Add documentation for barycentric coordinates
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@ -143,21 +143,22 @@ FILE_SCOPE void DrawTriangle(PlatformRenderBuffer *const renderBuffer,
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}
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/*
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NOTE(doyle): Given two points that form a line and an extra point
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to test, we can determine whether a point lies on the line, or is
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to the left or right of a the line.
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/////////////////////////////////////////////////////////////////////////
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// Rearranging the Determinant
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/////////////////////////////////////////////////////////////////////////
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Given two points that form a line and an extra point to test, we can
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determine whether a point lies on the line, or is to the left or right of
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a the line.
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First forming a 3x3 matrix of our terms and deriving a 2x2 matrix
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by subtracting the 1st column from the 2nd and 1st column from
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the third.
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First forming a 3x3 matrix of our terms and deriving a 2x2 matrix by
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subtracting the 1st column from the 2nd and 1st column from the third.
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| ax bx cx | | (bx - ax) (cx - ax) |
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| ax bx cx | | (bx - ax) (cx - ax) |
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m = | ay by cy | ==> | (by - ay) (cy - ay) |
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| 1 1 1 |
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| 1 1 1 |
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From our 2x2 representation we can calculate the determinant
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which gives us the signed area of the triangle extended into
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a parallelogram.
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From our 2x2 representation we can calculate the determinant which gives
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us the signed area of the triangle extended into a parallelogram.
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det(m) = (bx - ax)(cy - ay) - (by - ay)(cx - ax)
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@ -167,38 +168,77 @@ FILE_SCOPE void DrawTriangle(PlatformRenderBuffer *const renderBuffer,
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- CW and c(x,y) is outside the triangle, the signed area is positive
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- CW and c(x,y) is inside the triangle, the signed area is negative
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NOTE(doyle): The det(m) can be rearranged if expanded to be
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SignedArea(cx, cy) = (ay - by)cx + (bx - ay)cy + (ax*by + ay*bx)
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/////////////////////////////////////////////////////////////////////////
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// Optimising the Determinant Calculation
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/////////////////////////////////////////////////////////////////////////
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The det(m) can be rearranged if expanded to be
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SignedArea(cx, cy) = (ay - by)cx + (bx - ay)cy + (ax*by - ay*bx)
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When we scan to fill our triangle we go pixel by pixel, left to right,
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bottom to top, notice that this translates to +1 for x and +1 for y, i.e.
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The first pixel's signed area is cx, then cx+1, cx+2 .. etc
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SignedArea(cx, cy) = (ay - by)cx + (bx - ax)cy + (ax*by + ay*bx)
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SignedArea(cx+1, cy) = (ay - by)cx+1 + (bx - ax)cy + (ax*by + ay*bx)
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SignedArea(cx, cy) = (ay - by)cx + (bx - ax)cy + (ax*by - ay*bx)
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SignedArea(cx+1, cy) = (ay - by)cx+1 + (bx - ax)cy + (ax*by - ay*bx)
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Then
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SignedArea(cx+1, cy) - SignedArea(cx, cy) =
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(ay - by)cx+1 + (bx - ax)cy + (ax*by + ay*bx)
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- (ay - by)cx + (bx - ax)cy + (ax*by + ay*bx)
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(ay - by)cx+1 + (bx - ax)cy + (ax*by - ay*bx)
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- (ay - by)cx + (bx - ax)cy + (ax*by - ay*bx)
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= (ay - by)cx+1 - (ay - by)cx
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= (ay - by)(cx+1 - cx)
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= (ay - by)(1) = (ay - by)
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Similarly when progressing in y
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SignedArea(cx, cy) = (ay - by)cx + (bx - ay)cy + (ax*by + ay*bx)
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SignedArea(cx, cy+1) = (ay - by)cx + (bx - ay)cy+1 + (ax*by + ay*bx)
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SignedArea(cx, cy) = (ay - by)cx + (bx - ay)cy + (ax*by - ay*bx)
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SignedArea(cx, cy+1) = (ay - by)cx + (bx - ay)cy+1 + (ax*by - ay*bx)
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Then
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SignedArea(cx, cy+1) - SignedArea(cx, cy) =
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(ay - by)cx + (bx - ax)cy+1 + (ax*by + ay*bx)
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- (ay - by)cx + (bx - ax)cy + (ax*by + ay*bx)
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(ay - by)cx + (bx - ax)cy+1 + (ax*by - ay*bx)
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- (ay - by)cx + (bx - ax)cy + (ax*by - ay*bx)
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= (bx - ax)cy+1 - (bx - ax)cy
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= (bx - ax)(cy+1 - cy)
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= (bx - ax)(1) = (bx - ax)
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Then we can see that when we progress along x, we only need to change by
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the value of SignedArea by (ay - by) and similarly for y, (bx - ay)
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/////////////////////////////////////////////////////////////////////////
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// Barycentric Coordinates
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/////////////////////////////////////////////////////////////////////////
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At this point we have an equation that can be used to calculate the
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2x the signed area of a triangle, or the signed area of a parallelogram,
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the two of which are equivalent.
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det(m) = (bx - ax)(cy - ay) - (by - ay)(cx - ax)
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SignedArea(cx, cy) = (ay - by)cx + (bx - ay)cy + (ax*by - ay*bx)
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A barycentric coordinate is some coefficient on A, B, C that allows us to
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specify an arbitrary point in the triangle as a linear combination of the
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three usually with some coefficient [0, 1].
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The SignedArea turns out to be actually the barycentric coord for c(x, y)
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normalised to the sum of the parallelogram area. For example a triangle
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with points, A, B, C and an arbitrary point P inside the triangle. Then
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SignedArea(P) with vertex A and B = Barycentric Coordinate for C
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SignedArea(P) with vertex B and C = Barycentric Coordinate for A
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SignedArea(P) with vertex C and A = Barycentric Coordinate for B
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B
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/ \
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/ \
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/ P \
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/_______\
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A C
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This is normalised to the area's sum, but we can trivially turn this into
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a normalised version by dividing the area of the parallelogram, i.e.
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BaryCentricA(P) = (SignedArea(P) w. vertex C and B)/SignedArea(of the orig triangle)
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BaryCentricB(P) = (SignedArea(P) w. vertex A and C)/SignedArea(of the orig triangle)
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BaryCentricC(P) = (SignedArea(P) w. vertex A and B)/SignedArea(of the orig triangle)
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*/
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DqnV2i scanP = DqnV2i_2i(min.x, min.y);
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@ -263,7 +303,6 @@ FILE_SCOPE void DrawText(PlatformRenderBuffer *const renderBuffer,
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stbtt_aligned_quad alignedQuad = {};
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stbtt_GetPackedQuad(font.atlas, font.bitmapDim.w, font.bitmapDim.h,
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charIndex, &pos.x, &pos.y, &alignedQuad, true);
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stbtt_packedchar *charData = font.atlas + charIndex;
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DqnRect fontRect = {};
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fontRect.min = DqnV2_2f(alignedQuad.s0 * font.bitmapDim.w, alignedQuad.t1 * font.bitmapDim.h);
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@ -279,7 +318,13 @@ FILE_SCOPE void DrawText(PlatformRenderBuffer *const renderBuffer,
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u8 *fontPtr = font.bitmap + fontOffset;
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DQN_ASSERT(sizeof(u32) == renderBuffer->bytesPerPixel);
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f32 fontHeightOffset = charData->yoff2 + charData->yoff;
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// NOTE(doyle): This offset, yOffset and flipping t1, t0 is necessary
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// for reversing the order of the font since its convention is 0,0 top
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// left and -ve Y.
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stbtt_packedchar *const charData = font.atlas + charIndex;
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f32 fontHeightOffset = charData->yoff2 + charData->yoff;
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u32 screenOffset = (u32)(screenRect.min.x + (screenRect.min.y - fontHeightOffset) * renderBuffer->width);
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u32 *screenPtr = ((u32 *)renderBuffer->memory) + screenOffset;
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