Converting the general equation into a parametrized equation

Answer 1

In web novels, it was common to encounter situations where a general equation needed to be converted into a parametrized equation. This transformation not only made it easier for readers to understand, but also increased the interest and interaction of the article. Generally speaking, the method of transforming a general equation into a parametrized equation mainly depended on the specific form of the equation. The following are some common situations and conversion methods:

###Straight line equation

** General equation **:$Ax + By + C = 0$

** Param equation **: You can choose a param $t$and then represent the functions of $x$and $y$as $t$. For example, if the slope of a straight line exists, you can set $x = x_0 + t\cos\beta $,$y = y_0 + t\sin\beta $, where $(x_0, y_0)$is a point on the straight line, and $\beta $is the tilt angle of the straight line.

###The equation of a circle

** General equation **:$(x-a)^2 + (y-b)^2 = r ^2 $

** The equation of parameters **:$x = a + r ,$y = b + r , where $t$is the parameters representing the angle.

###Elliptic equation

** General equation **:$/frac{x ^2}{a ^2} +/frac{y ^2}{b ^2} = 1$

** The equation of parameters **:$x = a\cost $,$y = b\sin t$, where $t$is the parameters.

###Parabola equation

** General equation **:$y ^2 = 4px$(Take the right opening as an example)

** Paramenter equation **: You can choose $t$as the argument, let $y = 2pt$, then $x = t ^2 $.

###Hyperbola equation

** General equation **:$/frac{x ^2}{a ^2} -/frac{y ^2}{b ^2} = 1$

[** Parameric equation **: It can be expressed using a hyperbolic function, such as $x = a\cosh t$,$y = b\sinh t$.]

###Illustration

Suppose a character in a web novel needs to move along a specific path, which can be described by the general equation $y = x ^2 $. In order to increase the dynamic of the story, the author might choose to transform it into a mathematical equation.

** General equation **:$y = x ^2 $

** Paramenter equation **: You can choose $t$as the parameters, let $x = t$, then $y = t ^2 $. In this way, the position of the character could be described according to the change of $t$.

In general, to convert a general equation into a parametrized equation, one needed to choose the appropriate parameters and representation method according to the specific form of the equation. This transformation not only enhanced the visual effect of the online novel, but also made the readers more invested in the development of the story.

The Extraordinary Ordinary Life novel is equally exciting. Everyone is welcome to click and read it!