![]() ![]() Dividing by 2 makes it a little prettier:ĭc^2 (x-a)^2 - 2dcs (x-a)(y-b) ds^2 (y-b)^2 s(x-a) c(y-b) = 0. 3 point equation: The equation of a parabola passing through 3 two-dimensional points equation: a statement declaring two mathematical expressions are equal. Standard Form Equation The standard form of a parabola's equation is generally expressed: y a x 2 b x c The role of 'a' If a > 0, the parabola opens upwards If a < 0 it opens downwards. To translate that to have its "vertex" at a point $(a, b)$, you could do a similar matrix trick, or something far simpler:Ģdc^2 (x-a)^2 - 4dcs (x-a)(y-b) 2ds^2 (y-b)^2 2s(x-a) 2c(y-b) = 0Īnd that's your answer. In order to give a picture of the spatial focusing let us consider that surface waves are produced by a parabolic wave maker. The equation of a parabola can be expressed in either standard or vertex form as shown in the picture below. The good news is that if you're willing to write a point $(x, y)$ as a vector $v = \begin = 0.Ģdc^2 x^2 - 4dcs xy 2ds^2 y^2 2sx 2cy = 0Īnd you can substitute in $\cos t$ for $c$ and $\sin t$ for $s$. Where $y'$ and $x'$ are coordinates of a point in a rotated and transalted coordinate system. Tool to find the equation of a function from its points, its coordinates x, yf(x) according to some interpolation methods and equation finder algorithms.
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