{
  "aliases": [
    "reflection",
    "mirror-isometry"
  ],
  "backlinks": null,
  "body": "\n## Definition\n\nLet $a \\in \\mathbb{R}^n$ be a nonzero vector and $c$ a real number, and let $\\cdot$ be the dot product. Let $H = \\{\\, x \\in \\mathbb{R}^n : x \\cdot a = c \\,\\}$ be the hyperplane with the [[normal vector|0WQJJCCTC8]] $a$.\nThe reflection in $H$ is the map\n$$\\sigma_H \\colon \\mathbb{R}^n \\to \\mathbb{R}^n, \\qquad \\sigma_H(x) = x - 2\\, \\frac{x \\cdot a - c}{a \\cdot a}\\, a.$$\nFor $n = 2$, $H$ is a line; for $n = 3$, $H$ is a plane.\n\n## Remarks\n\nThe map $\\sigma_H$ is an indirect [[Euclidean isometry|0J1N3WTKM4]]. Its fixed points are the points of $H$, and $\\sigma_H \\circ \\sigma_H$ is the identity.\nFor $x \\notin H$, the hyperplane $H$ is the [[perpendicular bisector|055VNYT0PB]] of $x$ and $\\sigma_H(x)$.\nThe linear part of $\\sigma_H$ is an orthogonal matrix with determinant $-1$ and the eigenvalues $-1, 1, \\dots, 1$.\nReflections in hyperplanes generate the Euclidean group.\n\n## Explanations\n\n### L1 human\n\nA reflection is a mirror image. Each point goes to the point on the other side of the mirror, at the same distance from it. The reflection in the line $y = x$ in the plane sends $(x, y)$ to $(y, x)$.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "aliases": [
      "reflection",
      "mirror-isometry"
    ],
    "defines": "06BRDF3F0E",
    "kind": "definition",
    "msc": [
      "51F15"
    ],
    "name": "Reflection in a line or plane",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Reflection_(mathematics)"
      },
      {
        "locator": "Reflection across a line in the plane",
        "url": "https://en.wikipedia.org/wiki/Reflection_(mathematics)"
      },
      {
        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/Reflection_(mathematics)"
      }
    ],
    "tag": "027MBSVPY3",
    "uses": [
      "0WQJJCCTC8",
      "0J1N3WTKM4",
      "055VNYT0PB"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/027MBSVPY3/",
  "links": [
    {
      "field": "defines",
      "target": "06BRDF3F0E"
    },
    {
      "field": "uses",
      "target": "0WQJJCCTC8"
    },
    {
      "field": "uses",
      "target": "0J1N3WTKM4"
    },
    {
      "field": "uses",
      "target": "055VNYT0PB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/027MBSVPY3.md",
  "msc": [
    "51F15"
  ],
  "name": "Reflection in a line or plane",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/027/027MBSVPY3.md",
  "status": null,
  "tag": "027MBSVPY3",
  "uri": "m2://027MBSVPY3",
  "url": "https://math2.tamnd.com/wiki/reflection-in-a-line-or-plane/",
  "writing": {
    "tag": "027MBSVPY3",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}