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Reshaping dataflows

February 28, 2013

This post is part of a series – go here for the index.

Welcome back! So far, we’ve spent quite some time “zoomed in” on various components of the Software Occlusion Culling demo, looking at various micro-architectural pitfalls and individual loops. In the last two posts, we “zoomed out” and focused on the big picture: what work runs when, and how to keep all cores busy. Now, it’s time to look at what lies in between: the plumbing, if you will. We’ll be looking at the dataflows between subsystems and modules and how to improve them.

This is one of my favorite topics in optimization, and it’s somewhat under-appreciated. There’s plenty of material on how to make loops run fast (although a lot of it is outdated or just wrong, so beware), and at this point there’s plenty of ways of getting concurrency up and running: there’s OpenMP, Intel’s TBB, Apple’s GCD, Windows Thread Pools and ConcRT for CPU, there’s OpenCL, CUDA and DirectCompute for jobs that are GPU-suitable, and so forth; you get the idea. The point being that it’s not hard to find a shrink-wrap solution that gets you up and running, and a bit of profiling (like we just did) is usually enough to tell you what needs to be done to make it all go smoothly.

But back to the topic at hand: improving dataflow. The problem is that, unlike the other two aspects I mentioned, there’s really no recipe to follow; it’s very much context-dependent. It basically boils down to looking at both sides of the interface between systems and functions and figuring out if there’s a better way to handle that interaction. We’ve seen a bit of that earlier when talking about frustum culling; rather than trying to define it in words, I’ll just do it by example, so let’s dive right in!

A simple example

A good example is the member variable TransformedAABBoxSSE::mVisible, declared like this:

bool *mVisible;

A pointer to a bool. So where does that pointer come from?

inline void SetVisible(bool *visible){mVisible = visible;}

It turns out that the constructor initializes this pointer to NULL, and the only method that ever does anything with mVisible is RasterizeAndDepthTestAABBox, which executes *mVisible = true; if the bounding box is found to be visible. So how does this all get used?

mpVisible[i] = false;
mpTransformedAABBox[i].SetVisible(&mpVisible[i]);
if(...)
{
    mpTransformedAABBox[i].TransformAABBox();
    mpTransformedAABBox[i].RasterizeAndDepthTestAABBox(...);
}

That’s it. That’s the only call sites. There’s really no reason for mVisible to be state – semantically, it’s just a return value for RasterizeAndDepthTestAABBox, so that’s what it should be – always try to get rid of superfluous state. This doesn’t even have anything to do with optimization per se; explicit dataflow is easy for programmers to see and reason about, while implicit dataflow (through pointers, members and state) is hard to follow (both for humans and compilers!) and error-prone.

Anyway, making this return value explicit is really basic, so I’m not gonna walk through the details; you can always look at the corresponding commit. I won’t bother benchmarking this change either.

A more interesting case

In the depth test rasterizer, right after determining the bounding box, there’s this piece of code:

for(int vv = 0; vv < 3; vv++) 
{
    // If W (holding 1/w in our case) is not between 0 and 1,
    // then vertex is behind near clip plane (1.0 in our case).
    // If W < 1 (for W>0), and 1/W < 0 (for W < 0).
    VecF32 nearClipMask0 = cmple(xformedPos[vv].W, VecF32(0.0f));
    VecF32 nearClipMask1 = cmpge(xformedPos[vv].W, VecF32(1.0f));
    VecS32 nearClipMask = float2bits(or(nearClipMask0,
        nearClipMask1));

    if(!is_all_zeros(nearClipMask))
    {
        // All four vertices are behind the near plane (we're
        // processing four triangles at a time w/ SSE)
        return true;
    }
}

Okay. The transform code sets things up so that the “w” component of the screen-space positions actually contains 1/w; the first part of this code then tries to figure out whether the source vertex was in front of the near plane (i.e. outside the view frustum or not). An ugly wrinkle here is that the near plane is hard-coded to be at 1. Doing this after dividing by w adds extra complications since the code needs to be careful about the signs. And the second comment is outright wrong – it in fact early-outs when any of the four active triangles have vertex number vv outside the near-clip plane, not when all of them do. In other words, if any of the 4 active triangles get near-clipped, the test rasterizer will just punt and return true (“visible”).

So here’s the thing: there’s really no reason to do this check after we’re done with triangle setup. Nor do we even have to gather the 3 triangle vertices to discover that one of them is in front of the near plane. A box has 8 vertices, and we’ll know whether any of them are in front of the near plane as soon as we’re done transforming them, before we even think about triangle setup! So let’s look at the function that transforms the vertices:

void TransformedAABBoxSSE::TransformAABBox()
{
    for(UINT i = 0; i < AABB_VERTICES; i++)
    {
        mpXformedPos[i] = TransformCoords(&mpBBVertexList[i],
            mCumulativeMatrix);
        float oneOverW = 1.0f/max(mpXformedPos[i].m128_f32[3],
            0.0000001f);
        mpXformedPos[i] = mpXformedPos[i] * oneOverW;
        mpXformedPos[i].m128_f32[3] = oneOverW;
    }
}

As we can see, returning 1/w does in fact take a bit of extra work, so we’d like to avoid it, especially since that 1/w is really only referenced by the near-clip checking code. Also, the code seems to clamp w at some arbitrary small positive value – which means that the part of the near clip computation in the depth test rasterizer that worries about w<0 is actually unnecessary. This is the kind of thing I’m talking about – each piece of code in isolation seems reasonable, but once you look at both sides it becomes clear that the pieces don’t fit together all that well.

It turns out that after TransformCoords, we’re in “homogeneous viewport space”, i.e. we’re still in a homogeneous space, but unlike the homogeneous clip space you might be used to from vertex shaders, this one also has the viewport transform baked in. But our viewport transform leaves z alone (we fixed that in the previous post!), so we still have a D3D-style clip volume for z:

0 \le z \le w

Since we’re using a reversed clip volume, the z≤w constraint is the near-plane one. Note that this test doesn’t need any special cases for negative signs and also doesn’t have a hardcoded near-plane location any more: it just automatically uses whatever the projection matrix says, which is the right thing to do!

Even better, if we test for near-clip anyway, there’s no need to clamp w at all. We know that anything with w≤0 is outside the near plane, and if a vertex is outside the near plane we’re not gonna rasterize the box anyway. Now we might still end up dividing by 0, but since we’re dealing with floats, this is a well-defined operation (it might return infinities or NaNs, but that’s fine).

And on the subject of not rasterizing the box: as I said earlier, as soon as one vertex is outside the near-plane, we know we’re going to return true from the depth test rasterizer, so there’s no point even starting the operation. To facilitate this, we just make TransformAABBox return whether the box should be rasterized or not. Putting it all together:

bool TransformedAABBoxSSE::TransformAABBox()
{
    __m128 zAllIn = _mm_castsi128_ps(_mm_set1_epi32(~0));

    for(UINT i = 0; i < AABB_VERTICES; i++)
    {
        __m128 vert = TransformCoords(&mpBBVertexList[i],
            mCumulativeMatrix);

        // We have inverted z; z is inside of near plane iff z <= w.
        __m128 vertZ = _mm_shuffle_ps(vert, vert, 0xaa); //vert.zzzz
        __m128 vertW = _mm_shuffle_ps(vert, vert, 0xff); //vert.wwww
        __m128 zIn = _mm_cmple_ps(vertZ, vertW);
        zAllIn = _mm_and_ps(zAllIn, zIn);

        // project
        mpXformedPos[i] = _mm_div_ps(vert, vertW);
    }

    // return true if and only if all verts inside near plane
    return _mm_movemask_ps(zAllIn) == 0xf;
}

In case you’re wondering why this code uses raw SSE intrinsics and not VecF32, it’s because I’m purposefully trying to keep anything depending on the SIMD width out of VecF32, which makes it a lot easier to go to 8-wide AVX should we want to at some point. But this code really uses 4-vectors of (x,y,z,w) and needs to do shuffles, so it doesn’t fit in that model and I want to keep it separate. But the actual logic is just what I described.

And once we have this return value from TransformAABBox, we get to remove the near-clip test from the depth test rasterizer, and we get to move our early-out for near-clipped boxes all the way to the call site:

if(mpTransformedAABBox[i].TransformAABBox())
    mpVisible[i] = mpTransformedAABBox[i].RasterizeAndDepthTestAABBox(...);
else
    mpVisible[i] = true;

So, the oneOverW hack, the clamping hack and the hard-coded near plane are gone. That’s already a victory in terms of code quality, but did it improve the run time?

Change: Transform/early-out fixes

Depth test min 25th med 75th max mean sdev
Start 1.109 1.152 1.166 1.182 1.240 1.167 0.022
Transform fixes 1.054 1.092 1.102 1.112 1.146 1.102 0.016

Another 0.06ms off our median depth test time, which may not sound big but is over 5% of what’s left of it at this point.

Getting warmer

The bounding box rasterizer has one more method that’s called per-box though, and this is one that really deserves some special attention. Meet IsTooSmall:

bool TransformedAABBoxSSE::IsTooSmall(__m128 *pViewMatrix,
    __m128 *pProjMatrix, CPUTCamera *pCamera)
{
    float radius = mBBHalf.lengthSq(); // Use length-squared to
    // avoid sqrt().  Relative comparisons hold.

    float fov = pCamera->GetFov();
    float tanOfHalfFov = tanf(fov * 0.5f);

    MatrixMultiply(mWorldMatrix, pViewMatrix, mCumulativeMatrix);
    MatrixMultiply(mCumulativeMatrix, pProjMatrix,
        mCumulativeMatrix);
    MatrixMultiply(mCumulativeMatrix, mViewPortMatrix,
        mCumulativeMatrix);

    __m128 center = _mm_set_ps(1.0f, mBBCenter.z, mBBCenter.y,
        mBBCenter.x);
    __m128 mBBCenterOSxForm = TransformCoords(&center,
        mCumulativeMatrix);
    float w = mBBCenterOSxForm.m128_f32[3];
    if( w > 1.0f )
    {
        float radiusDivW = radius / w;
        float r2DivW2DivTanFov = radiusDivW / tanOfHalfFov;

        return r2DivW2DivTanFov <
            (mOccludeeSizeThreshold * mOccludeeSizeThreshold);
    }

    return false;
}

Note that MatrixMultiply(A, B, C) performs C = A * B; the rest should be easy enough to figure out from the code. Now there’s really several problems with this function, so let’s go straight to a list:

  • radius (which is really radius squared) only depends on mBBHalf, which is fixed at initialization time. There’s no need to recompute it every time.
  • Similarly, fov and tanOfHalfFov only depend on the camera, and absolutely do not need to be recomputed once for every box. This is what gave us the _tan_pentium4 cameo all the way back in “Frustum culling: turning the crank”, by the way.
  • The view matrix, projection matrix and viewport matrix are also all camera or global constants. Again, no need to multiply these together for every box – the only matrix that is different between boxes is the very first one, the world matrix, and since matrix multiplication is associative, we can just concatenate the other three once.
  • There’s also no need for mOccludeeSizeThreshold to be squared every time – we can do that once.
  • Nor is there a need for it to be stored per box, since it’s a global constant owned by the depth test rasterizer.
  • (radius / w) / tanOfHalfFov would be better computed as radius / (w * tanOfHalfFov).
  • But more importantly, since all we’re doing is a compare and both w and tanOfHalfFov are positive, we can just multiply through by them and get rid of the divide altogether.

All these things are common problems that I must have fixed a hundred times, but I have to admit that it’s pretty rare to see so many of them in a single page of code. Anyway, rather than fixing these one by one, let’s just cut to the chase: instead of all the redundant computations, we just move everything that only depends on the camera (or is global) into a single struct that holds our setup, which I dubbed BoxTestSetup. Here’s the code:

struct BoxTestSetup
{
    __m128 mViewProjViewport[4];
    float radiusThreshold;

    void Init(const __m128 viewMatrix[4],
        const __m128 projMatrix[4], CPUTCamera *pCamera,
        float occludeeSizeThreshold);
};

void BoxTestSetup::Init(const __m128 viewMatrix[4],
    const __m128 projMatrix[4], CPUTCamera *pCamera,
    float occludeeSizeThreshold)
{
    // viewportMatrix is a global float4x4; we need a __m128[4]
    __m128 viewPortMatrix[4];
    viewPortMatrix[0] = _mm_loadu_ps((float*)&viewportMatrix.r0);
    viewPortMatrix[1] = _mm_loadu_ps((float*)&viewportMatrix.r1);
    viewPortMatrix[2] = _mm_loadu_ps((float*)&viewportMatrix.r2);
    viewPortMatrix[3] = _mm_loadu_ps((float*)&viewportMatrix.r3);

    MatrixMultiply(viewMatrix, projMatrix, mViewProjViewport);
    MatrixMultiply(mViewProjViewport, viewPortMatrix,
        mViewProjViewport);

    float fov = pCamera->GetFov();
    float tanOfHalfFov = tanf(fov * 0.5f);
    radiusThreshold = occludeeSizeThreshold * occludeeSizeThreshold
        * tanOfHalfFov;
}

This is initialized once we start culling and simply kept on the stack. Then we just pass it to IsTooSmall, which after our surgery looks like this:

bool TransformedAABBoxSSE::IsTooSmall(const BoxTestSetup &setup)
{
    MatrixMultiply(mWorldMatrix, setup.mViewProjViewport,
        mCumulativeMatrix);

    __m128 center = _mm_set_ps(1.0f, mBBCenter.z, mBBCenter.y,
        mBBCenter.x);
    __m128 mBBCenterOSxForm = TransformCoords(&center,
        mCumulativeMatrix);
    float w = mBBCenterOSxForm.m128_f32[3];
    if( w > 1.0f )
    {
        return mRadiusSq < w * setup.radiusThreshold;
    }

    return false;
}

Wow, that method sure seems to have lost a few pounds. Let’s run the numbers:

Change: IsTooSmall cleanup

Depth test min 25th med 75th max mean sdev
Start 1.109 1.152 1.166 1.182 1.240 1.167 0.022
Transform fixes 1.054 1.092 1.102 1.112 1.146 1.102 0.016
IsTooSmall cleanup 0.860 0.893 0.908 0.917 0.954 0.905 0.018

Another 0.2ms off the median run time, bringing our total reduction for this post to about 22%. So are we done? Not yet!

The state police

Currently, each TransformedAABBoxSSE still keeps its own copy of the cumulative transform matrix and a copy of its transformed vertices. But it’s not necessary for these to be persistent – we compute them once, use them to rasterize the box, then don’t look at them again until the next frame. So, like mVisible earlier, there’s really no need to keep them around as state; instead, it’s better to just store them on the stack. Less pointers per TransformedAABBoxSSE, less cache misses, and – perhaps most important of all – it makes the bounding box objects themselves stateless. Granted, that’s the case only because our world is perfectly static and nothing is animated at runtime, but still, stateless is good! Stateless is easier to read, easier to debug, and easier to test.

Again, this is another change that is purely mechanical – just pass in a pointer to cumulativeMatrix and xformedPos to the functions that want them. So this time, I’m just going to refer you directly to the two commits that implement this idea, and skip straight to the results:

Change: Reduce amount of state

Depth test min 25th med 75th max mean sdev
Start 1.109 1.152 1.166 1.182 1.240 1.167 0.022
Transform fixes 1.054 1.092 1.102 1.112 1.146 1.102 0.016
IsTooSmall cleanup 0.860 0.893 0.908 0.917 0.954 0.905 0.018
Reduce state 0.834 0.862 0.873 0.886 0.938 0.875 0.017

Only about 0.03ms this time, but we also save 192 bytes (plus allocator overhead) worth of memory per box, which is a nice bonus. And anyway, we’re not done yet, because I have one more!

It’s more fun to compute

There’s one more piece of unnecessary data we currently store per bounding box: the vertex list, initialized in CreateAABBVertexIndexList:

float3 min = mBBCenter - bbHalf;
float3 max = mBBCenter + bbHalf;
	
//Top 4 vertices in BB
mpBBVertexList[0] = _mm_set_ps(1.0f, max.z, max.y, max.x);
mpBBVertexList[1] = _mm_set_ps(1.0f, max.z, max.y, min.x); 
mpBBVertexList[2] = _mm_set_ps(1.0f, min.z, max.y, min.x);
mpBBVertexList[3] = _mm_set_ps(1.0f, min.z, max.y, max.x);
// Bottom 4 vertices in BB
mpBBVertexList[4] = _mm_set_ps(1.0f, min.z, min.y, max.x);
mpBBVertexList[5] = _mm_set_ps(1.0f, max.z, min.y, max.x);
mpBBVertexList[6] = _mm_set_ps(1.0f, max.z, min.y, min.x);
mpBBVertexList[7] = _mm_set_ps(1.0f, min.z, min.y, min.x);

This is, in effect, just treating the bounding box as a general mesh. But that’s extremely wasteful – we already store center and half-extent, the min/max corner positions are trivial to reconstruct from that information, and all the other vertices can be constructed by splicing min/max together componentwise using a set of masks that is the same for all bounding boxes. So these 8*16 = 128 bytes of vertex data really don’t pay their way.

But more importantly, note that the we only ever use two distinct values for x, y and z each. Now TransformAABBox, which we already saw above, uses TransformCoords to compute the matrix-vector product v*M with the cumulative transform matrix, using the expression

v.x * M.row[0] + v.y * M.row[1] + v.z * M.row[2] + M.row[3] (v.w is assumed to be 1)

and because we know that v.x is either min.x or max.x, we can multiply both by M.row[0] once and store the result. Then the 8 individual vertices can skip the multiplies altogether. Putting it all together leads to the following new code for TransformAABBox:

// 0 = use min corner, 1 = use max corner
static const int sBBxInd[AABB_VERTICES] = { 1, 0, 0, 1, 1, 1, 0, 0 };
static const int sBByInd[AABB_VERTICES] = { 1, 1, 1, 1, 0, 0, 0, 0 };
static const int sBBzInd[AABB_VERTICES] = { 1, 1, 0, 0, 0, 1, 1, 0 };

bool TransformedAABBoxSSE::TransformAABBox(__m128 xformedPos[],
    const __m128 cumulativeMatrix[4])
{
    // w ends up being garbage, but it doesn't matter - we ignore
    // it anyway.
    __m128 vCenter = _mm_loadu_ps(&mBBCenter.x);
    __m128 vHalf   = _mm_loadu_ps(&mBBHalf.x);

    __m128 vMin    = _mm_sub_ps(vCenter, vHalf);
    __m128 vMax    = _mm_add_ps(vCenter, vHalf);

    // transforms
    __m128 xRow[2], yRow[2], zRow[2];
    xRow[0] = _mm_shuffle_ps(vMin, vMin, 0x00) * cumulativeMatrix[0];
    xRow[1] = _mm_shuffle_ps(vMax, vMax, 0x00) * cumulativeMatrix[0];
    yRow[0] = _mm_shuffle_ps(vMin, vMin, 0x55) * cumulativeMatrix[1];
    yRow[1] = _mm_shuffle_ps(vMax, vMax, 0x55) * cumulativeMatrix[1];
    zRow[0] = _mm_shuffle_ps(vMin, vMin, 0xaa) * cumulativeMatrix[2];
    zRow[1] = _mm_shuffle_ps(vMax, vMax, 0xaa) * cumulativeMatrix[2];

    __m128 zAllIn = _mm_castsi128_ps(_mm_set1_epi32(~0));

    for(UINT i = 0; i < AABB_VERTICES; i++)
    {
        // Transform the vertex
        __m128 vert = cumulativeMatrix[3];
        vert += xRow[sBBxInd[i]];
        vert += yRow[sBByInd[i]];
        vert += zRow[sBBzInd[i]];

        // We have inverted z; z is inside of near plane iff z <= w.
        __m128 vertZ = _mm_shuffle_ps(vert, vert, 0xaa); //vert.zzzz
        __m128 vertW = _mm_shuffle_ps(vert, vert, 0xff); //vert.wwww
        __m128 zIn = _mm_cmple_ps(vertZ, vertW);
        zAllIn = _mm_and_ps(zAllIn, zIn);

        // project
        xformedPos[i] = _mm_div_ps(vert, vertW);
    }

    // return true if and only if none of the verts are z-clipped
    return _mm_movemask_ps(zAllIn) == 0xf;
}

Admittedly, quite a bit longer than the original one, but that’s because we front-load a lot of the computation; most of the per-vertex work done in TransformCoords is gone. And here’s our reward:

Change: Get rid of per-box vertex list

Depth test min 25th med 75th max mean sdev
Start 1.109 1.152 1.166 1.182 1.240 1.167 0.022
Transform fixes 1.054 1.092 1.102 1.112 1.146 1.102 0.016
IsTooSmall cleanup 0.860 0.893 0.908 0.917 0.954 0.905 0.018
Reduce state 0.834 0.862 0.873 0.886 0.938 0.875 0.017
Remove vert list 0.801 0.823 0.830 0.839 0.867 0.831 0.012

This brings our total for this post to a nearly 25% reduction in median depth test time, plus about 320 bytes memory reduction per TransformedAABBoxSSE – which, since we have about 27000 of them, works out to well over 8 megabytes. Such are the rewards for widening the scope beyond optimizing functions by themselves.

And as usual, the code for this time (plus some changes I haven’t discussed yet) is up on Github. Until next time!

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8 Comments
  1. Nitpicker permalink

    Just a small nitpick: You have a code comment

    // return true if and only all verts inside near plane

    I think it should be

    // return true if and only if all verts inside near plane

  2. Is there any reason for this chaotic order of cube verticies?

    // 0 = use min corner, 1 = use max corner
    static const int sBBxInd[AABB_VERTICES] = { 1, 0, 0, 1, 1, 1, 0, 0 };
    static const int sBByInd[AABB_VERTICES] = { 1, 1, 1, 1, 0, 0, 0, 0 };
    static const int sBBzInd[AABB_VERTICES] = { 1, 1, 0, 0, 0, 1, 1, 0 };
    

    I found it very hard to verify at a glance that this contains all possible 3-bit patterns in the columns. Why not ordering them as 3-digit binary numbers? That would be a lot easier to recognize and to check:

    static const int sBBxInd[AABB_VERTICES] = { 1, 1, 1, 1, 0, 0, 0, 0 };
    static const int sBByInd[AABB_VERTICES] = { 1, 1, 0, 0, 1, 1, 0, 0 };
    static const int sBBzInd[AABB_VERTICES] = { 1, 0, 1, 0, 1, 0, 1, 0 };
    

    Would that work, or would that introduce some problem I’m not aware of?

    • The reason is that this is the order the original code used – see the initialization for mpBBVertexList I show in the article. You can of course use any vertex order you want, but changing it also involves changing the index list for the box (the list of which vertices make up which triangles). For this series I’m trying to keep the changes as small and focused as possible, so no renumbering as long as it doesn’t provide any serious advantages. :)

      That said, in my own code I tend to be fairly obsessive about this kind of thing. I always number box vertices such that 0=(minX,minY,minZ), 1=(maxX,minY,minZ), 2=(minX,maxY,minZ), …, 7=(maxX,maxY,maxZ). Easy to generate in code ((ind & 2) ? maxY : minY and the like) and I never have to think about it.

  3. Hey Fabian,

    Sweet!

    I suspect we can beat a rasterizer for testing the bounding boxes. Surely there’s a ray-intersect-box test that’s faster than rasterizing (up-to) six triangles. Even using a bbox-v-frustum test would be faster (i.e., the pixel view is a tiny frustum) The math surely simplifies if we degenerate the frustum to a line.

    At the least, we could rasterize quads instead of triangles – quad coverage test is similar to triangle coverage test (is point inside four edges instead of point inside three edges). And, since our quads are always planar, (I think) the interpolated Z is given from barycentrics of either of the quad’s triangles (i.e., point is on the triangle’s plane, but outside the boundaries. I think the barycentrics provide extrapolation in that case)

    • Yes, a Kay-Kajiya slab-based bounding box test should be fairly cheap per pixel. I haven’t implemented this though; I’m still trying to avoid changes that would produce different results, even if only slightly, although I have deviated from that on occasion (e.g. the incremental z-stepping). I might look into it later. :)

      Quads, I did fully implement weeks ago, and didn’t write about, not least because it’s a big change that’s a bit tricky to describe. But as usual, the devil’s in the details. Specifically, this is a place where the lack of subpixel correction is really problematic.

      Here are the two primary issues I ran into: yes, you can extrapolate from the barycentrics of one triangle just fine in general, but the problem is that the interpolation setup uses the pixel-snapped vertex coordinates, which introduces quite a bit of wobble, especially once the triangle you’re using as interpolation reference gets small. It also means that the planes determined by the two triangles of a quad really are quite a bit different. You could do a setup computation using the original vertex coordinates, but now you’re using barycentric coordinates for the wrong triangle to do the interpolation. You can compensate for that too, which basically boils down to adding a transform from the snapped to the un-snapped triangle, but it sure does get ugly. The right solution is to actually use proper sub-pixel accuracy.

      The second problem is related: once a quad gets small, you can get into a situation where the quad is non-degenerate, but one of the two triangles making it up is. Which is a serious problem if that’s the triangle you happened to use for interpolation setup. Again, you can detect that case and correct for it, but it sure ruins the simplicity of the algorithm quite thoroughly.

      Having proper sub-pixel would make both of these lots better, and using a consistent fill rule would help more still (for example, right now any tri that has a bounding box smaller than 2×2 pixels is automatically degenerate), but doing so requires “changing the rules” quite a bit from what I’ve done so far. Which is why I didn’t write about it. :)

  4. Nitpick: there seems to be some HTML markup mishap with

    // If W 1 (for W>0), …

    Was that an unescaped less-than sign that got removed?

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