Eraser tool»
The Eraser tool (Q key) is used to erase nodes and/or simplify paths:
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Clicking with the Eraser tool removes any nodes and handles that you click, while Shift-clicking will selectively not remove certain “strategic” nodes.
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Ctrl-drag to resize the Eraser (like adjusting the size of a brush or eraser in Photoshop).
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Ctrl-click a node, and then click another node on the same contour, and the Eraser will simplify the path between those two nodes: