When you don't want to use straight against curves, it is entirely possible, of course, to construct a great drawing out of just curves. But you have to be a little careful with them and use them right.
Curves that are parallel or even would either look like a vase or a macaroni noodle.
Both are lifeless. The basic key to drawing with curves is to avoid parallel curves or even curves. Use offset curves to create a drawing that has all the flow of nice curves without being lifeless. This section from "The Illusion of Life" pretty much sums it up:
As Frank and Ollie point out, nature already designed living forms in offset curves, making our jobs easier (I blurred out the racy parts for those of you who don't want to see that kind of stuff. Isn't Photoshop cool?)
For some reason this is one of those things that can be hard for our brains to comprehend easily. If I'm not careful, I find that in life drawing I tend to draw figures with the exact same curve on both sides of the body and it always looks mushy. I guess our brains have a natural tendency to even things out and make them balanced and static. So try to keep this in mind and train yourself to always avoid parallels in your drawing. Straight parallel lines in your drawing should be avoided as well - again, it tends to be lifeless. Even in the simplest of drawings, removing the parallel lines immediately improves the drawing.
This touches on yet another big secret of design - avoid symmetry. The human eye doesn't like symmetry. It's lifeless and boring. We'll talk about that more in a future post.
Fred Moore was a master of the offset curve and I think that's a big part of what gives his drawings their flow and rhythm. They're made of curves that flow into and oppose each other but they are never ever symmetrical.
Check out this one (recently brought to my attention by The Blackwing Diaries, of course). There's not a single straight line in the drawing (except, okay, maybe the one under her right arm). But nowhere are there any symmetrical curves in the drawing (other than the back of the chair). Every part of her body has a different curve on the left side of her body than it has on the right side. No shape of her body is symmetrical.
I never thought about it much before but it occurs to me that Walt Kelly never used straights much in his drawing either. He gets pretty good results with some offset curves and some nice thick and thin lines.
I'm going to make yet another rather rash and uneducated statement that just occurred to me and say that I think most Disney drawing is based on offset curves. I don't think there are too many Disney animators that use straights against curves very often - it's hard to animate with straight lines and keep the appearance of living forms. Straights are pretty rare in living forms (and even in man-made forms, if you think about it). So I would say most Disney drawing is based on offset curves, but I haven't ever thought about that before or done any research so I could be wrong!