Difference between revisions of "Math bits"

Revision as of 12:10, 25 April 2012

The nearest point on an ellipse to a given point

An ellipse in the coordinates orientated alone its major and minor axes is given as,

$\begin{cases} \begin{array}{c} x=a\cos t\\ y=b\sin t \end{array} & 0\le t<2\pi\end{cases}$

The squared distance from a point on ellipse to a given point $(x, y)$ ,

$\begin{array}{rcl} s^{2}&=&(x-a\cos t)^{2}+(y-b\sin t)^{2}\\ &=&x^{2}+y^{2}+a^{2}\cos^{2}t+b^{2}\sin^{2}t-2xa\cos t-2yb\sin t \end{array}$

The stationary points at the zero points of its first order derivative of $t$ ,

$\frac{d(s^{2})}{dt}=-a^{2}\sin2t+b^{2}\sin2t+2xa\sin t-2yb\cos t=0$

This stationary condition is a quartic equation of $cos t$ . With variable change $u = cos t$ ,

$γ 2 u 4 - 2αγ u 3 + (α 2 + β 2 - γ 2) u 2 + 2αγ u - α 2 = 0$

where $α = 2 a x$ , $β = 2 b y$ , and $γ = 2(a 2 - b 2)$ .

For all solutions from the quartic equation, the minimum distance point is identified by the convex condition,

$\frac{d^{2}(s^{2})}{dt^{2}}=\gamma(1-2u^{2})+\alpha u+\frac{\beta^{2}u}{\alpha-\gamma u}>0$

modular function in LibreCAD

the standard glibc fmod(x, a) is not convenient here, since we need a modular function work the same way for both positive and negative numbers, instead, we use,

$remainder(x - \frac{a}{2},a)+\frac{a}{2}$

@@ Line 33: / Line 33: @@
 where <math>\alpha=2 a x</math>, <math> \beta=2 b y</math>, and <math>\gamma=2(a^2-b^2)</math>.
-For all solutions from the quartic equation, the minimum distance point is identified the convex condition,
+For all solutions from the quartic equation, the minimum distance point is identified by the convex condition,
 <math>

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Difference between revisions of "Math bits"

Revision as of 12:10, 25 April 2012

The nearest point on an ellipse to a given point

modular function in LibreCAD

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