Curves a great. Curves are attractive. Most importantly curves are a key to design work. At Pratt Institute we learned about different curve types in our first 3D Design class at the Freshman year.
I find that curves can make or break a design or a piece of artwork for that matter. The most important part about curves is when used they should always be done perfectly. Nothing weakens a piece as much as a uncertain, wobbly curve (I am guilty of this myself) so make sure your curves are tight!
I put together this little overview of the different curves which you can see above.
The neutral curve: The neutral curve is likely the most well-known. It is the curve you find within circles. The neutral curve will always form a circle eventually.
The Ellipse: Another very common curve, the image above pretty much says all about it.
The Spiral: Perhaps the most popular. Spirals have throughout history been an element in many cultures art.
Parabola: The parabola curve, which can often be found in graphs was also the inspiration for the St. Louis Arch by E. Saarinen
Hyperbola: At first glance it looks like an upside down parabola, however the hyperbola is tighter than the parabola. It is also often found in graphs.
Catenary: The best way to describe the catenary curve is to imagine a necklace hold up by each end. The way it hangs is the catenary curve. It is a very beautiful curve that has been used for many bridges amongst other things.
The S-Curve: The name says it all.
The Trajectory Curve: The trajectory curve occurs when an object is sent into orbit. It is also a very beautiful curve used in many floral designs.
When working with art you will find that both line quality and curve quality make for very powerful designs. So use your curves but use them right.