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Create Gradient Mesh and a Mesh tool click estimated each border edge's handles from the surface's derivatives, so the mesh's border cut across a curved outline: a 1x1 mesh over a 400 x 300 pt ellipse came out up to 7 pt off it, with corners at its four mesh points. Each border edge now takes the handles of the cubic that best fits the outline between its points, and handles where two border edges meet along a side stay collinear. Deleting a mesh line (Alt-click with the Mesh tool) joined the two edges on either side with a straight line, which turned the ellipse's border into a diamond. The joined edge now keeps the outer handles, grown back to the length they had before the line split the edge.