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在直角坐标系中,点与坐标是一一对应的。若方程F_1(x,y)=0与方程F_2(x,y)=0同解时,这两个方程就表示同一曲线;反之,表示同一曲线的两个方程也必同解。但在极坐标系中,一个点对应无数个坐标((-1)~kρ,kπ+θ),其中k∈Z。方程f_1(ρ,θ)=0与f_2(ρ,θ)=0若同解就表示同一曲线,但表示同一曲线的两个方程却不一定同解。如方程ρ=θ与p=2π+θ表示同一曲线,但方程并不同解。我们在极坐标中把表示同一曲线的方程称为等价方程。显然所有的同解方程都是等价方程。
In Cartesian coordinates, points and coordinates are in one-to-one correspondence. If the equation F_1(x,y)=0 and the equation F_2(x,y)=0 have the same solution, the two equations represent the same curve; conversely, the two equations representing the same curve must also be the same solution. However, in the polar coordinate system, a point corresponds to an infinite number of coordinates ((-1)~kρ, kπ+θ), where k∈Z. The equation f_1(ρ,θ)=0 and f_2(ρ,θ)=0 indicate the same curve if the solutions are the same, but the two equations representing the same curve do not necessarily share the same solution. If the equation ρ=θ and p=2π+θ represent the same curve, the equations do not have the same solution. We call equations representing the same curve in polar coordinates as equivalent equations. Obviously all equations of the same equation are equivalent equations.