src/gpu/tessellate/GrWangsFormula.h

/*
 * Copyright 2020 Google Inc.
 *
 * Use of this source code is governed by a BSD-style license that can be
 * found in the LICENSE file.
 */

#ifndef GrWangsFormula_DEFINED
#define GrWangsFormula_DEFINED

#include "include/core/SkPoint.h"
#include "include/private/SkFloatingPoint.h"
#include "src/gpu/GrVx.h"
#include "src/gpu/tessellate/GrVectorXform.h"

// Wang's formula gives the minimum number of evenly spaced (in the parametric sense) line segments
// that a bezier curve must be chopped into in order to guarantee all lines stay within a distance
// of "1/precision" pixels from the true curve. Its definition for a bezier curve of degree "n" is
// as follows:
//
//     maxLength = max([length(p[i+2] - 2p[i+1] + p[i]) for (0 <= i <= n-2)])
//     numParametricSegments = sqrt(maxLength * precision * n*(n - 1)/8)
//
// (Goldman, Ron. (2003). 5.6.3 Wang's Formula. "Pyramid Algorithms: A Dynamic Programming Approach
// to Curves and Surfaces for Geometric Modeling". Morgan Kaufmann Publishers.)
namespace GrWangsFormula {

// Returns the value by which to multiply length in Wang's formula. (See above.)
template<int Degree> constexpr float length_term(float precision) {
    return (Degree * (Degree - 1) / 8.f) * precision;
}
template<int Degree> constexpr float length_term_pow2(float precision) {
    return ((Degree * Degree) * ((Degree - 1) * (Degree - 1)) / 64.f) * (precision * precision);
}

SK_ALWAYS_INLINE static float root4(float x) {
    return sqrtf(sqrtf(x));
}

// Returns nextlog2(sqrt(x)):
//
//   log2(sqrt(x)) == log2(x^(1/2)) == log2(x)/2 == log2(x)/log2(4) == log4(x)
//
SK_ALWAYS_INLINE static int nextlog4(float x) {
    return (sk_float_nextlog2(x) + 1) >> 1;
}

// Returns nextlog2(sqrt(sqrt(x))):
//
//   log2(sqrt(sqrt(x))) == log2(x^(1/4)) == log2(x)/4 == log2(x)/log2(16) == log16(x)
//
SK_ALWAYS_INLINE static int nextlog16(float x) {
    return (sk_float_nextlog2(x) + 3) >> 2;
}

// Returns Wang's formula, raised to the 4th power, specialized for a quadratic curve.
SK_ALWAYS_INLINE static float quadratic_pow4(float precision, const SkPoint pts[],
                                             const GrVectorXform& vectorXform = GrVectorXform()) {
    using grvx::float2, skvx::bit_pun;
    float2 p0 = bit_pun<float2>(pts[0]);
    float2 p1 = bit_pun<float2>(pts[1]);
    float2 p2 = bit_pun<float2>(pts[2]);
    float2 v = grvx::fast_madd<2>(-2, p1, p0) + p2;
    v = vectorXform(v);
    float2 vv = v*v;
    return (vv[0] + vv[1]) * length_term_pow2<2>(precision);
}

// Returns Wang's formula specialized for a quadratic curve.
SK_ALWAYS_INLINE static float quadratic(float precision, const SkPoint pts[],
                                        const GrVectorXform& vectorXform = GrVectorXform()) {
    return root4(quadratic_pow4(precision, pts, vectorXform));
}

// Returns the log2 value of Wang's formula specialized for a quadratic curve, rounded up to the
// next int.
SK_ALWAYS_INLINE static int quadratic_log2(float precision, const SkPoint pts[],
                                           const GrVectorXform& vectorXform = GrVectorXform()) {
    // nextlog16(x) == ceil(log2(sqrt(sqrt(x))))
    return nextlog16(quadratic_pow4(precision, pts, vectorXform));
}

// Returns Wang's formula, raised to the 4th power, specialized for a cubic curve.
SK_ALWAYS_INLINE static float cubic_pow4(float precision, const SkPoint pts[],
                                         const GrVectorXform& vectorXform = GrVectorXform()) {
    using grvx::float4;
    float4 p01 = float4::Load(pts);
    float4 p12 = float4::Load(pts + 1);
    float4 p23 = float4::Load(pts + 2);
    float4 v = grvx::fast_madd<4>(-2, p12, p01) + p23;
    v = vectorXform(v);
    float4 vv = v*v;
    return std::max(vv[0] + vv[1], vv[2] + vv[3]) * length_term_pow2<3>(precision);
}

// Returns Wang's formula specialized for a cubic curve.
SK_ALWAYS_INLINE static float cubic(float precision, const SkPoint pts[],
                                    const GrVectorXform& vectorXform = GrVectorXform()) {
    return root4(cubic_pow4(precision, pts, vectorXform));
}

// Returns the log2 value of Wang's formula specialized for a cubic curve, rounded up to the next
// int.
SK_ALWAYS_INLINE static int cubic_log2(float precision, const SkPoint pts[],
                                       const GrVectorXform& vectorXform = GrVectorXform()) {
    // nextlog16(x) == ceil(log2(sqrt(sqrt(x))))
    return nextlog16(cubic_pow4(precision, pts, vectorXform));
}

// Returns the maximum number of line segments a cubic with the given device-space bounding box size
// would ever need to be divided into. This is simply a special case of the cubic formula where we
// maximize its value by placing control points on specific corners of the bounding box.
SK_ALWAYS_INLINE static float worst_case_cubic(float precision, float devWidth, float devHeight) {
    float k = length_term<3>(precision);
    return sqrtf(2*k * SkVector::Length(devWidth, devHeight));
}

// Returns the maximum log2 number of line segments a cubic with the given device-space bounding box
// size would ever need to be divided into.
SK_ALWAYS_INLINE static int worst_case_cubic_log2(float precision, float devWidth,
                                                  float devHeight) {
    float kk = length_term_pow2<3>(precision);
    // nextlog16(x) == ceil(log2(sqrt(sqrt(x))))
    return nextlog16(4*kk * (devWidth * devWidth + devHeight * devHeight));
}

// Returns Wang's formula specialized for a conic curve, raised to the second power.
// Input points should be in projected space, and note tolerance parameter is not "precision".
//
// This is not actually due to Wang, but is an analogue from (Theorem 3, corollary 1):
//   J. Zheng, T. Sederberg. "Estimating Tessellation Parameter Intervals for
//   Rational Curves and Surfaces." ACM Transactions on Graphics 19(1). 2000.
SK_ALWAYS_INLINE static float conic_pow2(float tolerance, const SkPoint pts[], float w,
                                         const GrVectorXform& vectorXform = GrVectorXform()) {
    using grvx::dot, grvx::float2, grvx::float4, skvx::bit_pun;
    float2 p0 = vectorXform(bit_pun<float2>(pts[0]));
    float2 p1 = vectorXform(bit_pun<float2>(pts[1]));
    float2 p2 = vectorXform(bit_pun<float2>(pts[2]));

    // Compute center of bounding box in projected space
    const float2 C = 0.5f * (skvx::min(skvx::min(p0, p1), p2) + skvx::max(skvx::max(p0, p1), p2));

    // Translate by -C. This improves translation-invariance of the formula,
    // see Sec. 3.3 of cited paper
    p0 -= C;
    p1 -= C;
    p2 -= C;

    // Compute max length
    const float max_len = sqrtf(std::max(dot(p0, p0), std::max(dot(p1, p1), dot(p2, p2))));

    // Compute forward differences
    const float2 dp = grvx::fast_madd<2>(-2 * w, p1, p0) + p2;
    const float dw = fabsf(1 - 2 * w + 1);

    // Compute numerator and denominator for parametric step size of linearization
    const float r_minus_eps = std::max(0.f, max_len - tolerance);
    const float min_w = std::min(w, 1.f);
    const float numer = sqrtf(grvx::dot(dp, dp)) + r_minus_eps * dw;
    const float denom = 4 * min_w * tolerance;

    // Number of segments = sqrt(numer / denom).
    // This assumes parametric interval of curve being linearized is [t0,t1] = [0, 1].
    // If not, the number of segments is (tmax - tmin) / sqrt(denom / numer).
    return numer / denom;
}

// Returns the value of Wang's formula specialized for a conic curve.
SK_ALWAYS_INLINE static float conic(float tolerance, const SkPoint pts[], float w,
                                    const GrVectorXform& vectorXform = GrVectorXform()) {
    return sqrtf(conic_pow2(tolerance, pts, w, vectorXform));
}

// Returns the log2 value of Wang's formula specialized for a conic curve, rounded up to the next
// int.
SK_ALWAYS_INLINE static int conic_log2(float tolerance, const SkPoint pts[], float w,
                                       const GrVectorXform& vectorXform = GrVectorXform()) {
    // nextlog4(x) == ceil(log2(sqrt(x)))
    return nextlog4(conic_pow2(tolerance, pts, w, vectorXform));
}

}  // namespace GrWangsFormula

#endif