12; ++i) {
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point[boundaryGregory[i]] = Bs[boundaryBezSCol[i]] * Bt[boundaryBezTRow[i]];
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}
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for (int i = 0; i < 8; ++i) {
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point[interiorGregory[i]] = Bs[interiorBezSCol[i]] * Bt[interiorBezTRow[i]] * G[i];
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}
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//
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// For derivatives, the basis functions for the interior points are rational and ideally
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// require appropriate differentiation, i.e. product rule for the combination of B and G
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// and the quotient rule for the rational G itself. As initially proposed by Loop et al
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// though, the approximation using the 16 Bezier points arising from the G(s,t) has
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// proved adequate (and is what the GPU shaders use) so we continue to use that here.
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//
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// An implementation of the true derivatives is provided for future reference -- it is
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// unclear if the approximations will hold up under surface analysis involving higher
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// order differentiation.
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//
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if (deriv1 and deriv2) {
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bool find_second_partials = deriv1 and deriv12 and deriv22;
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// Remember to include derivative scaling in all assignments below:
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float dScale = (float)(1 << param.GetDepth());
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float d2Scale = dScale * dScale;
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// Combined weights for boundary points -- simple (scaled) tensor products:
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for (int i = 0; i < 12; ++i) {
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int iDst = boundaryGregory[i];
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int tRow = boundaryBezTRow[i];
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int sCol = boundaryBezSCol[i];
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deriv1[iDst] = Bds[sCol] * Bt[tRow] * dScale;
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deriv2[iDst] = Bdt[tRow] * Bs[sCol] * dScale;
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if (find_second_partials) {
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deriv11[iDst] = Bdss[sCol] * Bt[tRow] * d2Scale;
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deriv12[iDst] = Bds[sCol] * Bdt[tRow] * d2Scale;
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deriv22[iDst] = Bs[sCol] * Bdtt[tRow] * d2Scale;
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}
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}
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//#define _USE_BEZIER_PSEUDO_DERIVATIVES
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// dclyde's note: skipping half of the product rule like this does seem to change the result a lot in my tests.
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// This is not a runtime bottleneck for cloth sims anyway so I'm just using the accurate version.
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#ifdef _USE_BEZIER_PSEUDO_DERIVATIVES
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// Approximation to the true Gregory derivatives by differentiating the Bezier patch
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// unique to the given (s,t), i.e. having F = (g^+ * f^+) + (g^- * f^-) as its four
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// interior points:
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//
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// Combined weights for interior points -- (scaled) tensor products with G+ or G-:
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for (int i = 0; i < 8; ++i) {
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int iDst = interiorGregory[i];
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int tRow = interiorBezTRow[i];
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int sCol = interiorBezSCol[i];
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|
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deriv1[iDst] = Bds[sCol] * Bt[tRow] * G[i] * dScale;
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deriv2[iDst] = Bdt[tRow] * Bs[sCol] * G[i] * dScale;
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|
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|
if (find_second_partials) {
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deriv11[iDst] = Bdss[sCol] * Bt[tRow] * G[i] * d2Scale;
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|
deriv12[iDst] = Bds[sCol] * Bdt[tRow] * G[i] * d2Scale;
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|
deriv22[iDst] = Bs[sCol] * Bdtt[tRow] * G[i] * d2Scale;
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|
}
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|
}
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|
#else
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|
// True Gregory derivatives using appropriate differentiation of composite functions:
|
|
//
|
|
// Note that for G(s,t) = N(s,t) / D(s,t), all N' and D' are trivial constants (which
|
|
// simplifies things for higher order derivatives). And while each pair of functions
|
|
// G (i.e. the G+ and G- corresponding to points f+ and f-) must sum to 1 to ensure
|
|
// Bezier equivalence (when f+ = f-), the pairs of G' must similarly sum to 0. So we
|
|
// can potentially compute only one of the pair and negate the result for the other
|
|
// (and with 4 or 8 computations involving these constants, this is all very SIMD
|
|
// friendly...) but for now we treat all 8 independently for simplicity.
|
|
//
|
|
//float N[8] = { s, t, t, sC, sC, tC, tC, s };
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|
float D[8] = { df0, df0, df1, df1, df2, df2, df3, df3 };
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|
|
|
static float const Nds[8] = { 1.0f, 0.0f, 0.0f, -1.0f, -1.0f, 0.0f, 0.0f, 1.0f };
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|
static float const Ndt[8] = { 0.0f, 1.0f, 1.0f, 0.0f, 0.0f, -1.0f, -1.0f, 0.0f };
|
|
|
|
static float const Dds[8] = { 1.0f, 1.0f, -1.0f, -1.0f, -1.0f, -1.0f, 1.0f, 1.0f };
|
|
static float const Ddt[8] = { 1.0f, 1.0f, 1.0f, 1.0f, -1.0f, -1.0f, -1.0f, -1.0f };
|
|
|
|
// Combined weights for interior points -- (scaled) combinations of B, B', G and G':
|
|
for (int i = 0; i < 8; ++i) {
|
|
int iDst = interiorGregory[i];
|
|
int tRow = interiorBezTRow[i];
|
|
int sCol = interiorBezSCol[i];
|
|
|
|
// Quotient rule for G' (re-expressed in terms of G to simplify (and D = 1/D)):
|
|
float Gds = (Nds[i] - Dds[i] * G[i]) * D[i];
|
|
float Gdt = (Ndt[i] - Ddt[i] * G[i]) * D[i];
|
|
|
|
// Product rule combining B and B' with G and G' (and scaled):
|
|
deriv1[iDst] = (Bds[sCol] * G[i] + Bs[sCol] * Gds) * Bt[tRow] * dScale;
|
|
deriv2[iDst] = (Bdt[tRow] * G[i] + Bt[tRow] * Gdt) * Bs[sCol] * dScale;
|
|
|
|
if (find_second_partials) {
|
|
float Dsqr_inv = D[i]*D[i];
|
|
|
|
float Gdss = 2.0f * Dds[i] * Dsqr_inv * (G[i] * Dds[i] - Nds[i]);
|
|
float Gdst = Dsqr_inv * (2.0f * G[i] * Dds[i] * Ddt[i] - Nds[i] * Ddt[i] - Ndt[i] * Dds[i]);
|
|
float Gdtt = 2.0f * Ddt[i] * Dsqr_inv * (G[i] * Ddt[i] - Ndt[i]);
|
|
|
|
deriv11[iDst] = (Bdss[sCol] * G[i] + 2.0f * Bds[sCol] * Gds + Bs[sCol] * Gdss) * Bt[tRow] * d2Scale;
|
|
deriv12[iDst] = (Bt[tRow] * (Bs[sCol] * Gdst + Bds[sCol] * Gdt) + Bdt[tRow] * (Bds[sCol] * G[i] + Bs[sCol] * Gds)) * d2Scale;
|
|
deriv22[iDst] = (Bdtt[tRow] * G[i] + 2.0f * Bdt[tRow] * Gdt + Bt[tRow] * Gdtt) * Bs[sCol] * d2Scale;
|
|
}
|
|
}
|
|
#endif
|
|
}
|
|
}
|
|
|
|
} // end namespace internal
|
|
} // end namespace Far
|
|
|
|
} // end namespace OPENSUBDIV_VERSION
|
|
} // end namespace OpenSubdiv
|