| // Copyright 2016 The Go Authors. All rights reserved. |
| // Use of this source code is governed by a BSD-style |
| // license that can be found in the LICENSE file. |
| |
| package vector |
| |
| // This file contains a fixed point math implementation of the vector |
| // graphics rasterizer. |
| |
| const ( |
| // ϕ is the number of binary digits after the fixed point. |
| // |
| // For example, if ϕ == 10 (and int1ϕ is based on the int32 type) then we |
| // are using 22.10 fixed point math. |
| // |
| // When changing this number, also change the assembly code (search for ϕ |
| // in the .s files). |
| ϕ = 9 |
| |
| fxOne int1ϕ = 1 << ϕ |
| fxOneAndAHalf int1ϕ = 1<<ϕ + 1<<(ϕ-1) |
| fxOneMinusIota int1ϕ = 1<<ϕ - 1 // Used for rounding up. |
| ) |
| |
| // int1ϕ is a signed fixed-point number with 1*ϕ binary digits after the fixed |
| // point. |
| type int1ϕ int32 |
| |
| // int2ϕ is a signed fixed-point number with 2*ϕ binary digits after the fixed |
| // point. |
| // |
| // The Rasterizer's bufU32 field, nominally of type []uint32 (since that slice |
| // is also used by other code), can be thought of as a []int2ϕ during the |
| // fixedLineTo method. Lines of code that are actually like: |
| // buf[i] += uint32(etc) // buf has type []uint32. |
| // can be thought of as |
| // buf[i] += int2ϕ(etc) // buf has type []int2ϕ. |
| type int2ϕ int32 |
| |
| func fixedMax(x, y int1ϕ) int1ϕ { |
| if x > y { |
| return x |
| } |
| return y |
| } |
| |
| func fixedMin(x, y int1ϕ) int1ϕ { |
| if x < y { |
| return x |
| } |
| return y |
| } |
| |
| func fixedFloor(x int1ϕ) int32 { return int32(x >> ϕ) } |
| func fixedCeil(x int1ϕ) int32 { return int32((x + fxOneMinusIota) >> ϕ) } |
| |
| func (z *Rasterizer) fixedLineTo(bx, by float32) { |
| ax, ay := z.penX, z.penY |
| z.penX, z.penY = bx, by |
| dir := int1ϕ(1) |
| if ay > by { |
| dir, ax, ay, bx, by = -1, bx, by, ax, ay |
| } |
| // Horizontal line segments yield no change in coverage. Almost horizontal |
| // segments would yield some change, in ideal math, but the computation |
| // further below, involving 1 / (by - ay), is unstable in fixed point math, |
| // so we treat the segment as if it was perfectly horizontal. |
| if by-ay <= 0.000001 { |
| return |
| } |
| dxdy := (bx - ax) / (by - ay) |
| |
| ayϕ := int1ϕ(ay * float32(fxOne)) |
| byϕ := int1ϕ(by * float32(fxOne)) |
| |
| x := int1ϕ(ax * float32(fxOne)) |
| y := fixedFloor(ayϕ) |
| yMax := fixedCeil(byϕ) |
| if yMax > int32(z.size.Y) { |
| yMax = int32(z.size.Y) |
| } |
| width := int32(z.size.X) |
| |
| for ; y < yMax; y++ { |
| dy := fixedMin(int1ϕ(y+1)<<ϕ, byϕ) - fixedMax(int1ϕ(y)<<ϕ, ayϕ) |
| xNext := x + int1ϕ(float32(dy)*dxdy) |
| if y < 0 { |
| x = xNext |
| continue |
| } |
| buf := z.bufU32[y*width:] |
| d := dy * dir // d ranges up to ±1<<(1*ϕ). |
| x0, x1 := x, xNext |
| if x > xNext { |
| x0, x1 = x1, x0 |
| } |
| x0i := fixedFloor(x0) |
| x0Floor := int1ϕ(x0i) << ϕ |
| x1i := fixedCeil(x1) |
| x1Ceil := int1ϕ(x1i) << ϕ |
| |
| if x1i <= x0i+1 { |
| xmf := (x+xNext)>>1 - x0Floor |
| if i := clamp(x0i+0, width); i < uint(len(buf)) { |
| buf[i] += uint32(d * (fxOne - xmf)) |
| } |
| if i := clamp(x0i+1, width); i < uint(len(buf)) { |
| buf[i] += uint32(d * xmf) |
| } |
| } else { |
| oneOverS := x1 - x0 |
| twoOverS := 2 * oneOverS |
| x0f := x0 - x0Floor |
| oneMinusX0f := fxOne - x0f |
| oneMinusX0fSquared := oneMinusX0f * oneMinusX0f |
| x1f := x1 - x1Ceil + fxOne |
| x1fSquared := x1f * x1f |
| |
| // These next two variables are unused, as rounding errors are |
| // minimized when we delay the division by oneOverS for as long as |
| // possible. These lines of code (and the "In ideal math" comments |
| // below) are commented out instead of deleted in order to aid the |
| // comparison with the floating point version of the rasterizer. |
| // |
| // a0 := ((oneMinusX0f * oneMinusX0f) >> 1) / oneOverS |
| // am := ((x1f * x1f) >> 1) / oneOverS |
| |
| if i := clamp(x0i, width); i < uint(len(buf)) { |
| // In ideal math: buf[i] += uint32(d * a0) |
| D := oneMinusX0fSquared // D ranges up to ±1<<(2*ϕ). |
| D *= d // D ranges up to ±1<<(3*ϕ). |
| D /= twoOverS |
| buf[i] += uint32(D) |
| } |
| |
| if x1i == x0i+2 { |
| if i := clamp(x0i+1, width); i < uint(len(buf)) { |
| // In ideal math: buf[i] += uint32(d * (fxOne - a0 - am)) |
| // |
| // (x1i == x0i+2) and (twoOverS == 2 * (x1 - x0)) implies |
| // that twoOverS ranges up to +1<<(1*ϕ+2). |
| D := twoOverS<<ϕ - oneMinusX0fSquared - x1fSquared // D ranges up to ±1<<(2*ϕ+2). |
| D *= d // D ranges up to ±1<<(3*ϕ+2). |
| D /= twoOverS |
| buf[i] += uint32(D) |
| } |
| } else { |
| // This is commented out for the same reason as a0 and am. |
| // |
| // a1 := ((fxOneAndAHalf - x0f) << ϕ) / oneOverS |
| |
| if i := clamp(x0i+1, width); i < uint(len(buf)) { |
| // In ideal math: |
| // buf[i] += uint32(d * (a1 - a0)) |
| // or equivalently (but better in non-ideal, integer math, |
| // with respect to rounding errors), |
| // buf[i] += uint32(A * d / twoOverS) |
| // where |
| // A = (a1 - a0) * twoOverS |
| // = a1*twoOverS - a0*twoOverS |
| // Noting that twoOverS/oneOverS equals 2, substituting for |
| // a0 and then a1, given above, yields: |
| // A = a1*twoOverS - oneMinusX0fSquared |
| // = (fxOneAndAHalf-x0f)<<(ϕ+1) - oneMinusX0fSquared |
| // = fxOneAndAHalf<<(ϕ+1) - x0f<<(ϕ+1) - oneMinusX0fSquared |
| // |
| // This is a positive number minus two non-negative |
| // numbers. For an upper bound on A, the positive number is |
| // P = fxOneAndAHalf<<(ϕ+1) |
| // < (2*fxOne)<<(ϕ+1) |
| // = fxOne<<(ϕ+2) |
| // = 1<<(2*ϕ+2) |
| // |
| // For a lower bound on A, the two non-negative numbers are |
| // N = x0f<<(ϕ+1) + oneMinusX0fSquared |
| // ≤ x0f<<(ϕ+1) + fxOne*fxOne |
| // = x0f<<(ϕ+1) + 1<<(2*ϕ) |
| // < x0f<<(ϕ+1) + 1<<(2*ϕ+1) |
| // ≤ fxOne<<(ϕ+1) + 1<<(2*ϕ+1) |
| // = 1<<(2*ϕ+1) + 1<<(2*ϕ+1) |
| // = 1<<(2*ϕ+2) |
| // |
| // Thus, A ranges up to ±1<<(2*ϕ+2). It is possible to |
| // derive a tighter bound, but this bound is sufficient to |
| // reason about overflow. |
| D := (fxOneAndAHalf-x0f)<<(ϕ+1) - oneMinusX0fSquared // D ranges up to ±1<<(2*ϕ+2). |
| D *= d // D ranges up to ±1<<(3*ϕ+2). |
| D /= twoOverS |
| buf[i] += uint32(D) |
| } |
| dTimesS := uint32((d << (2 * ϕ)) / oneOverS) |
| for xi := x0i + 2; xi < x1i-1; xi++ { |
| if i := clamp(xi, width); i < uint(len(buf)) { |
| buf[i] += dTimesS |
| } |
| } |
| |
| // This is commented out for the same reason as a0 and am. |
| // |
| // a2 := a1 + (int1ϕ(x1i-x0i-3)<<(2*ϕ))/oneOverS |
| |
| if i := clamp(x1i-1, width); i < uint(len(buf)) { |
| // In ideal math: |
| // buf[i] += uint32(d * (fxOne - a2 - am)) |
| // or equivalently (but better in non-ideal, integer math, |
| // with respect to rounding errors), |
| // buf[i] += uint32(A * d / twoOverS) |
| // where |
| // A = (fxOne - a2 - am) * twoOverS |
| // = twoOverS<<ϕ - a2*twoOverS - am*twoOverS |
| // Noting that twoOverS/oneOverS equals 2, substituting for |
| // am and then a2, given above, yields: |
| // A = twoOverS<<ϕ - a2*twoOverS - x1f*x1f |
| // = twoOverS<<ϕ - a1*twoOverS - (int1ϕ(x1i-x0i-3)<<(2*ϕ))*2 - x1f*x1f |
| // = twoOverS<<ϕ - a1*twoOverS - int1ϕ(x1i-x0i-3)<<(2*ϕ+1) - x1f*x1f |
| // Substituting for a1, given above, yields: |
| // A = twoOverS<<ϕ - ((fxOneAndAHalf-x0f)<<ϕ)*2 - int1ϕ(x1i-x0i-3)<<(2*ϕ+1) - x1f*x1f |
| // = twoOverS<<ϕ - (fxOneAndAHalf-x0f)<<(ϕ+1) - int1ϕ(x1i-x0i-3)<<(2*ϕ+1) - x1f*x1f |
| // = B<<ϕ - x1f*x1f |
| // where |
| // B = twoOverS - (fxOneAndAHalf-x0f)<<1 - int1ϕ(x1i-x0i-3)<<(ϕ+1) |
| // = (x1-x0)<<1 - (fxOneAndAHalf-x0f)<<1 - int1ϕ(x1i-x0i-3)<<(ϕ+1) |
| // |
| // Re-arranging the defintions given above: |
| // x0Floor := int1ϕ(x0i) << ϕ |
| // x0f := x0 - x0Floor |
| // x1Ceil := int1ϕ(x1i) << ϕ |
| // x1f := x1 - x1Ceil + fxOne |
| // combined with fxOne = 1<<ϕ yields: |
| // x0 = x0f + int1ϕ(x0i)<<ϕ |
| // x1 = x1f + int1ϕ(x1i-1)<<ϕ |
| // so that expanding (x1-x0) yields: |
| // B = (x1f-x0f + int1ϕ(x1i-x0i-1)<<ϕ)<<1 - (fxOneAndAHalf-x0f)<<1 - int1ϕ(x1i-x0i-3)<<(ϕ+1) |
| // = (x1f-x0f)<<1 + int1ϕ(x1i-x0i-1)<<(ϕ+1) - (fxOneAndAHalf-x0f)<<1 - int1ϕ(x1i-x0i-3)<<(ϕ+1) |
| // A large part of the second and fourth terms cancel: |
| // B = (x1f-x0f)<<1 - (fxOneAndAHalf-x0f)<<1 - int1ϕ(-2)<<(ϕ+1) |
| // = (x1f-x0f)<<1 - (fxOneAndAHalf-x0f)<<1 + 1<<(ϕ+2) |
| // = (x1f - fxOneAndAHalf)<<1 + 1<<(ϕ+2) |
| // The first term, (x1f - fxOneAndAHalf)<<1, is a negative |
| // number, bounded below by -fxOneAndAHalf<<1, which is |
| // greater than -fxOne<<2, or -1<<(ϕ+2). Thus, B ranges up |
| // to ±1<<(ϕ+2). One final simplification: |
| // B = x1f<<1 + (1<<(ϕ+2) - fxOneAndAHalf<<1) |
| const C = 1<<(ϕ+2) - fxOneAndAHalf<<1 |
| D := x1f<<1 + C // D ranges up to ±1<<(1*ϕ+2). |
| D <<= ϕ // D ranges up to ±1<<(2*ϕ+2). |
| D -= x1fSquared // D ranges up to ±1<<(2*ϕ+3). |
| D *= d // D ranges up to ±1<<(3*ϕ+3). |
| D /= twoOverS |
| buf[i] += uint32(D) |
| } |
| } |
| |
| if i := clamp(x1i, width); i < uint(len(buf)) { |
| // In ideal math: buf[i] += uint32(d * am) |
| D := x1fSquared // D ranges up to ±1<<(2*ϕ). |
| D *= d // D ranges up to ±1<<(3*ϕ). |
| D /= twoOverS |
| buf[i] += uint32(D) |
| } |
| } |
| |
| x = xNext |
| } |
| } |
| |
| func fixedAccumulateOpOver(dst []uint8, src []uint32) { |
| // Sanity check that len(dst) >= len(src). |
| if len(dst) < len(src) { |
| return |
| } |
| |
| acc := int2ϕ(0) |
| for i, v := range src { |
| acc += int2ϕ(v) |
| a := acc |
| if a < 0 { |
| a = -a |
| } |
| a >>= 2*ϕ - 16 |
| if a > 0xffff { |
| a = 0xffff |
| } |
| // This algorithm comes from the standard library's image/draw package. |
| dstA := uint32(dst[i]) * 0x101 |
| maskA := uint32(a) |
| outA := dstA*(0xffff-maskA)/0xffff + maskA |
| dst[i] = uint8(outA >> 8) |
| } |
| } |
| |
| func fixedAccumulateOpSrc(dst []uint8, src []uint32) { |
| // Sanity check that len(dst) >= len(src). |
| if len(dst) < len(src) { |
| return |
| } |
| |
| acc := int2ϕ(0) |
| for i, v := range src { |
| acc += int2ϕ(v) |
| a := acc |
| if a < 0 { |
| a = -a |
| } |
| a >>= 2*ϕ - 8 |
| if a > 0xff { |
| a = 0xff |
| } |
| dst[i] = uint8(a) |
| } |
| } |
| |
| func fixedAccumulateMask(buf []uint32) { |
| acc := int2ϕ(0) |
| for i, v := range buf { |
| acc += int2ϕ(v) |
| a := acc |
| if a < 0 { |
| a = -a |
| } |
| a >>= 2*ϕ - 16 |
| if a > 0xffff { |
| a = 0xffff |
| } |
| buf[i] = uint32(a) |
| } |
| } |
