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CoplanarPoints[{p1,p2,p3,p4,…,pn}]

点 p1,p2,p3,p4,…,pnが共面かどうかを調べる．

詳細
Details and Options Details and Options
例題  
例  
スコープ  
アプリケーション  
基本的なアプリケーション  
幾何  
特性と関係  
関連項目
関連するガイド
履歴
引用書式

CoplanarPoints

CoplanarPoints[{p1,p2,p3,p4,…,pn}]

点 p1,p2,p3,p4,…,pnが共面かどうかを調べる．

詳細

  • CoplanarPointsは線形従属としても知られている．
  • 点集合が同一平面上にあるかどうかを調べる際に使われることが多い．
  • CoplanarPoints[{p1,p2,p3,p4,…,pn}]は，点 p4,…,pnが点 p1，p2，p3を通る平面上にある場合はTrueを返す．
  • 点 p1，p2，p3，p4が共面であるとき，行列{p2-p1,p3-p1,p4-p1}の階数は2以下である．

例題

すべて開く すべて閉じる

例  (2)

点{0,0,0}, {1,1,-2}, {-1,2,-1}, {3,-4,1}は同一面上にある：

Wolfram Language code: pts = {{0, 0, 0}, {1, 1, -2}, {-1, 2, -1}, {3, -4, 1}};
Wolfram Language code: CoplanarPoints[pts]

点をプロットする：

Wolfram Language code: Graphics3D[{InfinitePlane[pts[[1 ;; 3]]], Red, Point[pts]}]

点{0,0,0}, {1,1,-2}と{-1,2,-1}を含む平面の方程式を求める：

Wolfram Language code: CoplanarPoints[{{0, 0, 0}, {1, 1, -2}, {-1, 2, -1}, {x, y, z}}]

スコープ  (4)

CoplanarPointsは二次元の点に使うことができる：

Wolfram Language code: CoplanarPoints[{{1, 2}, {3, 4}, {5, 6}}]

三次元の点：

Wolfram Language code: CoplanarPoints[{{1, 2, 1}, {3, 4, 1}, {5, 6, 1}, {7, 8, 1}}]

n 次元の点：

Wolfram Language code: CoplanarPoints[{{0, 0, 1, 1}, {1, 0, 1, 0}, {0, 1, 0, 1}, {0, 1, 1, 1}, {0, 0, 0, 1}}]

CoplanarPointsは数値座標に使うことができる：

Wolfram Language code: CoplanarPoints[{{1, 2}, {3, 4}, {5, 6}}]

記号座標：

Wolfram Language code: CoplanarPoints[{{1, a, 1}, {a, 2, 1}, {1, 2, 1}, {a, b, c}}]

座標集合上のCoplanarPoints：

Wolfram Language code: CoplanarPoints[{{1, 2, 1}, {3, 4, 1}, {5, 6, 1}, {7, 8, 1}}]

点のリスト：

Wolfram Language code: CoplanarPoints[{Point[{b, 0, 1}], Point[{a, 1, 1}], Point[{a, 3, 1}], Point[{a, b, c}]}]

複数の点：

Wolfram Language code: CoplanarPoints[Point[{{b, 0, 1}, {a, 1, 1}, {a, 3, 1}, {a, b, c}}]]

CoplanarPointsは大きい集合に使うことができる：

Wolfram Language code: pts = RandomPoint[Polygon[{{0, 0, 0}, {1, 0, 0}, {1, 1, 0}}], 10 ^ 6];
Wolfram Language code: CoplanarPoints[pts]//AbsoluteTiming

アプリケーション  (5)

基本的なアプリケーション  (4)

2つの点が原点を通る同一平面上にあるための条件を求める：

Wolfram Language code: CoplanarPoints[{{0, 0, 0}, {0, 1, 0}, {a, 2, b}, {c, 3, d}}]

明示的な例：

Wolfram Language code: {{a, 2, b}, {c, 3, d}} /. FindInstance[% && a ≠ 0, {a, b, c, d}, Reals, 3]

2Dの点は同じ平面上にある：

Wolfram Language code: CoplanarPoints[{{1, 2}, {3, 4}, {5, 6}}]
Wolfram Language code: CoplanarPoints[RandomPoint[Disk[], 5000]]
Wolfram Language code: CoplanarPoints[RandomReal[1, {5000, 2}]]

点集合を含む平面の方程式を求める：

Wolfram Language code: pts = {{0, 0, 0}, {1, 1, -2}, {-1, 2, -1}};
Wolfram Language code: CoplanarPoints[Append[pts, {x, y, z}]]

共面の点を描画する：

Wolfram Language code: pts = {{0, 0, 0}, {1, 0, 0}, {1, 1, 1}};
Wolfram Language code: CoplanarPoints[pts]
Wolfram Language code: Graphics3D[{InfinitePlane[pts], Red, PointSize[Large], Point[pts]}]

幾何  (1)

多面体の面は平面上にある：

Wolfram Language code: poly = OuterPolyhedron[Polyhedron[{{-1/2, -1/2, -1/2}, {-1/2, -1/2, 1/2}, {-1/2, 1/2, -1/2}, {-1/2, 1/2, 1/2}, {1/2, -1/2, -1/2}, {1/2, -1/2, 1/2}, {1/2, 1/2, -1/2}, {1/2, 1/2, 1/2}}, {{8, 4, 2, 6}, {8, 6, 5, 7}, {8, 7, 3, 4}, {4, 3, 1, 2}, {1, 3, 7, 5}, {2, 1, 5, 6}}]];
Wolfram Language code: faces = poly[[2]]
Wolfram Language code: CoplanarPoints[PolyhedronCoordinates[poly][[#]]]& /@ faces

特性と関係  (5)

PositivelyOrientedPointsは共面の点に対してはFalseを返す：

Wolfram Language code: pts = {{0, 0, 0}, {1, 1, -2}, {-1, 2, -1}, {3, -4, 1}};
Wolfram Language code: {CoplanarPoints[pts], PositivelyOrientedPoints[pts]}

NegativelyOrientedPointsは共面の点に対してはFalseを返す：

Wolfram Language code: pts = {{0, 0, 0}, {1, 1, -2}, {-1, 2, -1}, {3, -4, 1}};
Wolfram Language code: {CoplanarPoints[pts], NegativelyOrientedPoints[pts]}

共線の点は共面である：

Wolfram Language code: pts = RandomPoint[Line[{{0, 0, 0}, {1, 1, 1}}], 5];
Wolfram Language code: {CollinearPoints[pts], CoplanarPoints[pts]}

RegionMemberを使って点が共面かどうか調べる：

Wolfram Language code: RegionMember[Line[{{0, 0, 1}, {1, 1, 1}}], {2, 2, 1}]
Wolfram Language code: CoplanarPoints[{{0, 0, 1}, {1, 1, 1}, {2, 2, 1}}]

InfinitePlaneを使ってグラフィック画像を描画する：

Wolfram Language code: pts = {{0, 0, 0}, {1, 0, 0}, {1, 1, 1}};
Wolfram Language code: Graphics3D[{InfinitePlane[pts], Red, PointSize[Large], Point[pts]}]
Wolfram Research (2020), CoplanarPoints, Wolfram言語関数， https://reference.wolfram.com/language/ref/CoplanarPoints.html.

テキスト

Wolfram Research (2020), CoplanarPoints, Wolfram言語関数， https://reference.wolfram.com/language/ref/CoplanarPoints.html.

CMS

Wolfram Language. 2020. "CoplanarPoints." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CoplanarPoints.html.

APA

Wolfram Language. (2020). CoplanarPoints. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CoplanarPoints.html

BibTeX

@misc{reference.wolfram_2026_coplanarpoints, author="Wolfram Research", title="{CoplanarPoints}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/CoplanarPoints.html}", note=[Accessed: 16-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_coplanarpoints, organization={Wolfram Research}, title={CoplanarPoints}, year={2020}, url={https://reference.wolfram.com/language/ref/CoplanarPoints.html}, note=[Accessed: 16-June-2026]}