1st Step for All

Science: 천문학 Astronomy; 생물학 Biology; 화학 Chemistry; 지질학 Geology; 수학 Mathematics; 물리 Physics

Engineering: 항공우주 Aerospace; 농업 Agriculture; 생물 Biological; 화학 Chemical; 토목공학 Civil; 전기공학 Electrical; 환경 Environmental; 산업 Industrial; 해양 Marine; 재료 Materials; 기계 Mechanical; 원자력 Nuclear

Architecture: 건축 Architecture

Health: 건강보건 Health

ETC: 기타 ETC

현재 위치

Calculating the coordinates of an arbitrary point by the distances to three known points

작성자 Uploader : billy 작성일 Upload Date: 2019-11-13변경일 Update Date: 2020-02-19조회수 View : 210

Type1 Type2

If the coordinates of three points, P1 (x1, y1), P2 (x2, y2) and P3 (x3, y3), and the distances of an arbitrary point to each of these three points, d1, d2 and d3, are known, coordinates of the arbitrary point A (x, y) is obtained as follows.

The following relation is established between point P1 and point A.

d1 = ((x-x1)^2+(y-y1)^2)^(1/2)

d1^2 = (x-x1)^2+(y-y1)^2

= x^2-2*x1*x+x1^2 + y^2-2*y1*y+y1^2

① x^2-2*x1*x + y^2-2*y1*y = d1^2 - x1^2 - y1^2 = C1

Applying the same as above between P2 and A and between P3 and A, the following two equations are obtained.

② x^2-2*x2*x + y^2-2*y2*y = d2^2 - x2^2 - y2^2 = C2

③ x^2-2*x3*x + y^2-2*y3*y = d3^2 - x3^2 - y3^2 = C3

Let ② - ① = ④, then,

④ 2*(x1-x2)*x + 2*(y1-y2)*y = C2-C1

Let ③ - ① = ⑤, then,

⑤ 2*(x1-x3)*x + 2*(y1-y3)*y = C3-C1

To erase y, let ④ - ⑤*(y1-y2)/(y1-y3) = ⑥, then,

⑥ 2*(x1-x2)*x - 2*((x1-x3)*(y1-y2)/(y1-y3))*x = (C2-C1)-(C3-C1)*(y1-y2)/(y1-y3)

2*((x1-x2)-(x1-x3)*(y1-y2)/(y1-y3))*x = (C2-C1)-(C3-C1)*(y1-y2)/(y1-y3)

x = ((C2-C1)-(C3-C1)*(y1-y2)/(y1-y3))/(2*((x1-x2)-(x1-x3)*(y1-y2)/(y1-y3)))

= ((C2-C1)*(y1-y3)-(C3-C1)*(y1-y2))/(2*((x1-x2)*(y1-y3)-(x1-x3)*(y1-y2)))

To erase x, let ④ - ⑤*(x1-x2)/(x1-x3) = ⑦, then in a same way,

⑦ 2*(y1-y2)*y - 2*((y1-y3)*(x1-x2)/(x1-x3))*y = (C2-C1)-(C3-C1)*(x1-x2)/(x1-x3)

2*((y1-y2)-(y1-y3)*(x1-x2)/(x1-x3))*y = (C2-C1)-(C3-C1)*(x1-x2)/(x1-x3)

y = ((C2-C1)*(x1-x3)-(C3-C1)*(x1-x2))/(2*((y1-y2)*(x1-x3)-(y1-y3)*(x1-x2)))

▶ Data input

□ Check of the errors in distance to each point

※ 이 사이트는 광고수익으로 운영됩니다.

※ 빨간색 밑줄부분에 자료를 입력하세요.
※ Input data on red underlines.

No Reference

★ 로그인 후 즐겨찾기에 추가할 수 있습니다.
★ To make new formula or to add this formula in your bookmark, log on please.

1st Step for All

현재 위치

Calculating the coordinates of an arbitrary point by the distances to three known points

코멘트