2. Spherical equations and spherical coordinates

2. Spherical equations and spherical coordinates

ISpherical Equation

1Definition: In the space Cartesian coordinate system, equation (X-a) ² + (Y-B) ² + (Z-C) ² = R (R is a real number)

The representation of the image is called a (generalized) sphere, where (a, B, c) is called its center.

It is called its radius.

It is not hard to see that the generalized sphere includes a common sphere, one point and a virtual sphere.

2Feature of the equation

Theorem:In the spatial Cartesian coordinate system, f (x, y, z) = 0 is the equation of a sphere 〉

This equation is the same as a ternary quadratic equation with the same square coefficient and the disappearance of cross product terms.

Certificate:"Linear>" f (x, y, z) = 0 is an equation of a sphere Σ. By definition, the Σ equation can be

Table

When F (x, y, z) = 0 is the same as their solutions, and it is equivalent to a square coefficient, cross Multiplication

The three-element quadratic equation with the product term disappearing.

"<=" Let f (x, y, z) = 0 be equal to the square coefficient of a square, and the three-element quadratic equation with the elimination of the cross product term (it may be set that the square coefficient is 1)

(1)

The same solution, that is

(2)

The same solution, while (2) indicates the spherical surface, and f (x, y, z) also indicates the spherical surface.

That is, f (x, y, z) = 0 is the equation of a ball surface.

IISpherical Coordinates (space Polar Coordinates)

Definition: After a Cartesian coordinate system is created in a space, set the cost om Limit = P for any M (x, y, z) Point in the space.

Then M is in the Sphere centered on O and with the radius of P, and thus there is a Phi, θ

(*)

On the contrary, we can determine a point of M (X, Y, y, z), we call the order three arrays P, Phi, θ as the M point spherical coordinate, recorded as M (p, Phi, θ)

Note: The points in the 1 ° Space do not have a one-to-one correspondence with their spherical coordinates.

2 ° known M point spherical coordinate, through (*) can obtain its Cartesian coordinate, and if known M Cartesian coordinate, through

(**)

The spherical coordinates can be obtained.