what is the different between Positive curvature and negative curvature?

Question

Can somebody tell me, if i rotate the image,will the Positive and negative change?For example, shape just like "O" in the image coordinate system used in opencv, does the upper part and lower part have same sign? and how about rotate the "O",will the sign change?

Answer 1

If you are given a photograph of a curve $\gamma$ you may discern parts having curvatures of opposite signs, but you cannot tell which parts should be called "positively curved" and which parts "negatively curved". This is because the photograph does not tell you the sense of direction of the curve.

A parametrized curve $$\gamma:\quad s\mapsto{\bf z}(s)=\bigl(x(s),y(s)\bigr)\ ,$$ however, has a "sense of direction": the sense of increasing $s$. This "sense of direction" is encoded in the tangent vector $${\bf t}(s):=\bigl(\dot x(s),\dot y(s)\bigr)\ .$$ Assume, for simplicity, that $s$ denotes arc length along $\gamma$, and let $$\theta(s):={\rm arg}\bigl( {\bf t}(s)\bigr)$$ denote the polar angle of the unit vector ${\bf t}(s)$. Then $$\kappa(s):=\dot\theta(s)=\dot x\ddot y-\dot y\ddot x$$ represents the "angular velocity" with which ${\bf t}(s)$ turns along $\gamma$, and is called the curvature of $\gamma$ at the point ${\bf z}(s)$. This curvature has a sign: It is positive if ${\bf t}$ turns counterclockwise, and negative if ${\bf t}$ turns clockwise when $s$ increases.

The curvature is unchanged under a global rotation of $\gamma$, but changes its sign when $\gamma$ undergoes a reflection with respect to a line.