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一、引言复数是用来表达平面上点的位置的数:z=x++(-1)~(1/2)y,x,y是实数,(x,y)即是点的笛儿直角坐标,或z=ρe~(iθ),ρ,θ是实数(i-(-1)~(1/2),(ρ,O)乃是点的极坐标。把一个数乘上z=ρe~(iθ),就是把这个数所表达的点沿这点与坐标原点的联线伸縮ρ倍,并从这联线起按反时針方向旋轉一个角度θ;把一个数加上复数z=x+iy,就是把这个数所表达的点沿横軸移动有向距离x,沿纵軸移动有向距离y。这样,利用复数的运算,初等平面几何上的許多定理可以化简其証明。同时,通过复数的运用可以对初等平面几何作概括的叙述,如全等形的理論是討論簡单图形在刚体运动(平移和旋轉)z→az+b(这里|a|=1)下不变的性貭,相似形的理論是討論在变換z→az+b(a,b是任意复数) 下不变的性貭。掌握了这些变換,不但能对初等平面几何学以簡叙繁,而且对复数的了解也更深刻。二、初等几何变換簡介变換理論是几伺作图的主要依据。如果借助于任何規則或规律对于某个图形,的每一个点A,在某个图形F＇有一个确定的点B与之对应,那么我們說,图形F被变換到图形F＇。Ⅰ.合同变換 假設有一个图形F,經过某种变換而变为与自己合同的图形F＇,那么这个变換叫做合同变換。合同变換分下列三种:
I. INTRODUCTION The plural is the number used to express the position of a point on a plane: z = x + + (-1) ~ (1/2) y, where x, y are real numbers, and (x, y) is the Cartesian coordinate of the point. , Or z = ρe ~ (iθ), ρ, θ is a real number (i-(-1) ~ (1/2), (ρ, O) is the polar coordinates of the point. Multiply a number by z = ρe ~ (iθ) is to scale the point expressed by this number along the line of this point with the origin of the coordinate by ρ times, and rotate it by an angle θ counterclockwise from this line; add a number to the number z=x +iy is to move the point expressed by this number along the horizontal axis with the directed distance x and along the vertical axis with the directed distance y.In this way, many theorems in elementary plane geometry can be used to simplify the proof using complex operations. The generalized geometry of elementary plane geometry can be described by the use of complex numbers. For example, the theory of congruent shapes is to discuss that simple graphs do not change under rigid body motion (translation and rotation) z→az+b (here |a|=1). The theory of similarity is to discuss the invariant nature of the transformation z → az + b (a, b is any complex number). Mastering these transformations can not only simplify the geometry of elementary planes. , And more profound understanding of the plural. Second, the introduction of elementary geometric transformation transformation If you use any rule or rule for each point A of a graph, and a certain point B corresponds to a graph F’, then we say that graph F is Transformation to the graph F’ I. Contract transformation Assume that there is a graph F, which after some transformation becomes a graph F’ with itself contract, then this transformation is called contract transformation. Contract transformation is divided into the following three Kind: