矢 (幾何)

圓弧的矢、弦和半徑

圓弧(sagitta,有時縮寫成sag[1])或弓形高[2]是指該圓弧對應的之中點到之中點的距離[3]。 在建築學中,矢廣泛用於計算跨越一定高度和距離所需的弧度,並且在光學中用於評估球面鏡或透鏡的厚度。矢的英文sagitta來自拉丁文sagitta意思是「箭」。

三角函數的正矢函數正是得名於[4][5],在割圓八線中,矢對應到正矢

矢與弓形高是相似的概念,差別僅在矢專指一個弧中點到弧兩端連線之中點的那條線,而弓形高是弓形的高。矢與弧相關,而弓形高與弓形相關。

公式

在下列等式中,代表矢(弓形高),為圓的半徑,為圓弧兩端點的距離,也就是弦長。其中半弦和弓心距正好是直角三角形的兩條邊,半徑剛好是其斜邊,根據勾股定理則有:

由此可以推導出矢、弦和半徑的關係式:

矢也可以透過正矢函數來計算出來。若圓弧對應的圓心角為Δ,令Δ = 2θ,則矢為:

近似值

當矢相對於半徑很小時,可以使用以下公式來近似計算[3]

或者,如果矢長(弓形高)很小,且已知矢長、半徑和弦長,則可以透過以下公式來估計計弧長:

其中,a弧長。這個公式為中國數學家沈括所知,兩個世紀後,郭守敬提出了一個更準確的公式。[6]

參見

參考文獻

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