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Picture here I unwrap UV from sphere mesh to grid and put texture to align with UV grid too. When white and black border of texture fall between UV columns, I get jagged edge like in the picture. This only happens with double-curve mesh. I want to know what cause this to happens?

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It looks like distortion because the trapezoids are rendered as two triangles by the GPU. This leads to incorrect UV interpolation.

Try this: draw a horizontally symmetrical trapezoid with the top being much shorter than the bottom. Then draw one of its diagonals to split it into two triangles. Place a mark at the midpoint of that diagonal. It should be obvious the mark is not at the horizontal center of the trapezoid. Yet that mark is where the middle of the interpolated texture coordinates will be. Your border will go through the mark but also through the center of the top and bottom edges.

You should not have that problem when producing a final image with any decent renderer.

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