§ 3 cylindrical equations and Cylindrical Coordinates

§ 3 cylindrical equations and Cylindrical Coordinates

ICylindrical equation of the busbar parallel to the coordinate axis

1Definition: A straight line l is always parallel to a certain direction V during motion ., And the total intersection with a curve c, then the trajectory of l is calledCylindrical, Where V. -- The orientation of the cylinder, c -- the quasi-line of the cylinder, and any position of l -- the bus of the cylinder.

2Equations and features:

Theorem: In the spatial coordinate system, F (x, y, z) = 0 is a bus, and the equation of the cylindrical parallel to the z axis <strong> is the same as that of x, the binary equation f (x, y) of y = 0

Certificate:"Cosine>" let's set the ternary equation F (x, y, z) as an equation of the column Σ parallel to the z axis, then the section c: <cosine> 〉

F (x, y) ≡ F (x, y, 0) indicates M (x, y, z) the coordinates of point x (x, y) = 0 and coordinate f (x, y) = 0 are the equations of Σ, so that f (x, y, z) = 0 and

F (x, y) = 0.

"<Strong" if F (x, y, z) = 0 is the same as f (x, y) = 0, take the c: line as the standard, if the cylinder parallel to the z axis is Σ, it can be proved that M (x, y, z) ε Σ <cosine> M coordinates meet f (x, y) = 0

∴ F (x, y) = 0 indicates the cylindrical Σ, and F (x, y, z) = 0 also indicates the cylindrical Σ

 

Example:In the Cartesian coordinate system, the following figure shows the cylindrical surface, hyperbolic cylindrical surface, plane, and parabolic cylindrical surface:

(Fig. 2.4)

(Fig. 2.5)

(Figure 2.6) (Figure 2.7)

 

IICylindrical Coordinates:

1Parameter Equation of the cylindrical surface:

Set the center axis of Σ on the cylindrical surface to coincide with the Z axis, radius = R

For P ε Σ, the projection of P on x. y is P ′

θ = percentile (I, OP '), then

R = + = rcos θ I + rsin θ J + UK ---- vector type parameter equation

0 θ <2 π, ∣ U equation <------ coordinate equation

2Definition: After the Cartesian coordinate system is established in the space, for M (x, y, z), set its distance to the Z axis to P, then M falls into the central axis with the Z axis, returns the cylindrical surface with the radius of P, θ, U

(*)

On the contrary, for the given P (p ≥ 0), θ (0 ≦ θ <2 π), u (∣ u then <), according to the (*) Formula

It can also be determined that a point of M (x, y, z) in the space is called the cylindrical coordinates of the order three arrays P, θ, and U as M points, u)

Note: The points in the 1 ° Space do not correspond to the cylindrical coordinates one by one.

Use (*) to obtain the Cartesian coordinates of a 2 ° curved Cylindrical Coordinate. Use the Cartesian coordinates to obtain the cylindrical coordinates.