User:Xovimat

email: xovimat@comcast.net

some of my web pages stored outside wikipedia.

A particular cause of congestion at a roundabout is when many motorists want to make a turn that crosses oncoming traffic. The two images below, which presume right-hand traffic, show how opposing drivers making left turns will cross each other's path twice in a roundabout, but will not cross paths at a conventional intersection.

Derivatives
Each first derivative below was calculated using implicit differentiation.

With $$y$$ as a function of $$x$$

 * $$\frac{dy}{dx} = \begin{cases}

\mbox{unbounded} & \mbox{when } y = 0 \land x \ne 0 \\ \pm1 & \mbox{when } y = 0 \land x = 0 \\ \frac{x(a^2 - 2x^2 - 2y^2)}{y(a^2 + 2x^2 + 2y^2)} & \mbox{when } y \ne 0 \end{cases}$$


 * $$\frac{d^2y}{dx^2} = \begin{cases}

\mbox{unbounded} & \mbox{when } y = 0 \land x \ne 0 \\ 0 & \mbox{when } y = 0 \land x = 0 \\ \frac{3a^6(y^2 - x^2)}{y^3(a^2 + 2x^2 + 2y^2)^3} & \mbox{when } y \ne 0 \end{cases}$$

With $$x$$ as a function of $$y$$

 * $$\frac{dx}{dy} = \begin{cases}

\mbox{unbounded} & \mbox{when } 2x^2 + 2y^2 = a^2 \\ \pm1 & \mbox{when } x = 0 \land y = 0 \\ \frac{y(a^2 + 2x^2 + 2y^2)}{x(a^2 - 2x^2 - 2y^2)}  & \mbox{otherwise } \end{cases}$$


 * $$\frac{d^2x}{dy^2} = \begin{cases}

\mbox{unbounded} & \mbox{when } 2x^2 + 2y^2 = a^2 \\ 0 & \mbox{when } x = 0 \land y = 0 \\ \frac{3a^6(x^2 - y^2)}{x^3(a^2 - 2x^2 - 2y^2)^3} & \mbox{otherwise } \end{cases}$$

Curvature
Once the first two derivatives are known, curvature is easily calculated:


 * $$\kappa = \pm3(x^2 + y^2)^{1/2}a^{-2}$$

the sign being chosen according to the direction of motion along the curve. The lemniscate has the property that the magnitude of the curvature at any point is proportional to that point's distance from the origin.