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The pin cushion shape may be created using the Effect>Warp>Bulge with negative value, first horizontally then vertically, or in reverse order. Positive value for actual bulging.
Bulge does not work as nicely with a triangle. Maybe new options have emerged since 10.
Found what I (at least) need....
Object -> Envelope Distort -> Make with Warp
On the Triangle try the Warp>Inflate and use a neg. number.
As far as I can see, all options result in different curvatures for horizontal/vertical and slanting sides, at least in 10, so that it will look funny for a triangle even if you use different valus for horizontal and vertical bend/whatever, including different curvatures at either end of slanting lines.
Has that changed?
To find the centroid (geometric center) of a triangle (or any other path, for that matter):
2. Paste In Front.
4. Snap a horizontal and a vertical Ruler Guide to the position of the points.
6.Use the intersection of the two Guides as the center of scale transformations on the points and/or the handles (Object>Select>Handles).
Here's a method for making your bulging triangle using Transform Effect:
After step 7, an alternative way to join the figure into one path is: Expand Appearance, Ungroup, Divide Pathfinder. (The Pathfinder will stupidly remove the fill and stroke.)
When will step 7 be released?
Edit: It appeared when I had posted, after being invisible. Strange.