Giuseppe Insana

The tightest-curved section, out of all sections of curve that make a complete turn (360 degrees), contains an x-crossing point vs. not so.

The tightest-curved section, out of all continuous sections of curve that make a complete turn (360 degrees), contains an x-crossing point vs. not so.

The tightest-curved section, out of all continuous sections of curve that make a complete turn (360 degrees), contains an x-crossing point vs. not so.

Out of any continuous length of curve that makes a complete turn (360 degrees), the tightest-curved contains an x-crossing point vs. not so.

Of any section of the curve that makes a complete turn (360 degrees), the tightest-curved section contains an x-crossing point vs. not so.

All examples are connected smooth curves allowed to self-intersect that must curve in only one direction (starting at one end, either clockwise or counter-clockwise), i.e. there is no inflection point.

All examples are connected smooth curves allowed to self-intersect that must curve in only one direction (starting at one end, either clockwise or counter-clockwise).

All examples are connected smooth curves allowed to self-intersect that must curve in only one direction.

Of any section of the curve that makes a complete turn (360 degrees), the tightest-curved section contains an x-crossing point vs. vs. not so.

Of any section of the curve that makes a complete turn (360 degrees), the tightest-curved section crosses through itself vs. vs. not so.

Of any subsection of the curve that makes a complete turn (360 degrees), the tightest-curved section crosses through itself vs. vs. not so.

Of any subsection of the curve that makes a complete turn (360 degrees), the tightest-curved crosses through itself vs. vs. not so.

Of any subsection of the curve that makes a complete turn (360 degrees), the subsection with tightest (average) curvature crosses through itself vs. vs. not so.

Out of the subsections of the curve that make a complete turn (360 degrees), the subsection with tightest (average) curvature crosses through itself vs. vs. not so.

Out of any subsection of the curve that makes a complete turn (360 degrees), the subsection with tightest (average) curvature crosses through itself vs. vs. not so.

Out of any section of the curve that makes a complete turn 360 degrees, the section with tightest (average) curvature crosses through itself vs. vs. not so.

Out of all sections of the curve that turn 360 degrees, the one with tightest (average) curvature crosses through itself vs. vs. not so.

The 360-degree arc in the curve that has the tightest (average) curvature crosses through itself vs. vs. not so.

The tightest-curved section of the curve is a loop crossing through itself vs. vs. not so.

There a loop crossing through itself on the generally tighter-curved end of the curve vs. not so.

There a loop crossing through itself on the intuitively tighter-curved end of the curve vs. not so.

Curve intersects itself near the end with greater curvature vs. not so.

Curve intersects itself at the point of greatest curvature vs. not so.