libs/graphics/sgl/SkGeometry.cpp

/* libs/graphics/sgl/SkGeometry.cpp
**
** Copyright 2006, Google Inc.
**
** Licensed under the Apache License, Version 2.0 (the "License"); 
** you may not use this file except in compliance with the License. 
** You may obtain a copy of the License at 
**
**     http://www.apache.org/licenses/LICENSE-2.0 
**
** Unless required by applicable law or agreed to in writing, software 
** distributed under the License is distributed on an "AS IS" BASIS, 
** WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 
** See the License for the specific language governing permissions and 
** limitations under the License.
*/

#include "SkGeometry.h"
#include "Sk64.h"
#include "SkMatrix.h"

/** If defined, this makes eval_quad and eval_cubic do more setup (sometimes
    involving integer multiplies by 2 or 3, but fewer calls to SkScalarMul.
    May also introduce overflow of fixed when we compute our setup.
*/
#ifdef SK_SCALAR_IS_FIXED
    #define DIRECT_EVAL_OF_POLYNOMIALS
#endif

////////////////////////////////////////////////////////////////////////

#if defined(SK_SCALAR_IS_FIXED) && !defined(SK_CPU_HAS_CONDITIONAL_INSTR)
    static int is_not_monotonic(int a, int b, int c, int d)
    {
        return (((a - b) | (b - c) | (c - d)) & ((b - a) | (c - b) | (d - c))) >> 31;
    }
    static int is_not_monotonic(int a, int b, int c)
    {
        return (((a - b) | (b - c)) & ((b - a) | (c - b))) >> 31;
    }
#else   // scalar-is-float or we have fast if/then instructions
    static int is_not_monotonic(SkScalar a, SkScalar b, SkScalar c, SkScalar d)
    {
        int neg = 0, pos = 0;

        if (a < b) neg = 1;
        if (a > b) pos = 1;
        if (b < c) neg = 1;
        if (b > c) pos = 1;
        if (c < d) neg = 1;
        if (c > d) pos = 1;

        return neg & pos;
    }
    static int is_not_monotonic(SkScalar a, SkScalar b, SkScalar c)
    {
        int neg = 0, pos = 0;

        if (a < b) neg = 1;
        if (a > b) pos = 1;
        if (b < c) neg = 1;
        if (b > c) pos = 1;

        return neg & pos;
    }
#endif

////////////////////////////////////////////////////////////////////////

static bool is_unit_interval(SkScalar x)
{
    return x > 0 && x < SK_Scalar1;
}

static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio)
{
    if (numer < 0)
    {
        numer = -numer;
        denom = -denom;
    }

    if (denom == 0 || numer == 0 || numer >= denom)
        return 0;

    if (ratio)
    {
        SkScalar r = SkScalarDiv(numer, denom);
        SkASSERT(r >= 0 && r < SK_Scalar1);
        if (r == 0) // catch underflow if numer <<<< denom
            return 0;
        *ratio = r;
    }
    return 1;
}

/** From Numerical Recipes in C.

    Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C])
    x1 = Q / A
    x2 = C / Q
*/
int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2])
{
    SkScalar* r = roots;

    if (A == 0)
        return valid_unit_divide(-C, B, roots);

#ifdef SK_SCALAR_IS_FLOAT
    float R = B*B - 4*A*C;
    if (R < 0)  // complex roots
        return 0;
    R = sk_float_sqrt(R);
#else
    Sk64    RR, tmp;

    RR.setMul(B,B);
    tmp.setMul(A,C);
    tmp.shiftLeft(2);
    RR.sub(tmp);
    if (RR.isNeg())
        return 0;
    SkFixed R = RR.getSqrt();
#endif

    SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2;
    r += valid_unit_divide(Q, A, r);
    r += valid_unit_divide(C, Q, r);
    if (r - roots == 2)
    {
        if (roots[0] > roots[1])
            SkTSwap<SkScalar>(roots[0], roots[1]);
        else if (roots[0] == roots[1])  // nearly-equal?
            r -= 1; // skip the double root
    }
    return (int)(r - roots);
}

#ifdef SK_SCALAR_IS_FIXED
/** Trim A/B/C down so that they are all <= 32bits
    and then call SkFindUnitQuadRoots()
*/
static int Sk64FindFixedQuadRoots(const Sk64& A, const Sk64& B, const Sk64& C, SkFixed roots[2])
{
    int na = A.shiftToMake32();
    int nb = B.shiftToMake32();
    int nc = C.shiftToMake32();

    int shift = SkMax32(na, SkMax32(nb, nc));
    SkASSERT(shift >= 0);

    return SkFindUnitQuadRoots(A.getShiftRight(shift), B.getShiftRight(shift), C.getShiftRight(shift), roots);
}
#endif

/////////////////////////////////////////////////////////////////////////////////////
/////////////////////////////////////////////////////////////////////////////////////

static SkScalar eval_quad(const SkScalar src[], SkScalar t)
{
    SkASSERT(src);
    SkASSERT(t >= 0 && t <= SK_Scalar1);

#ifdef DIRECT_EVAL_OF_POLYNOMIALS
    SkScalar    C = src[0];
    SkScalar    A = src[4] - 2 * src[2] + C;
    SkScalar    B = 2 * (src[2] - C);
    return SkScalarMul(SkScalarMul(A, t) + B, t) + C;
#else
    SkScalar    ab = SkScalarInterp(src[0], src[2], t);
    SkScalar    bc = SkScalarInterp(src[2], src[4], t); 
    return SkScalarInterp(ab, bc, t);
#endif
}

static SkScalar eval_quad_derivative(const SkScalar src[], SkScalar t)
{
    SkScalar A = src[4] - 2 * src[2] + src[0];
    SkScalar B = src[2] - src[0];

    return 2 * (SkScalarMul(A, t) + B);
}

static SkScalar eval_quad_derivative_at_half(const SkScalar src[])
{
    SkScalar A = src[4] - 2 * src[2] + src[0];
    SkScalar B = src[2] - src[0];
    return A + 2 * B;
}

void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent)
{
    SkASSERT(src);
    SkASSERT(t >= 0 && t <= SK_Scalar1);

    if (pt)
        pt->set(eval_quad(&src[0].fX, t), eval_quad(&src[0].fY, t));
    if (tangent)
        tangent->set(eval_quad_derivative(&src[0].fX, t),
                     eval_quad_derivative(&src[0].fY, t));
}

void SkEvalQuadAtHalf(const SkPoint src[3], SkPoint* pt, SkVector* tangent)
{
    SkASSERT(src);

    if (pt)
    {
        SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
        SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
        SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
        SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
        pt->set(SkScalarAve(x01, x12), SkScalarAve(y01, y12));
    }
    if (tangent)
        tangent->set(eval_quad_derivative_at_half(&src[0].fX),
                     eval_quad_derivative_at_half(&src[0].fY));
}

static void interp_quad_coords(const SkScalar* src, SkScalar* dst, SkScalar t)
{
    SkScalar    ab = SkScalarInterp(src[0], src[2], t);
    SkScalar    bc = SkScalarInterp(src[2], src[4], t);

    dst[0] = src[0];
    dst[2] = ab;
    dst[4] = SkScalarInterp(ab, bc, t);
    dst[6] = bc;
    dst[8] = src[4];
}

void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t)
{
    SkASSERT(t > 0 && t < SK_Scalar1);

    interp_quad_coords(&src[0].fX, &dst[0].fX, t);
    interp_quad_coords(&src[0].fY, &dst[0].fY, t);
}

void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5])
{
    SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
    SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
    SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
    SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);

    dst[0] = src[0];
    dst[1].set(x01, y01);
    dst[2].set(SkScalarAve(x01, x12), SkScalarAve(y01, y12));
    dst[3].set(x12, y12);
    dst[4] = src[2];
}

/** Quad'(t) = At + B, where
    A = 2(a - 2b + c)
    B = 2(b - a)
    Solve for t, only if it fits between 0 < t < 1
*/
int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1])
{
    /*  At + B == 0
        t = -B / A
    */
#ifdef SK_SCALAR_IS_FIXED
    return is_not_monotonic(a, b, c) && valid_unit_divide(a - b, a - b - b + c, tValue);
#else
    return valid_unit_divide(a - b, a - b - b + c, tValue);
#endif
}

static void flatten_double_quad_extrema(SkScalar coords[14])
{
    coords[2] = coords[6] = coords[4];
}

static void force_quad_monotonic_in_y(SkPoint pts[3])
{
    // zap pts[1].fY to the nearest value
    SkScalar ab = SkScalarAbs(pts[0].fY - pts[1].fY);
    SkScalar bc = SkScalarAbs(pts[1].fY - pts[2].fY);
    pts[1].fY = ab < bc ? pts[0].fY : pts[2].fY;
}

/*  Returns 0 for 1 quad, and 1 for two quads, either way the answer is
    stored in dst[]. Guarantees that the 1/2 quads will be monotonic.
*/
int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5])
{
#if 0
    static bool once = true;
    if (once)
    {
        once = false;
        SkPoint s[3] = { 0, 26398, 0, 26331, 0, 20621428 };
        SkPoint d[6];
        
        int n = SkChopQuadAtYExtrema(s, d);
        SkDebugf("chop=%d, Y=[%x %x %x %x %x %x]\n", n, d[0].fY, d[1].fY, d[2].fY, d[3].fY, d[4].fY, d[5].fY);
    }
#endif

    SkScalar    tValue;
    int         roots = SkFindQuadExtrema(src[0].fY, src[1].fY, src[2].fY, &tValue);

    if (dst)
    {
        if (roots == 0) // nothing to chop
        {
            memcpy(dst, src, 3*sizeof(SkPoint));
            // check if valid_unit_divide gave up but we're still not monotonic
            // can happen if valid_unit_divide can't see the t-value (underflow)
            // e.g. SkPoint s[3] = { 0, 26398, 0, 26331, 0, 20621428 };
            if (is_not_monotonic(src[0].fY, src[1].fY, src[2].fY))
                force_quad_monotonic_in_y(dst);
        }
        else
        {
            SkChopQuadAt(src, dst, tValue);
            flatten_double_quad_extrema(&dst[0].fY);
        }
    }
    return roots;
}

//  F(t)    = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2
//  F'(t)   = 2 (b - a) + 2 (a - 2b + c) t
//  F''(t)  = 2 (a - 2b + c)
//
//  A = 2 (b - a)
//  B = 2 (a - 2b + c)
//
//  Maximum curvature for a quadratic means solving
//  Fx' Fx'' + Fy' Fy'' = 0
//
//  t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2)
//
int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5])
{
    SkScalar    Ax = src[1].fX - src[0].fX;
    SkScalar    Ay = src[1].fY - src[0].fY;
    SkScalar    Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX;
    SkScalar    By = src[0].fY - src[1].fY - src[1].fY + src[2].fY;
    SkScalar    t = 0;  // 0 means don't chop

#ifdef SK_SCALAR_IS_FLOAT
    (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t);
#else
    // !!! should I use SkFloat here? seems like it
    Sk64    numer, denom, tmp;

    numer.setMul(Ax, -Bx);
    tmp.setMul(Ay, -By);
    numer.add(tmp);

    if (numer.isPos())  // do nothing if numer <= 0
    {
        denom.setMul(Bx, Bx);
        tmp.setMul(By, By);
        denom.add(tmp);
        SkASSERT(!denom.isNeg());
        if (numer < denom)
        {
            t = numer.getFixedDiv(denom);
            SkASSERT(t >= 0 && t <= SK_Fixed1);     // assert that we're numerically stable (ha!)
            if ((unsigned)t >= SK_Fixed1)           // runtime check for numerical stability
                t = 0;  // ignore the chop
        }
    }
#endif

    if (t == 0)
    {
        memcpy(dst, src, 3 * sizeof(SkPoint));
        return 1;
    }
    else
    {
        SkChopQuadAt(src, dst, t);
        return 2;
    }
}

////////////////////////////////////////////////////////////////////////////////////////
///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS /////
////////////////////////////////////////////////////////////////////////////////////////

static void get_cubic_coeff(const SkScalar pt[], SkScalar coeff[4])
{
    coeff[0] = pt[6] + 3*(pt[2] - pt[4]) - pt[0];
    coeff[1] = 3*(pt[4] - pt[2] - pt[2] + pt[0]);
    coeff[2] = 3*(pt[2] - pt[0]);
    coeff[3] = pt[0];
}

void SkGetCubicCoeff(const SkPoint pts[4], SkScalar cx[4], SkScalar cy[4])
{
    SkASSERT(pts);

    if (cx)
        get_cubic_coeff(&pts[0].fX, cx);
    if (cy)
        get_cubic_coeff(&pts[0].fY, cy);
}

static SkScalar eval_cubic(const SkScalar src[], SkScalar t)
{
    SkASSERT(src);
    SkASSERT(t >= 0 && t <= SK_Scalar1);

    if (t == 0)
        return src[0];

#ifdef DIRECT_EVAL_OF_POLYNOMIALS
    SkScalar D = src[0];
    SkScalar A = src[6] + 3*(src[2] - src[4]) - D;
    SkScalar B = 3*(src[4] - src[2] - src[2] + D);
    SkScalar C = 3*(src[2] - D);

    return SkScalarMul(SkScalarMul(SkScalarMul(A, t) + B, t) + C, t) + D;
#else
    SkScalar    ab = SkScalarInterp(src[0], src[2], t);
    SkScalar    bc = SkScalarInterp(src[2], src[4], t);
    SkScalar    cd = SkScalarInterp(src[4], src[6], t);
    SkScalar    abc = SkScalarInterp(ab, bc, t);
    SkScalar    bcd = SkScalarInterp(bc, cd, t);
    return SkScalarInterp(abc, bcd, t);
#endif
}

/** return At^2 + Bt + C
*/
static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t)
{
    SkASSERT(t >= 0 && t <= SK_Scalar1);

    return SkScalarMul(SkScalarMul(A, t) + B, t) + C;
}

static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t)
{
    SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
    SkScalar B = 2*(src[4] - 2 * src[2] + src[0]);
    SkScalar C = src[2] - src[0];

    return eval_quadratic(A, B, C, t);
}

static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t)
{
    SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
    SkScalar B = src[4] - 2 * src[2] + src[0];

    return SkScalarMul(A, t) + B;
}

void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc, SkVector* tangent, SkVector* curvature)
{
    SkASSERT(src);
    SkASSERT(t >= 0 && t <= SK_Scalar1);

    if (loc)
        loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t));
    if (tangent)
        tangent->set(eval_cubic_derivative(&src[0].fX, t),
                     eval_cubic_derivative(&src[0].fY, t));
    if (curvature)
        curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t),
                       eval_cubic_2ndDerivative(&src[0].fY, t));
}

/** Cubic'(t) = At^2 + Bt + C, where
    A = 3(-a + 3(b - c) + d)
    B = 6(a - 2b + c)
    C = 3(b - a)
    Solve for t, keeping only those that fit betwee 0 < t < 1
*/
int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d, SkScalar tValues[2])
{
#ifdef SK_SCALAR_IS_FIXED
    if (!is_not_monotonic(a, b, c, d))
        return 0;
#endif

    // we divide A,B,C by 3 to simplify
    SkScalar A = d - a + 3*(b - c);
    SkScalar B = 2*(a - b - b + c);
    SkScalar C = b - a;

    return SkFindUnitQuadRoots(A, B, C, tValues);
}

static void interp_cubic_coords(const SkScalar* src, SkScalar* dst, SkScalar t)
{
    SkScalar    ab = SkScalarInterp(src[0], src[2], t);
    SkScalar    bc = SkScalarInterp(src[2], src[4], t);
    SkScalar    cd = SkScalarInterp(src[4], src[6], t);
    SkScalar    abc = SkScalarInterp(ab, bc, t);
    SkScalar    bcd = SkScalarInterp(bc, cd, t);
    SkScalar    abcd = SkScalarInterp(abc, bcd, t);

    dst[0] = src[0];
    dst[2] = ab;
    dst[4] = abc;
    dst[6] = abcd;
    dst[8] = bcd;
    dst[10] = cd;
    dst[12] = src[6];
}

void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t)
{
    SkASSERT(t > 0 && t < SK_Scalar1);

    interp_cubic_coords(&src[0].fX, &dst[0].fX, t);
    interp_cubic_coords(&src[0].fY, &dst[0].fY, t);
}

void SkChopCubicAt(const SkPoint src[4], SkPoint dst[], const SkScalar tValues[], int roots)
{
#ifdef SK_DEBUG
    {
        for (int i = 0; i < roots - 1; i++)
        {
            SkASSERT(is_unit_interval(tValues[i]));
            SkASSERT(is_unit_interval(tValues[i+1]));
            SkASSERT(tValues[i] < tValues[i+1]);
        }
    }
#endif

    if (dst)
    {
        if (roots == 0) // nothing to chop
            memcpy(dst, src, 4*sizeof(SkPoint));
        else
        {
            SkScalar    t = tValues[0];
            SkPoint     tmp[4];

            for (int i = 0; i < roots; i++)
            {
                SkChopCubicAt(src, dst, t);
                if (i == roots - 1)
                    break;

                SkDEBUGCODE(int valid =) valid_unit_divide(tValues[i+1] - tValues[i], SK_Scalar1 - tValues[i], &t);
                SkASSERT(valid);

                dst += 3;
                memcpy(tmp, dst, 4 * sizeof(SkPoint));
                src = tmp;
            }
        }
    }
}

void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7])
{
    SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
    SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
    SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
    SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
    SkScalar x23 = SkScalarAve(src[2].fX, src[3].fX);
    SkScalar y23 = SkScalarAve(src[2].fY, src[3].fY);

    SkScalar x012 = SkScalarAve(x01, x12);
    SkScalar y012 = SkScalarAve(y01, y12);
    SkScalar x123 = SkScalarAve(x12, x23);
    SkScalar y123 = SkScalarAve(y12, y23);

    dst[0] = src[0];
    dst[1].set(x01, y01);
    dst[2].set(x012, y012);
    dst[3].set(SkScalarAve(x012, x123), SkScalarAve(y012, y123));
    dst[4].set(x123, y123);
    dst[5].set(x23, y23);
    dst[6] = src[3];
}

static void flatten_double_cubic_extrema(SkScalar coords[14])
{
    coords[4] = coords[8] = coords[6];
}

/** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that
    the resulting beziers are monotonic in Y. This is called by the scan converter.
    Depending on what is returned, dst[] is treated as follows
    0   dst[0..3] is the original cubic
    1   dst[0..3] and dst[3..6] are the two new cubics
    2   dst[0..3], dst[3..6], dst[6..9] are the three new cubics
    If dst == nil, it is ignored and only the count is returned.
*/
int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10])
{
    SkScalar    tValues[2];
    int         roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY, src[3].fY, tValues);

    SkChopCubicAt(src, dst, tValues, roots);
    if (dst && roots > 0)
    {
        // we do some cleanup to ensure our Y extrema are flat
        flatten_double_cubic_extrema(&dst[0].fY);
        if (roots == 2)
            flatten_double_cubic_extrema(&dst[3].fY);
    }
    return roots;
}

/** http://www.faculty.idc.ac.il/arik/quality/appendixA.html

    Inflection means that curvature is zero.
    Curvature is [F' x F''] / [F'^3]
    So we solve F'x X F''y - F'y X F''y == 0
    After some canceling of the cubic term, we get
    A = b - a
    B = c - 2b + a
    C = d - 3c + 3b - a
    (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0
*/
int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[])
{
    SkScalar    Ax = src[1].fX - src[0].fX;
    SkScalar    Ay = src[1].fY - src[0].fY;
    SkScalar    Bx = src[2].fX - 2 * src[1].fX + src[0].fX;
    SkScalar    By = src[2].fY - 2 * src[1].fY + src[0].fY;
    SkScalar    Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX;
    SkScalar    Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY;
    int         count;

#ifdef SK_SCALAR_IS_FLOAT
    count = SkFindUnitQuadRoots(Bx*Cy - By*Cx, Ax*Cy - Ay*Cx, Ax*By - Ay*Bx, tValues);
#else
    Sk64    A, B, C, tmp;

    A.setMul(Bx, Cy);
    tmp.setMul(By, Cx);
    A.sub(tmp);

    B.setMul(Ax, Cy);
    tmp.setMul(Ay, Cx);
    B.sub(tmp);

    C.setMul(Ax, By);
    tmp.setMul(Ay, Bx);
    C.sub(tmp);

    count = Sk64FindFixedQuadRoots(A, B, C, tValues);
#endif

    return count;
}

int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10])
{
    SkScalar    tValues[2];
    int         count = SkFindCubicInflections(src, tValues);

    if (dst)
    {
        if (count == 0)
            memcpy(dst, src, 4 * sizeof(SkPoint));
        else
            SkChopCubicAt(src, dst, tValues, count);
    }
    return count + 1;
}

template <typename T> void bubble_sort(T array[], int count)
{
    for (int i = count - 1; i > 0; --i)
        for (int j = i; j > 0; --j)
            if (array[j] < array[j-1])
            {
                T   tmp(array[j]);
                array[j] = array[j-1];
                array[j-1] = tmp;
            }
}

#include "SkFP.h"

// newton refinement
#if 0
static SkScalar refine_cubic_root(const SkFP coeff[4], SkScalar root)
{
    //  x1 = x0 - f(t) / f'(t)

    SkFP    T = SkScalarToFloat(root);
    SkFP    N, D;

    // f' = 3*coeff[0]*T^2 + 2*coeff[1]*T + coeff[2]
    D = SkFPMul(SkFPMul(coeff[0], SkFPMul(T,T)), 3);
    D = SkFPAdd(D, SkFPMulInt(SkFPMul(coeff[1], T), 2));
    D = SkFPAdd(D, coeff[2]);

    if (D == 0)
        return root;

    // f = coeff[0]*T^3 + coeff[1]*T^2 + coeff[2]*T + coeff[3]
    N = SkFPMul(SkFPMul(SkFPMul(T, T), T), coeff[0]);
    N = SkFPAdd(N, SkFPMul(SkFPMul(T, T), coeff[1]));
    N = SkFPAdd(N, SkFPMul(T, coeff[2]));
    N = SkFPAdd(N, coeff[3]);

    if (N)
    {
        SkScalar delta = SkFPToScalar(SkFPDiv(N, D));

        if (delta)
            root -= delta;
    }
    return root;
}
#endif

#if defined _WIN32 && _MSC_VER >= 1300  && defined SK_SCALAR_IS_FIXED // disable warning : unreachable code if building fixed point for windows desktop
#pragma warning ( disable : 4702 )
#endif

/*  Solve coeff(t) == 0, returning the number of roots that
    lie withing 0 < t < 1.
    coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3]
*/
static int solve_cubic_polynomial(const SkFP coeff[4], SkScalar tValues[3])
{
#ifndef SK_SCALAR_IS_FLOAT
    return 0;   // this is not yet implemented for software float
#endif

    if (SkScalarNearlyZero(coeff[0]))   // we're just a quadratic
    {
        return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues);
    }

    SkFP    a, b, c, Q, R;

    {
        SkASSERT(coeff[0] != 0);

        SkFP inva = SkFPInvert(coeff[0]);
        a = SkFPMul(coeff[1], inva);
        b = SkFPMul(coeff[2], inva);
        c = SkFPMul(coeff[3], inva);
    }
    Q = SkFPDivInt(SkFPSub(SkFPMul(a,a), SkFPMulInt(b, 3)), 9);
//  R = (2*a*a*a - 9*a*b + 27*c) / 54;
    R = SkFPMulInt(SkFPMul(SkFPMul(a, a), a), 2);
    R = SkFPSub(R, SkFPMulInt(SkFPMul(a, b), 9));
    R = SkFPAdd(R, SkFPMulInt(c, 27));
    R = SkFPDivInt(R, 54);

    SkFP Q3 = SkFPMul(SkFPMul(Q, Q), Q);
    SkFP R2MinusQ3 = SkFPSub(SkFPMul(R,R), Q3);
    SkFP adiv3 = SkFPDivInt(a, 3);

    SkScalar*   roots = tValues;
    SkScalar    r;

    if (SkFPLT(R2MinusQ3, 0))   // we have 3 real roots
    {
#ifdef SK_SCALAR_IS_FLOAT
        float theta = sk_float_acos(R / sk_float_sqrt(Q3));
        float neg2RootQ = -2 * sk_float_sqrt(Q);

        r = neg2RootQ * sk_float_cos(theta/3) - adiv3;
        if (is_unit_interval(r))
            *roots++ = r;

        r = neg2RootQ * sk_float_cos((theta + 2*SK_ScalarPI)/3) - adiv3;
        if (is_unit_interval(r))
            *roots++ = r;

        r = neg2RootQ * sk_float_cos((theta - 2*SK_ScalarPI)/3) - adiv3;
        if (is_unit_interval(r))
            *roots++ = r;

        // now sort the roots
        bubble_sort(tValues, (int)(roots - tValues));
#endif
    }
    else                // we have 1 real root
    {
        SkFP A = SkFPAdd(SkFPAbs(R), SkFPSqrt(R2MinusQ3));
        A = SkFPCubeRoot(A);
        if (SkFPGT(R, 0))
            A = SkFPNeg(A);

        if (A != 0)
            A = SkFPAdd(A, SkFPDiv(Q, A));
        r = SkFPToScalar(SkFPSub(A, adiv3));
        if (is_unit_interval(r))
            *roots++ = r;
    }

    return (int)(roots - tValues);
}

/*  Looking for F' dot F'' == 0
    
    A = b - a
    B = c - 2b + a
    C = d - 3c + 3b - a

    F' = 3Ct^2 + 6Bt + 3A
    F'' = 6Ct + 6B

    F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
*/
static void formulate_F1DotF2(const SkScalar src[], SkFP coeff[4])
{
    SkScalar    a = src[2] - src[0];
    SkScalar    b = src[4] - 2 * src[2] + src[0];
    SkScalar    c = src[6] + 3 * (src[2] - src[4]) - src[0];

    SkFP    A = SkScalarToFP(a);
    SkFP    B = SkScalarToFP(b);
    SkFP    C = SkScalarToFP(c);

    coeff[0] = SkFPMul(C, C);
    coeff[1] = SkFPMulInt(SkFPMul(B, C), 3);
    coeff[2] = SkFPMulInt(SkFPMul(B, B), 2);
    coeff[2] = SkFPAdd(coeff[2], SkFPMul(C, A));
    coeff[3] = SkFPMul(A, B);
}

// EXPERIMENTAL: can set this to zero to accept all t-values 0 < t < 1
//#define kMinTValueForChopping (SK_Scalar1 / 256)
#define kMinTValueForChopping   0

/*  Looking for F' dot F'' == 0
    
    A = b - a
    B = c - 2b + a
    C = d - 3c + 3b - a

    F' = 3Ct^2 + 6Bt + 3A
    F'' = 6Ct + 6B

    F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
*/
int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3])
{
    SkFP    coeffX[4], coeffY[4];
    int     i;

    formulate_F1DotF2(&src[0].fX, coeffX);
    formulate_F1DotF2(&src[0].fY, coeffY);

    for (i = 0; i < 4; i++)
        coeffX[i] = SkFPAdd(coeffX[i],coeffY[i]);

    SkScalar    t[3];
    int         count = solve_cubic_polynomial(coeffX, t);
    int         maxCount = 0;

    // now remove extrema where the curvature is zero (mins)
    // !!!! need a test for this !!!!
    for (i = 0; i < count; i++)
    {
        // if (not_min_curvature())
        if (t[i] > kMinTValueForChopping && t[i] < SK_Scalar1 - kMinTValueForChopping)
            tValues[maxCount++] = t[i];
    }
    return maxCount;
}

int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13], SkScalar tValues[3])
{
    SkScalar    t_storage[3];

    if (tValues == nil)
        tValues = t_storage;

    int count = SkFindCubicMaxCurvature(src, tValues);

    if (dst)
    {
        if (count == 0)
            memcpy(dst, src, 4 * sizeof(SkPoint));
        else
            SkChopCubicAt(src, dst, tValues, count);
    }
    return count + 1;
}

////////////////////////////////////////////////////////////////////////////////

/*  Find t value for quadratic [a, b, c] = d.
    Return 0 if there is no solution
*/
static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d)
{
    // At^2 + Bt + C = d
    SkScalar A = a - 2 * b + c;
    SkScalar B = 2 * (b - a);
    SkScalar C = a - d;

    SkScalar    roots[2];
    int         count = SkFindUnitQuadRoots(A, B, C, roots);

    SkASSERT(count <= 1);
    return count == 1 ? roots[0] : 0;
}

/* given a quad-curve and a point (x,y), chop the quad at that point and return
    the new quad's offCurve point.
*/
static bool quad_pt2OffCurve(const SkPoint quad[3], SkScalar x, SkScalar y, SkPoint* offCurve)
{
    SkScalar t;
    SkPoint tmp[5];

    if (SkScalarAbs(x) < SkScalarAbs(y))
        t = quad_solve(quad[0].fX, quad[1].fX, quad[2].fX, x);
    else
        t = quad_solve(quad[0].fY, quad[1].fY, quad[2].fY, y);

    if (t > 0)
    {
        SkChopQuadAt(quad, tmp, t);
        *offCurve = tmp[1];
        return true;
    }
    return false;
}

static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = {
    { SK_Scalar1,           0               },
    { SK_Scalar1,           SK_ScalarTanPIOver8 },
    { SK_ScalarRoot2Over2,  SK_ScalarRoot2Over2 },
    { SK_ScalarTanPIOver8,  SK_Scalar1          },

    { 0,                    SK_Scalar1      },
    { -SK_ScalarTanPIOver8, SK_Scalar1  },
    { -SK_ScalarRoot2Over2, SK_ScalarRoot2Over2 },
    { -SK_Scalar1,          SK_ScalarTanPIOver8 },

    { -SK_Scalar1,          0               },
    { -SK_Scalar1,          -SK_ScalarTanPIOver8    },
    { -SK_ScalarRoot2Over2, -SK_ScalarRoot2Over2    },
    { -SK_ScalarTanPIOver8, -SK_Scalar1     },

    { 0,                    -SK_Scalar1     },
    { SK_ScalarTanPIOver8,  -SK_Scalar1     },
    { SK_ScalarRoot2Over2,  -SK_ScalarRoot2Over2    },
    { SK_Scalar1,           -SK_ScalarTanPIOver8    },

    { SK_Scalar1,           0   }
};

int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop,
                   SkRotationDirection dir, const SkMatrix* userMatrix,
                   SkPoint quadPoints[])
{
    // check for (effectively) coincident vectors
    {
        SkScalar dot = SkScalarMul(uStart.fX, uStop.fX) + SkScalarMul(uStart.fY, uStop.fY);
        if (SkScalarAbs(dot - SK_Scalar1) <= SK_ScalarNearlyZero)
            return 0;
    }
    // rotate unitStop so that unitStart is at (1,0)
    SkScalar x = SkScalarMul(uStop.fX, uStart.fX) + SkScalarMul(uStop.fY, uStart.fY);
    SkScalar y = SkScalarMul(uStop.fY, uStart.fX) - SkScalarMul(uStop.fX, uStart.fY);

    if (dir == kCCW_SkRotationDirection)
        y = -y;

    // what octant (quadratic curve) is [xy] in?
    int oct = 0;
    bool sameSign = true;

    if (y < 0)
        oct += 4;
    if ((x < 0) != (y < 0))
    {
        oct += 2;
        sameSign = false;
    }
    if ((SkScalarAbs(x) < SkScalarAbs(y)) == sameSign)
        oct += 1;

    if (SkScalarAbs(y) >= SK_Scalar1 || x <= -SK_Scalar1)
        oct += 1;

    int wholeCount = oct << 1;
    memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint));

    const SkPoint* arc = &gQuadCirclePts[wholeCount];
    if (quad_pt2OffCurve(arc, x, y, &quadPoints[wholeCount + 1]))
    {
        quadPoints[wholeCount + 2].set(x, y);
        wholeCount += 2;
    }
    wholeCount += 1;

    // now handle counter-clockwise and the initial unitStart rotation
    SkMatrix    matrix;
    matrix.setSinCos(uStart.fY, uStart.fX);
    if (dir == kCCW_SkRotationDirection)
        matrix.preScale(SK_Scalar1, -SK_Scalar1);
    if (userMatrix)
        matrix.postConcat(*userMatrix);
    matrix.mapPoints(quadPoints, wholeCount);
    return wholeCount;
}


/////////////////////////////////////////////////////////////////////////////////////////
/////////////////////////////////////////////////////////////////////////////////////////

#ifdef SK_DEBUG

void SkGeometry::UnitTest()
{
#ifdef SK_SUPPORT_UNITTEST
    SkPoint pts[3], dst[5];

    pts[0].set(0, 0);
    pts[1].set(100, 50);
    pts[2].set(0, 100);

    int count = SkChopQuadAtMaxCurvature(pts, dst);
    SkASSERT(count == 1 || count == 2);
#endif
}

#endif