Involutes

An involute is a curve generated by unwinding a taut string from a given curve, typically a circle. The end of the string traces the involute as it is unwound. 

Involutes can be drawn for a variety of base curves. Here are some common examples: 

1. Circle:

The most common base curve for drawing an involute. The involute of a circle is widely used in gear design.

2. Polygon:

Involutes can be drawn for regular polygons (e.g., triangles, squares) by unwinding a string from each side. The resulting curve will have a series of connected arcs.

3. Ellipse:

The involute of an ellipse is more complex, but it can still be traced by unwinding a string from the ellipse's perimeter.

4. Parabola:

An involute can be drawn for a parabolic curve, with the string unwinding from the parabolic arc.

5. Straight Line:

When the base curve is a straight line, the involute is a spiral curve, where the distance between turns of the spiral increases progressively.

6. Other Curves:

In theory, involutes can be drawn for any smooth curve, including hyperbolas, catenaries, and arbitrary curves, though the resulting involute might be complex.