Version 0121+. Not tested on earlier builds.

I know you've had a couple of reports like this, and I see they have been marked
RESOLVED and DUPLICATE. Well, it's not RESOLVED.

The following code draws nothing:
beginShape();
curveVertex(10,20);
curveVertex(100,20);
curveVertex(50,100);
endShape(CLOSE);

The following code draws a curved triangle, like the above code should:
beginShape();
curveVertex(10,20);
curveVertex(100,20);
curveVertex(50,100);
curveVertex(10,20); //repeat 3 vertices
curveVertex(100,20);
curveVertex(50,100);
endShape(CLOSE);

With four vertices and no repetition, a small sliver is drawn. Through other tests, I have
determined that the first three curveVertex() commands aren't drawn. All vertices following
it draw correctly.

i don't think you're understanding the reference:
http://processing.org/reference/curveVertex_.html
the curve() function has additional details:
http://processing.org/reference/curve_.html
please read it again and if it still doesn't make sense, post a question
about it on the discussion board.