To make an extended double crochet stitch: Yarn over once, insert hook, yarn over and draw a loop through one loop. Yarn over and draw through two loops. Yarn over and again draw through two loops.
This swatch of edc looks like a swatch of double crochet, but it's actually a step taller than the double crochet, making its height fit between the double and treble crochet stitches.
(From left to right) Double crochet, extended double crochet, and treble crochet.
Extended stitches could easily make a smoother line. (i.e. If you used the pattern of: sc, dsc, dc, edc, tr you would have a smooth curve instead of a stepped or zig zag one.