Because? θ+sin？ θ= 1

ρ=x？ +y？

ρcosθ=x

ρsinθ=y

A parametric equation is very similar to a function: it consists of some numbers in a specified set, called parameters or independent variables, which determine the result of the dependent variable. For example, kinematics, the parameter is usually "time", and the result of the equation is speed, position and so on.

Generally speaking, in the plane rectangular coordinate system, if the coordinates x and y of any point on the curve are functions of a variable t:?

And for each allowable value of t, the point (x, y) determined by the equations is on this curve, then this equation is called the parametric equation of the curve, and the variable T connecting the variables x and y is called the parametric variable, which is called the parameter for short. Relatively speaking, the equation that directly gives the point coordinate relationship is called the constant equation.

Extended data:

In the proof of Cauchy mean value theorem, parametric equation is also applied.

Cauchy mean value theorem

If the functions f(x) and F(x) satisfy:

(1) is continuous on the closed interval [a, b];

(2) Derivable in the open interval (a, b);

(3) For any x∈(a, b), F'(x)≠0.

Then at least one zeta in (a, b) constitutes an equation.

[f (b)-f (a)]/[f (b)-f (a)] = f' (zeta)/f' (zeta) holds.

Cauchy concisely and strictly proved the basic theorem of calculus, namely Newton-Leibniz formula. He strictly proved Taylor formula with remainder by definite integral, expressed the area of curved trapezoid by differential and integral mean value theorem, and deduced the formulas of graphic area, curved surface area and solid volume between plane curves.

A parametric curve can also be a function of multiple parameters. For example, a parametric surface is a function of two parameters (s, t) or (u, v).

For example, a cylinder:

r(u，v)=[x(u，v)，y(u，v)，z(u，v)]=[acos(u)，asin(u)，v]

Parameter is the abbreviation of parameter variable. It comes from the study of sports and other problems. When a particle moves, its position must be related to time. That is to say, there is a functional relationship between the coordinates of mass x and y and time t, x=f(t) and y=g(t). The variable T in these two functions is a "participating variable" relative to the variables X and Y representing the geometric position of particles. Parameter variables in such practical problems are abstracted into mathematics and become parameters. The task of parameters in the parametric equation we have learned is to communicate the relationship between variables X, Y and some constants, which provides convenience for studying the shape and properties of curves.

When describing the law of motion with parametric equations, it is often more direct and simple than using ordinary equations. It is very suitable for solving a series of problems such as maximum voyage, maximum altitude, flight time or trajectory. For some important but complex curves (such as the involute of a circle), it is difficult or even impossible to establish their ordinary equations, and the listed equations are complex and difficult to understand.

Drawing curves according to equations is very time-consuming; However, it is often easy to indirectly relate two variables X and Y by using parametric equation, and the equation is simple and clear, and drawing is not too difficult.

References:

Baidu encyclopedia-parameter equation