core/SkPoint.cpp - platform/external/chromium_org/third_party/skia/src - Git at Google


 /*
  * Copyright 2008 The Android Open Source Project
  *
  * Use of this source code is governed by a BSD-style license that can be
  * found in the LICENSE file.
  */


 #include "SkPoint.h"

 void SkIPoint::rotateCW(SkIPoint* dst) const {
     SkASSERT(dst);

     // use a tmp in case this == dst
     int32_t tmp = fX;
     dst->fX = -fY;
     dst->fY = tmp;
 }

 void SkIPoint::rotateCCW(SkIPoint* dst) const {
     SkASSERT(dst);

     // use a tmp in case this == dst
     int32_t tmp = fX;
     dst->fX = fY;
     dst->fY = -tmp;
 }

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

 void SkPoint::setIRectFan(int l, int t, int r, int b, size_t stride) {
     SkASSERT(stride >= sizeof(SkPoint));

     ((SkPoint*)((intptr_t)this + 0 * stride))->set(SkIntToScalar(l),
                                                    SkIntToScalar(t));
     ((SkPoint*)((intptr_t)this + 1 * stride))->set(SkIntToScalar(l),
                                                    SkIntToScalar(b));
     ((SkPoint*)((intptr_t)this + 2 * stride))->set(SkIntToScalar(r),
                                                    SkIntToScalar(b));
     ((SkPoint*)((intptr_t)this + 3 * stride))->set(SkIntToScalar(r),
                                                    SkIntToScalar(t));
 }

 void SkPoint::setRectFan(SkScalar l, SkScalar t, SkScalar r, SkScalar b,
                          size_t stride) {
     SkASSERT(stride >= sizeof(SkPoint));

     ((SkPoint*)((intptr_t)this + 0 * stride))->set(l, t);
     ((SkPoint*)((intptr_t)this + 1 * stride))->set(l, b);
     ((SkPoint*)((intptr_t)this + 2 * stride))->set(r, b);
     ((SkPoint*)((intptr_t)this + 3 * stride))->set(r, t);
 }

 void SkPoint::rotateCW(SkPoint* dst) const {
     SkASSERT(dst);

     // use a tmp in case this == dst
     SkScalar tmp = fX;
     dst->fX = -fY;
     dst->fY = tmp;
 }

 void SkPoint::rotateCCW(SkPoint* dst) const {
     SkASSERT(dst);

     // use a tmp in case this == dst
     SkScalar tmp = fX;
     dst->fX = fY;
     dst->fY = -tmp;
 }

 void SkPoint::scale(SkScalar scale, SkPoint* dst) const {
     SkASSERT(dst);
     dst->set(SkScalarMul(fX, scale), SkScalarMul(fY, scale));
 }

 bool SkPoint::normalize() {
     return this->setLength(fX, fY, SK_Scalar1);
 }

 bool SkPoint::setNormalize(SkScalar x, SkScalar y) {
     return this->setLength(x, y, SK_Scalar1);
 }

 bool SkPoint::setLength(SkScalar length) {
     return this->setLength(fX, fY, length);
 }

 #ifdef SK_SCALAR_IS_FLOAT

 // Returns the square of the Euclidian distance to (dx,dy).
 static inline float getLengthSquared(float dx, float dy) {
     return dx * dx + dy * dy;
 }

 // Calculates the square of the Euclidian distance to (dx,dy) and stores it in
 // *lengthSquared.  Returns true if the distance is judged to be "nearly zero".
 //
 // This logic is encapsulated in a helper method to make it explicit that we
 // always perform this check in the same manner, to avoid inconsistencies
 // (see http://code.google.com/p/skia/issues/detail?id=560 ).
 static inline bool isLengthNearlyZero(float dx, float dy,
                                       float *lengthSquared) {
     *lengthSquared = getLengthSquared(dx, dy);
     return *lengthSquared <= (SK_ScalarNearlyZero * SK_ScalarNearlyZero);
 }

 SkScalar SkPoint::Normalize(SkPoint* pt) {
     float x = pt->fX;
     float y = pt->fY;
     float mag2;
     if (isLengthNearlyZero(x, y, &mag2)) {
         return 0;
     }

     float mag, scale;
     if (SkScalarIsFinite(mag2)) {
         mag = sk_float_sqrt(mag2);
         scale = 1 / mag;
     } else {
         // our mag2 step overflowed to infinity, so use doubles instead.
         // much slower, but needed when x or y are very large, other wise we
         // divide by inf. and return (0,0) vector.
         double xx = x;
         double yy = y;
         double magmag = sqrt(xx * xx + yy * yy);
         mag = (float)magmag;
         // we perform the divide with the double magmag, to stay exactly the
         // same as setLength. It would be faster to perform the divide with
         // mag, but it is possible that mag has overflowed to inf. but still
         // have a non-zero value for scale (thanks to denormalized numbers).
         scale = (float)(1 / magmag);
     }
     pt->set(x * scale, y * scale);
     return mag;
 }

 SkScalar SkPoint::Length(SkScalar dx, SkScalar dy) {
     float mag2 = dx * dx + dy * dy;
     if (SkScalarIsFinite(mag2)) {
         return sk_float_sqrt(mag2);
     } else {
         double xx = dx;
         double yy = dy;
         return (float)sqrt(xx * xx + yy * yy);
     }
 }

 /*
  *  We have to worry about 2 tricky conditions:
  *  1. underflow of mag2 (compared against nearlyzero^2)
  *  2. overflow of mag2 (compared w/ isfinite)
  *
  *  If we underflow, we return false. If we overflow, we compute again using
  *  doubles, which is much slower (3x in a desktop test) but will not overflow.
  */
 bool SkPoint::setLength(float x, float y, float length) {
     float mag2;
     if (isLengthNearlyZero(x, y, &mag2)) {
         return false;
     }

     float scale;
     if (SkScalarIsFinite(mag2)) {
         scale = length / sk_float_sqrt(mag2);
     } else {
         // our mag2 step overflowed to infinity, so use doubles instead.
         // much slower, but needed when x or y are very large, other wise we
         // divide by inf. and return (0,0) vector.
         double xx = x;
         double yy = y;
         scale = (float)(length / sqrt(xx * xx + yy * yy));
     }
     fX = x * scale;
     fY = y * scale;
     return true;
 }

 #else

 #include "Sk64.h"

 // Returns the square of the Euclidian distance to (dx,dy) in *result.
 static inline void getLengthSquared(SkScalar dx, SkScalar dy, Sk64 *result) {
     Sk64    dySqr;

     result->setMul(dx, dx);
     dySqr.setMul(dy, dy);
     result->add(dySqr);
 }

 // Calculates the square of the Euclidian distance to (dx,dy) and stores it in
 // *lengthSquared.  Returns true if the distance is judged to be "nearly zero".
 //
 // This logic is encapsulated in a helper method to make it explicit that we
 // always perform this check in the same manner, to avoid inconsistencies
 // (see http://code.google.com/p/skia/issues/detail?id=560 ).
 static inline bool isLengthNearlyZero(SkScalar dx, SkScalar dy,
                                       Sk64 *lengthSquared) {
     Sk64 tolSqr;
     getLengthSquared(dx, dy, lengthSquared);

     // we want nearlyzero^2, but to compute it fast we want to just do a
     // 32bit multiply, so we require that it not exceed 31bits. That is true
     // if nearlyzero is <= 0xB504, which should be trivial, since usually
     // nearlyzero is a very small fixed-point value.
     SkASSERT(SK_ScalarNearlyZero <= 0xB504);

     tolSqr.set(0, SK_ScalarNearlyZero * SK_ScalarNearlyZero);
     return *lengthSquared <= tolSqr;
 }

 SkScalar SkPoint::Normalize(SkPoint* pt) {
     Sk64 mag2;
     if (!isLengthNearlyZero(pt->fX, pt->fY, &mag2)) {
         SkScalar mag = mag2.getSqrt();
         SkScalar scale = SkScalarInvert(mag);
         pt->fX = SkScalarMul(pt->fX, scale);
         pt->fY = SkScalarMul(pt->fY, scale);
         return mag;
     }
     return 0;
 }

 bool SkPoint::CanNormalize(SkScalar dx, SkScalar dy) {
     Sk64 mag2_unused;
     return !isLengthNearlyZero(dx, dy, &mag2_unused);
 }

 SkScalar SkPoint::Length(SkScalar dx, SkScalar dy) {
     Sk64    tmp;
     getLengthSquared(dx, dy, &tmp);
     return tmp.getSqrt();
 }

 #ifdef SK_DEBUGx
 static SkFixed fixlen(SkFixed x, SkFixed y) {
     float fx = (float)x;
     float fy = (float)y;

     return (int)floorf(sqrtf(fx*fx + fy*fy) + 0.5f);
 }
 #endif

 static inline uint32_t squarefixed(unsigned x) {
     x >>= 16;
     return x*x;
 }

 #if 1   // Newton iter for setLength

 static inline unsigned invsqrt_iter(unsigned V, unsigned U) {
     unsigned x = V * U >> 14;
     x = x * U >> 14;
     x = (3 << 14) - x;
     x = (U >> 1) * x >> 14;
     return x;
 }

 static const uint16_t gInvSqrt14GuessTable[] = {
     0x4000, 0x3c57, 0x393e, 0x3695, 0x3441, 0x3235, 0x3061,
     0x2ebd, 0x2d41, 0x2be7, 0x2aaa, 0x2987, 0x287a, 0x2780,
     0x2698, 0x25be, 0x24f3, 0x2434, 0x2380, 0x22d6, 0x2235,
     0x219d, 0x210c, 0x2083, 0x2000, 0x1f82, 0x1f0b, 0x1e99,
     0x1e2b, 0x1dc2, 0x1d5d, 0x1cfc, 0x1c9f, 0x1c45, 0x1bee,
     0x1b9b, 0x1b4a, 0x1afc, 0x1ab0, 0x1a67, 0x1a20, 0x19dc,
     0x1999, 0x1959, 0x191a, 0x18dd, 0x18a2, 0x1868, 0x1830,
     0x17fa, 0x17c4, 0x1791, 0x175e, 0x172d, 0x16fd, 0x16ce
 };

 #define BUILD_INVSQRT_TABLEx
 #ifdef BUILD_INVSQRT_TABLE
 static void build_invsqrt14_guess_table() {
     for (int i = 8; i <= 63; i++) {
         unsigned x = SkToU16((1 << 28) / SkSqrt32(i << 25));
         printf("0x%x, ", x);
     }
     printf("\n");
 }
 #endif

 static unsigned fast_invsqrt(uint32_t x) {
 #ifdef BUILD_INVSQRT_TABLE
     unsigned top2 = x >> 25;
     SkASSERT(top2 >= 8 && top2 <= 63);

     static bool gOnce;
     if (!gOnce) {
         build_invsqrt14_guess_table();
         gOnce = true;
     }
 #endif

     unsigned V = x >> 14;   // make V .14

     unsigned top = x >> 25;
     SkASSERT(top >= 8 && top <= 63);
     SkASSERT(top - 8 < SK_ARRAY_COUNT(gInvSqrt14GuessTable));
     unsigned U = gInvSqrt14GuessTable[top - 8];

     U = invsqrt_iter(V, U);
     return invsqrt_iter(V, U);
 }

 /*  We "normalize" x,y to be .14 values (so we can square them and stay 32bits.
     Then we Newton-iterate this in .14 space to compute the invser-sqrt, and
     scale by it at the end. The .14 space means we can execute our iterations
     and stay in 32bits as well, making the multiplies much cheaper than calling
     SkFixedMul.
 */
 bool SkPoint::setLength(SkFixed ox, SkFixed oy, SkFixed length) {
     if (ox == 0) {
         if (oy == 0) {
             return false;
         }
         this->set(0, SkApplySign(length, SkExtractSign(oy)));
         return true;
     }
     if (oy == 0) {
         this->set(SkApplySign(length, SkExtractSign(ox)), 0);
         return true;
     }

     unsigned x = SkAbs32(ox);
     unsigned y = SkAbs32(oy);
     int zeros = SkCLZ(x | y);

     // make x,y 1.14 values so our fast sqr won't overflow
     if (zeros > 17) {
         x <<= zeros - 17;
         y <<= zeros - 17;
     } else {
         x >>= 17 - zeros;
         y >>= 17 - zeros;
     }
     SkASSERT((x | y) <= 0x7FFF);

     unsigned invrt = fast_invsqrt(x*x + y*y);

     x = x * invrt >> 12;
     y = y * invrt >> 12;

     if (length != SK_Fixed1) {
         x = SkFixedMul(x, length);
         y = SkFixedMul(y, length);
     }
     this->set(SkApplySign(x, SkExtractSign(ox)),
               SkApplySign(y, SkExtractSign(oy)));
     return true;
 }
 #else
 /*
     Normalize x,y, and then scale them by length.

     The obvious way to do this would be the following:
         S64 tmp1, tmp2;
         tmp1.setMul(x,x);
         tmp2.setMul(y,y);
         tmp1.add(tmp2);
         len = tmp1.getSqrt();
         x' = SkFixedDiv(x, len);
         y' = SkFixedDiv(y, len);
     This is fine, but slower than what we do below.

     The present technique does not compute the starting length, but
     rather fiddles with x,y iteratively, all the while checking its
     magnitude^2 (avoiding a sqrt).

     We normalize by first shifting x,y so that at least one of them
     has bit 31 set (after taking the abs of them).
     Then we loop, refining x,y by squaring them and comparing
     against a very large 1.0 (1 << 28), and then adding or subtracting
     a delta (which itself is reduced by half each time through the loop).
     For speed we want the squaring to be with a simple integer mul. To keep
     that from overflowing we shift our coordinates down until we are dealing
     with at most 15 bits (2^15-1)^2 * 2 says withing 32 bits)
     When our square is close to 1.0, we shift x,y down into fixed range.
 */
 bool SkPoint::setLength(SkFixed ox, SkFixed oy, SkFixed length) {
     if (ox == 0) {
         if (oy == 0)
             return false;
         this->set(0, SkApplySign(length, SkExtractSign(oy)));
         return true;
     }
     if (oy == 0) {
         this->set(SkApplySign(length, SkExtractSign(ox)), 0);
         return true;
     }

     SkFixed x = SkAbs32(ox);
     SkFixed y = SkAbs32(oy);

     // shift x,y so that the greater of them is 15bits (1.14 fixed point)
     {
         int shift = SkCLZ(x | y);
         // make them .30
         x <<= shift - 1;
         y <<= shift - 1;
     }

     SkFixed dx = x;
     SkFixed dy = y;

     for (int i = 0; i < 17; i++) {
         dx >>= 1;
         dy >>= 1;

         U32 len2 = squarefixed(x) + squarefixed(y);
         if (len2 >> 28) {
             x -= dx;
             y -= dy;
         } else {
             x += dx;
             y += dy;
         }
     }
     x >>= 14;
     y >>= 14;

 #ifdef SK_DEBUGx    // measure how far we are from unit-length
     {
         static int gMaxError;
         static int gMaxDiff;

         SkFixed len = fixlen(x, y);
         int err = len - SK_Fixed1;
         err = SkAbs32(err);

         if (err > gMaxError) {
             gMaxError = err;
             SkDebugf("gMaxError %d\n", err);
         }

         float fx = SkAbs32(ox)/65536.0f;
         float fy = SkAbs32(oy)/65536.0f;
         float mag = sqrtf(fx*fx + fy*fy);
         fx /= mag;
         fy /= mag;
         SkFixed xx = (int)floorf(fx * 65536 + 0.5f);
         SkFixed yy = (int)floorf(fy * 65536 + 0.5f);
         err = SkMax32(SkAbs32(xx-x), SkAbs32(yy-y));
         if (err > gMaxDiff) {
             gMaxDiff = err;
             SkDebugf("gMaxDiff %d\n", err);
         }
     }
 #endif

     x = SkApplySign(x, SkExtractSign(ox));
     y = SkApplySign(y, SkExtractSign(oy));
     if (length != SK_Fixed1) {
         x = SkFixedMul(x, length);
         y = SkFixedMul(y, length);
     }

     this->set(x, y);
     return true;
 }
 #endif

 #endif

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

 SkScalar SkPoint::distanceToLineBetweenSqd(const SkPoint& a,
                                            const SkPoint& b,
                                            Side* side) const {

     SkVector u = b - a;
     SkVector v = *this - a;

     SkScalar uLengthSqd = u.lengthSqd();
     SkScalar det = u.cross(v);
     if (NULL != side) {
         SkASSERT(-1 == SkPoint::kLeft_Side &&
                   0 == SkPoint::kOn_Side &&
                   1 == kRight_Side);
         *side = (Side) SkScalarSignAsInt(det);
     }
     return SkScalarMulDiv(det, det, uLengthSqd);
 }

 SkScalar SkPoint::distanceToLineSegmentBetweenSqd(const SkPoint& a,
                                                   const SkPoint& b) const {
     // See comments to distanceToLineBetweenSqd. If the projection of c onto
     // u is between a and b then this returns the same result as that
     // function. Otherwise, it returns the distance to the closer of a and
     // b. Let the projection of v onto u be v'.  There are three cases:
     //    1. v' points opposite to u. c is not between a and b and is closer
     //       to a than b.
     //    2. v' points along u and has magnitude less than y. c is between
     //       a and b and the distance to the segment is the same as distance
     //       to the line ab.
     //    3. v' points along u and has greater magnitude than u. c is not
     //       not between a and b and is closer to b than a.
     // v' = (u dot v) * u / |u|. So if (u dot v)/|u| is less than zero we're
     // in case 1. If (u dot v)/|u| is > |u| we are in case 3. Otherwise
     // we're in case 2. We actually compare (u dot v) to 0 and |u|^2 to
     // avoid a sqrt to compute |u|.

     SkVector u = b - a;
     SkVector v = *this - a;

     SkScalar uLengthSqd = u.lengthSqd();
     SkScalar uDotV = SkPoint::DotProduct(u, v);

     if (uDotV <= 0) {
         return v.lengthSqd();
     } else if (uDotV > uLengthSqd) {
         return b.distanceToSqd(*this);
     } else {
         SkScalar det = u.cross(v);
         return SkScalarMulDiv(det, det, uLengthSqd);
     }
 }

	/*
	* Copyright 2008 The Android Open Source Project
	*
	* Use of this source code is governed by a BSD-style license that can be
	* found in the LICENSE file.
	*/


	#include "SkPoint.h"

	void SkIPoint::rotateCW(SkIPoint* dst) const {
	SkASSERT(dst);

	// use a tmp in case this == dst
	int32_t tmp = fX;
	dst->fX = -fY;
	dst->fY = tmp;
	}

	void SkIPoint::rotateCCW(SkIPoint* dst) const {
	SkASSERT(dst);

	// use a tmp in case this == dst
	int32_t tmp = fX;
	dst->fX = fY;
	dst->fY = -tmp;
	}

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

	void SkPoint::setIRectFan(int l, int t, int r, int b, size_t stride) {
	SkASSERT(stride >= sizeof(SkPoint));

	((SkPoint)((intptr_t)this + 0 stride))->set(SkIntToScalar(l),
	SkIntToScalar(t));
	((SkPoint)((intptr_t)this + 1 stride))->set(SkIntToScalar(l),
	SkIntToScalar(b));
	((SkPoint)((intptr_t)this + 2 stride))->set(SkIntToScalar(r),
	SkIntToScalar(b));
	((SkPoint)((intptr_t)this + 3 stride))->set(SkIntToScalar(r),
	SkIntToScalar(t));
	}

	void SkPoint::setRectFan(SkScalar l, SkScalar t, SkScalar r, SkScalar b,
	size_t stride) {
	SkASSERT(stride >= sizeof(SkPoint));

	((SkPoint)((intptr_t)this + 0 stride))->set(l, t);
	((SkPoint)((intptr_t)this + 1 stride))->set(l, b);
	((SkPoint)((intptr_t)this + 2 stride))->set(r, b);
	((SkPoint)((intptr_t)this + 3 stride))->set(r, t);
	}

	void SkPoint::rotateCW(SkPoint* dst) const {
	SkASSERT(dst);

	// use a tmp in case this == dst
	SkScalar tmp = fX;
	dst->fX = -fY;
	dst->fY = tmp;
	}

	void SkPoint::rotateCCW(SkPoint* dst) const {
	SkASSERT(dst);

	// use a tmp in case this == dst
	SkScalar tmp = fX;
	dst->fX = fY;
	dst->fY = -tmp;
	}

	void SkPoint::scale(SkScalar scale, SkPoint* dst) const {
	SkASSERT(dst);
	dst->set(SkScalarMul(fX, scale), SkScalarMul(fY, scale));
	}

	bool SkPoint::normalize() {
	return this->setLength(fX, fY, SK_Scalar1);
	}

	bool SkPoint::setNormalize(SkScalar x, SkScalar y) {
	return this->setLength(x, y, SK_Scalar1);
	}

	bool SkPoint::setLength(SkScalar length) {
	return this->setLength(fX, fY, length);
	}

	#ifdef SK_SCALAR_IS_FLOAT

	// Returns the square of the Euclidian distance to (dx,dy).
	static inline float getLengthSquared(float dx, float dy) {
	return dx * dx + dy * dy;
	}

	// Calculates the square of the Euclidian distance to (dx,dy) and stores it in
	// *lengthSquared. Returns true if the distance is judged to be "nearly zero".
	//
	// This logic is encapsulated in a helper method to make it explicit that we
	// always perform this check in the same manner, to avoid inconsistencies
	// (see http://code.google.com/p/skia/issues/detail?id=560 ).
	static inline bool isLengthNearlyZero(float dx, float dy,
	float *lengthSquared) {
	*lengthSquared = getLengthSquared(dx, dy);
	return lengthSquared <= (SK_ScalarNearlyZero SK_ScalarNearlyZero);
	}

	SkScalar SkPoint::Normalize(SkPoint* pt) {
	float x = pt->fX;
	float y = pt->fY;
	float mag2;
	if (isLengthNearlyZero(x, y, &mag2)) {
	return 0;
	}

	float mag, scale;
	if (SkScalarIsFinite(mag2)) {
	mag = sk_float_sqrt(mag2);
	scale = 1 / mag;
	} else {
	// our mag2 step overflowed to infinity, so use doubles instead.
	// much slower, but needed when x or y are very large, other wise we
	// divide by inf. and return (0,0) vector.
	double xx = x;
	double yy = y;
	double magmag = sqrt(xx * xx + yy * yy);
	mag = (float)magmag;
	// we perform the divide with the double magmag, to stay exactly the
	// same as setLength. It would be faster to perform the divide with
	// mag, but it is possible that mag has overflowed to inf. but still
	// have a non-zero value for scale (thanks to denormalized numbers).
	scale = (float)(1 / magmag);
	}
	pt->set(x * scale, y * scale);
	return mag;
	}

	SkScalar SkPoint::Length(SkScalar dx, SkScalar dy) {
	float mag2 = dx * dx + dy * dy;
	if (SkScalarIsFinite(mag2)) {
	return sk_float_sqrt(mag2);
	} else {
	double xx = dx;
	double yy = dy;
	return (float)sqrt(xx * xx + yy * yy);
	}
	}

	/*
	* We have to worry about 2 tricky conditions:
	* 1. underflow of mag2 (compared against nearlyzero^2)
	* 2. overflow of mag2 (compared w/ isfinite)
	*
	* If we underflow, we return false. If we overflow, we compute again using
	* doubles, which is much slower (3x in a desktop test) but will not overflow.
	*/
	bool SkPoint::setLength(float x, float y, float length) {
	float mag2;
	if (isLengthNearlyZero(x, y, &mag2)) {
	return false;
	}

	float scale;
	if (SkScalarIsFinite(mag2)) {
	scale = length / sk_float_sqrt(mag2);
	} else {
	// our mag2 step overflowed to infinity, so use doubles instead.
	// much slower, but needed when x or y are very large, other wise we
	// divide by inf. and return (0,0) vector.
	double xx = x;
	double yy = y;
	scale = (float)(length / sqrt(xx * xx + yy * yy));
	}
	fX = x * scale;
	fY = y * scale;
	return true;
	}

	#else

	#include "Sk64.h"

	// Returns the square of the Euclidian distance to (dx,dy) in *result.
	static inline void getLengthSquared(SkScalar dx, SkScalar dy, Sk64 *result) {
	Sk64 dySqr;

	result->setMul(dx, dx);
	dySqr.setMul(dy, dy);
	result->add(dySqr);
	}

	// Calculates the square of the Euclidian distance to (dx,dy) and stores it in
	// *lengthSquared. Returns true if the distance is judged to be "nearly zero".
	//
	// This logic is encapsulated in a helper method to make it explicit that we
	// always perform this check in the same manner, to avoid inconsistencies
	// (see http://code.google.com/p/skia/issues/detail?id=560 ).
	static inline bool isLengthNearlyZero(SkScalar dx, SkScalar dy,
	Sk64 *lengthSquared) {
	Sk64 tolSqr;
	getLengthSquared(dx, dy, lengthSquared);

	// we want nearlyzero^2, but to compute it fast we want to just do a
	// 32bit multiply, so we require that it not exceed 31bits. That is true
	// if nearlyzero is <= 0xB504, which should be trivial, since usually
	// nearlyzero is a very small fixed-point value.
	SkASSERT(SK_ScalarNearlyZero <= 0xB504);

	tolSqr.set(0, SK_ScalarNearlyZero * SK_ScalarNearlyZero);
	return *lengthSquared <= tolSqr;
	}

	SkScalar SkPoint::Normalize(SkPoint* pt) {
	Sk64 mag2;
	if (!isLengthNearlyZero(pt->fX, pt->fY, &mag2)) {
	SkScalar mag = mag2.getSqrt();
	SkScalar scale = SkScalarInvert(mag);
	pt->fX = SkScalarMul(pt->fX, scale);
	pt->fY = SkScalarMul(pt->fY, scale);
	return mag;
	}
	return 0;
	}

	bool SkPoint::CanNormalize(SkScalar dx, SkScalar dy) {
	Sk64 mag2_unused;
	return !isLengthNearlyZero(dx, dy, &mag2_unused);
	}

	SkScalar SkPoint::Length(SkScalar dx, SkScalar dy) {
	Sk64 tmp;
	getLengthSquared(dx, dy, &tmp);
	return tmp.getSqrt();
	}

	#ifdef SK_DEBUGx
	static SkFixed fixlen(SkFixed x, SkFixed y) {
	float fx = (float)x;
	float fy = (float)y;

	return (int)floorf(sqrtf(fxfx + fyfy) + 0.5f);
	}
	#endif

	static inline uint32_t squarefixed(unsigned x) {
	x >>= 16;
	return x*x;
	}

	#if 1 // Newton iter for setLength

	static inline unsigned invsqrt_iter(unsigned V, unsigned U) {
	unsigned x = V * U >> 14;
	x = x * U >> 14;
	x = (3 << 14) - x;
	x = (U >> 1) * x >> 14;
	return x;
	}

	static const uint16_t gInvSqrt14GuessTable[] = {
	0x4000, 0x3c57, 0x393e, 0x3695, 0x3441, 0x3235, 0x3061,
	0x2ebd, 0x2d41, 0x2be7, 0x2aaa, 0x2987, 0x287a, 0x2780,
	0x2698, 0x25be, 0x24f3, 0x2434, 0x2380, 0x22d6, 0x2235,
	0x219d, 0x210c, 0x2083, 0x2000, 0x1f82, 0x1f0b, 0x1e99,
	0x1e2b, 0x1dc2, 0x1d5d, 0x1cfc, 0x1c9f, 0x1c45, 0x1bee,
	0x1b9b, 0x1b4a, 0x1afc, 0x1ab0, 0x1a67, 0x1a20, 0x19dc,
	0x1999, 0x1959, 0x191a, 0x18dd, 0x18a2, 0x1868, 0x1830,
	0x17fa, 0x17c4, 0x1791, 0x175e, 0x172d, 0x16fd, 0x16ce
	};

	#define BUILD_INVSQRT_TABLEx
	#ifdef BUILD_INVSQRT_TABLE
	static void build_invsqrt14_guess_table() {
	for (int i = 8; i <= 63; i++) {
	unsigned x = SkToU16((1 << 28) / SkSqrt32(i << 25));
	printf("0x%x, ", x);
	}
	printf("\n");
	}
	#endif

	static unsigned fast_invsqrt(uint32_t x) {
	#ifdef BUILD_INVSQRT_TABLE
	unsigned top2 = x >> 25;
	SkASSERT(top2 >= 8 && top2 <= 63);

	static bool gOnce;
	if (!gOnce) {
	build_invsqrt14_guess_table();
	gOnce = true;
	}
	#endif

	unsigned V = x >> 14; // make V .14

	unsigned top = x >> 25;
	SkASSERT(top >= 8 && top <= 63);
	SkASSERT(top - 8 < SK_ARRAY_COUNT(gInvSqrt14GuessTable));
	unsigned U = gInvSqrt14GuessTable[top - 8];

	U = invsqrt_iter(V, U);
	return invsqrt_iter(V, U);
	}

	/* We "normalize" x,y to be .14 values (so we can square them and stay 32bits.
	Then we Newton-iterate this in .14 space to compute the invser-sqrt, and
	scale by it at the end. The .14 space means we can execute our iterations
	and stay in 32bits as well, making the multiplies much cheaper than calling
	SkFixedMul.
	*/
	bool SkPoint::setLength(SkFixed ox, SkFixed oy, SkFixed length) {
	if (ox == 0) {
	if (oy == 0) {
	return false;
	}
	this->set(0, SkApplySign(length, SkExtractSign(oy)));
	return true;
	}
	if (oy == 0) {
	this->set(SkApplySign(length, SkExtractSign(ox)), 0);
	return true;
	}

	unsigned x = SkAbs32(ox);
	unsigned y = SkAbs32(oy);
	int zeros = SkCLZ(x \| y);

	// make x,y 1.14 values so our fast sqr won't overflow
	if (zeros > 17) {
	x <<= zeros - 17;
	y <<= zeros - 17;
	} else {
	x >>= 17 - zeros;
	y >>= 17 - zeros;
	}
	SkASSERT((x \| y) <= 0x7FFF);

	unsigned invrt = fast_invsqrt(xx + yy);

	x = x * invrt >> 12;
	y = y * invrt >> 12;

	if (length != SK_Fixed1) {
	x = SkFixedMul(x, length);
	y = SkFixedMul(y, length);
	}
	this->set(SkApplySign(x, SkExtractSign(ox)),
	SkApplySign(y, SkExtractSign(oy)));
	return true;
	}
	#else
	/*
	Normalize x,y, and then scale them by length.

	The obvious way to do this would be the following:
	S64 tmp1, tmp2;
	tmp1.setMul(x,x);
	tmp2.setMul(y,y);
	tmp1.add(tmp2);
	len = tmp1.getSqrt();
	x' = SkFixedDiv(x, len);
	y' = SkFixedDiv(y, len);
	This is fine, but slower than what we do below.

	The present technique does not compute the starting length, but
	rather fiddles with x,y iteratively, all the while checking its
	magnitude^2 (avoiding a sqrt).

	We normalize by first shifting x,y so that at least one of them
	has bit 31 set (after taking the abs of them).
	Then we loop, refining x,y by squaring them and comparing
	against a very large 1.0 (1 << 28), and then adding or subtracting
	a delta (which itself is reduced by half each time through the loop).
	For speed we want the squaring to be with a simple integer mul. To keep
	that from overflowing we shift our coordinates down until we are dealing
	with at most 15 bits (2^15-1)^2 * 2 says withing 32 bits)
	When our square is close to 1.0, we shift x,y down into fixed range.
	*/
	bool SkPoint::setLength(SkFixed ox, SkFixed oy, SkFixed length) {
	if (ox == 0) {
	if (oy == 0)
	return false;
	this->set(0, SkApplySign(length, SkExtractSign(oy)));
	return true;
	}
	if (oy == 0) {
	this->set(SkApplySign(length, SkExtractSign(ox)), 0);
	return true;
	}

	SkFixed x = SkAbs32(ox);
	SkFixed y = SkAbs32(oy);

	// shift x,y so that the greater of them is 15bits (1.14 fixed point)
	{
	int shift = SkCLZ(x \| y);
	// make them .30
	x <<= shift - 1;
	y <<= shift - 1;
	}

	SkFixed dx = x;
	SkFixed dy = y;

	for (int i = 0; i < 17; i++) {
	dx >>= 1;
	dy >>= 1;

	U32 len2 = squarefixed(x) + squarefixed(y);
	if (len2 >> 28) {
	x -= dx;
	y -= dy;
	} else {
	x += dx;
	y += dy;
	}
	}
	x >>= 14;
	y >>= 14;

	#ifdef SK_DEBUGx // measure how far we are from unit-length
	{
	static int gMaxError;
	static int gMaxDiff;

	SkFixed len = fixlen(x, y);
	int err = len - SK_Fixed1;
	err = SkAbs32(err);

	if (err > gMaxError) {
	gMaxError = err;
	SkDebugf("gMaxError %d\n", err);
	}

	float fx = SkAbs32(ox)/65536.0f;
	float fy = SkAbs32(oy)/65536.0f;
	float mag = sqrtf(fxfx + fyfy);
	fx /= mag;
	fy /= mag;
	SkFixed xx = (int)floorf(fx * 65536 + 0.5f);
	SkFixed yy = (int)floorf(fy * 65536 + 0.5f);
	err = SkMax32(SkAbs32(xx-x), SkAbs32(yy-y));
	if (err > gMaxDiff) {
	gMaxDiff = err;
	SkDebugf("gMaxDiff %d\n", err);
	}
	}
	#endif

	x = SkApplySign(x, SkExtractSign(ox));
	y = SkApplySign(y, SkExtractSign(oy));
	if (length != SK_Fixed1) {
	x = SkFixedMul(x, length);
	y = SkFixedMul(y, length);
	}

	this->set(x, y);
	return true;
	}
	#endif

	#endif

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

	SkScalar SkPoint::distanceToLineBetweenSqd(const SkPoint& a,
	const SkPoint& b,
	Side* side) const {

	SkVector u = b - a;
	SkVector v = *this - a;

	SkScalar uLengthSqd = u.lengthSqd();
	SkScalar det = u.cross(v);
	if (NULL != side) {
	SkASSERT(-1 == SkPoint::kLeft_Side &&
	0 == SkPoint::kOn_Side &&
	1 == kRight_Side);
	*side = (Side) SkScalarSignAsInt(det);
	}
	return SkScalarMulDiv(det, det, uLengthSqd);
	}

	SkScalar SkPoint::distanceToLineSegmentBetweenSqd(const SkPoint& a,
	const SkPoint& b) const {
	// See comments to distanceToLineBetweenSqd. If the projection of c onto
	// u is between a and b then this returns the same result as that
	// function. Otherwise, it returns the distance to the closer of a and
	// b. Let the projection of v onto u be v'. There are three cases:
	// 1. v' points opposite to u. c is not between a and b and is closer
	// to a than b.
	// 2. v' points along u and has magnitude less than y. c is between
	// a and b and the distance to the segment is the same as distance
	// to the line ab.
	// 3. v' points along u and has greater magnitude than u. c is not
	// not between a and b and is closer to b than a.
	// v' = (u dot v) * u / \|u\|. So if (u dot v)/\|u\| is less than zero we're
	// in case 1. If (u dot v)/\|u\| is > \|u\| we are in case 3. Otherwise
	// we're in case 2. We actually compare (u dot v) to 0 and \|u\|^2 to
	// avoid a sqrt to compute \|u\|.

	SkVector u = b - a;
	SkVector v = *this - a;

	SkScalar uLengthSqd = u.lengthSqd();
	SkScalar uDotV = SkPoint::DotProduct(u, v);

	if (uDotV <= 0) {
	return v.lengthSqd();
	} else if (uDotV > uLengthSqd) {
	return b.distanceToSqd(*this);
	} else {
	SkScalar det = u.cross(v);
	return SkScalarMulDiv(det, det, uLengthSqd);
	}
	}