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Sum of a sequence, Sum of lengths of connected straight lines perpendicular to rotated lines at a certain angle

작성자 Uploader : billy 작성일 Upload Date: 2019-11-19변경일 Update Date: 2020-02-19조회수 View : 1703

Sum of the lengths of the connected lines that are perpendicular to lines rotated at a constant angle around the original point.

If the length of the line OA1 is L, the length of line OA2 is L * cos(θ) and a the length of the line A1A2 is L * sin(θ). Using this relationship, we get :

With n=1, L : L*cos(θ) : L*sin(θ)
With n=2, L*cos(θ) : L*(cos(θ))^2 : L*cos(θ)*sin(θ)
With n=3, L*(cos(θ))^2 : L*(cos(θ))^3 : L*(cos(θ))^2*sin(θ)
With n=4, L*(cos(θ))^3 : L*(cos(θ))^4 : L*(cos(θ))^3*sin(θ)
  …
With n=k, L*(cos(θ))^(k-1) : L*(cos(θ))^k : L*(cos(θ))^(k-1)*sin(θ)

Then, the sum of lengths of lines is :

Sum = A1A2 + A2A3 + A3A4 + …

  = L*sin(θ)+L*cos(θ)*sin(θ)+L*(cos(θ))^2*sin(θ)+L*(cos(θ))^3*sin(θ)+…

This is equal to the sum of a geometric sequence of which the first term is L*sin(θ) and the common ratio is cos(θ).

From Sum = L*sin(θ)*(1-(cos(θ))^n)/(1-cos(θ)), if n → ∞, (cos(θ))^n = 0, then,

Sum = L*sin(θ)/(1-cos(θ))


*** 참고문헌[References] ***
Sum = L*sin(θ)/(1-cos(θ))
작성자의 수식그림이 없습니다. No picture for this formula
변수명 Variable 변수값 Value 변 수 설 명 Description of the variable


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