How to determine the parameter equation by traditional methods

The method of transforming space curve into parametric equation is as follows: Let the general equation of space curve be F(x, y, z)=0 and G(x, y, z)=0, so that any one of x, y or z can get a suitable parametric equation for simplification.

For example, z=f(t), and then bring back the general equation of F(x, y, z)=0, and get F 1(x, y)=f 1(t), g1(.

Polar coordinate is also a form of parametric equation. For example, let x=rcosθ and y=rsinθ in the curve, and get the parameter equation r=f(θ).

Mathematical parameter equation formula

1, the parametric equation of the circle

X=a+r, cosθy=b+r, sinθ(a, b) is the center coordinate, r is the radius of the circle, and θ is the parameter.

2. Parametric equation of ellipse

X=a, cosθy=b, sinθa is the length of the long semi-axis, b is the length of the short semi-axis, and θ is the parameter.

3. Parametric equation of hyperbola

X=a, secθ (secθ) y=b, tanθa is the real semi-axis length, b is the imaginary semi-axis length, and θ is the parameter.

4. Parametric equation of parabola

X = 2pt 2, y=2pt, p represents the distance from the focus to the directrix, and t is the parameter.

5. Parametric equation of straight line

X=x'+tcosa, y=y'+tsina, x', y' and a represent straight lines passing through (x', y'), the inclination angle is a, and t is the parameter.