The simplest polar curves simply leave one of the variables constant and then vary the other. Keeping r constant and varying theta creates a circle centered at the origin of the graph with radius r. Keeping theta constant and varying r generates a straight line that passes through the origin of the graph.
If r and theta are made to be proportional, r = theta * 5, for instance, the spiral of Archimedes results.
The conic sections are any curve that can be created by intersecting a cone with a plane. The curve results from tracing the perimeter of the cone on the plane. The conic sections are the circle, parabola, ellipse and hyperbola. All of the conic sections can be expressed as equations relating r and theta so they are all polar curves.
Trigonometric relations between r and theta can create surprising and sometimes beautiful shapes. The equation r = cos(n * theta), where n is a constant, is called the rose because it has symmetric petals around the center of the graph. If n is an odd integer it will have an odd number of petals, if it is an even integer the rose will have an even number of petals. If n is a fraction, then instead of the petals the graph will have intersecting circles and loops. And if n is irrational, the number pi for instance, then it will have an infinite number of petals, creating a dense curve.
The butterfly is a polar curve that is famous because it looks so much like its name. The complicated equation, involving multiple trigonometric functions, including one as an exponent, creates a polar curve with seven distinct sections. Six symmetric sections looks like the wings of the butterfly while the seventh appears to form the head.
Not all polar curves are famous for how they look. The Maclauring trisectrix served in the study of angle trisection, how to split an angle into three equal parts. Angle trisection has been an important geometric topic since ancient times, and while the Maclauring trisectrix looks like a simple loop, it furthered understanding of this topic.