{"id":216,"date":"2024-03-26T21:44:00","date_gmt":"2024-03-26T13:44:00","guid":{"rendered":"https:\/\/blog.bexonbai.com\/?p=216"},"modified":"2026-03-02T21:54:16","modified_gmt":"2026-03-02T13:54:16","slug":"2d-planar-coordinate-system-conversion","status":"publish","type":"post","link":"https:\/\/blog.bexonbai.com\/?p=216","title":{"rendered":"2D planar coordinate system conversion"},"content":{"rendered":"\n

The 2D planar coordinate system conversion I use at work, the demo code is implemented in Rust, and other code should be implemented in roughly the same way. \ud83e\udd14<\/p>\n\n\n\n

If you are using the standard right-handed Cartesian coordinate system you may need to transform the y-axis appropriately.<\/p>\n\n\n\n

Rotation<\/h2>\n\n\n\n

clockwise<\/h3>\n\n\n\n
A<\/mi>=<\/mo>[<\/mo>c<\/mi>o<\/mi>s<\/mi>(<\/mo>\u03b8<\/mi>)<\/mo><\/mrow><\/mtd>s<\/mi>i<\/mi>n<\/mi>(<\/mo>\u03b8<\/mi>)<\/mo><\/mrow><\/mtd>0<\/mn><\/mtd><\/mtr>s<\/mi>i<\/mi>n<\/mi>(<\/mo>\u03b8<\/mi>)<\/mo><\/mrow><\/mtd>c<\/mi>o<\/mi>s<\/mi>(<\/mo>\u03b8<\/mi>)<\/mo><\/mrow><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>0<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow>A=\n\\begin{gathered}\n\\begin{bmatrix} \ncos(\\theta) & sin(\\theta) & 0 \\\\ \nsin(\\theta) & cos(\\theta) & 0 \\\\\n0 & 0 & 1\n\\end{bmatrix}\n\\end{gathered}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

counterclockwise<\/h3>\n\n\n\n
A<\/mi>=<\/mo>[<\/mo>c<\/mi>o<\/mi>s<\/mi>(<\/mo>\u03b8<\/mi>)<\/mo><\/mrow><\/mtd>s<\/mi>i<\/mi>n<\/mi>(<\/mo>\u2212<\/mo>\u03b8<\/mi>)<\/mo><\/mrow><\/mtd>0<\/mn><\/mtd><\/mtr>s<\/mi>i<\/mi>n<\/mi>(<\/mo>\u03b8<\/mi>)<\/mo><\/mrow><\/mtd>c<\/mi>o<\/mi>s<\/mi>(<\/mo>\u03b8<\/mi>)<\/mo><\/mrow><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>0<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow>A=\n\\begin{gathered}\n\\begin{bmatrix} \ncos(\\theta) & sin(-\\theta) & 0 \\\\ \nsin(\\theta) & cos(\\theta) & 0 \\\\\n0 & 0 & 1\n\\end{bmatrix}\n\\end{gathered}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

matrix equation<\/h3>\n\n\n\n
[<\/mo>x<\/mi>1<\/mn><\/msub><\/mtd><\/mtr>y<\/mi>1<\/mn><\/msub><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable>=<\/mo>A<\/mi>\u22c5<\/mo>[<\/mo>x<\/mi><\/mtd><\/mtr>y<\/mi><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow>\\begin{gathered}\n\\begin{bmatrix} \nx_1 \\\\ \ny_1 \\\\\n1\n\\end{bmatrix}\n\\end{gathered}\n=\nA\n\\cdot\n\\begin{gathered}\n\\begin{bmatrix} \nx \\\\ \ny \\\\\n1\n\\end{bmatrix}\n\\end{gathered}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

Rust demo code<\/h3>\n\n\n\n
use<\/strong> std::f64::consts::PI;\nuse<\/strong> ndarray::arr2;\n\nfn<\/strong> rotation<\/strong>() {\n    \/\/ 2D point rotation 45 degrees<\/em>\n    let<\/strong> theta =<\/strong> 45.0 *<\/strong> (PI \/<\/strong> 180.0);\n\n    \/\/ Convert Ordinary Coordinate (x, y) to Homogeneous Coordinate (x, y, 1)<\/em>\n    let<\/strong> point =<\/strong> arr2<\/strong>(&<\/strong>[[0.0], [10.0], [1.0]]);\n\n    \/\/ 2D rotated matrices<\/em>\n    let<\/strong> a =<\/strong> arr2<\/strong>(&<\/strong>[\n        [f64::cos<\/strong>(theta), f64::sin<\/strong>(theta), 0.0],\n        [f64::sin<\/strong>(theta), f64::cos<\/strong>(theta), 0.0],\n        [0.0, 0.0, 1.0],\n    ]);\n\n    let<\/strong> new_point =<\/strong> a.dot<\/strong>(&<\/strong>point);\n\n    println!(\"{}\", &<\/strong>new_point);\n}\n<\/code><\/pre>\n\n\n\n
    Finished dev [<\/strong>unoptimized + debuginfo] target(<\/strong>s)<\/strong> in <\/strong>0.11s\n     Running `target\/debug\/two_d_cover`\n[[<\/strong>7.071067811865475],\n [<\/strong>7.0710678118654755],\n [<\/strong>1]]\n<\/code><\/pre>\n\n\n\n
Rotate a point around another point<\/strong><\/h2>\n\n\n\n
    \n
  1. Translate the specified point to the origin to obtain the translation matrix T<\/mi>1<\/mn><\/msub>T_1<\/annotation><\/semantics><\/math><\/li>\n\n\n\n
  2. Perform a rotation to obtain the rotation matrix R<\/mi>R<\/annotation><\/semantics><\/math><\/li>\n\n\n\n
  3. Translate to the specified point to get the translation matrix T<\/mi>2<\/mn><\/msub>T_2<\/annotation><\/semantics><\/math>.<\/li>\n<\/ol>\n\n\n\n
    the point (<\/mo>x<\/mi>0<\/mn><\/msub>,<\/mo>y<\/mi>0<\/mn><\/msub>)<\/mo><\/mrow>(x_0, y_0)<\/annotation><\/semantics><\/math> is rotated around the point (<\/mo>x<\/mi>s<\/mi><\/msub>,<\/mo>y<\/mi>s<\/mi><\/msub>)<\/mo><\/mrow>(x_s, y_s)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n
    T<\/mi>1<\/mn><\/msub>=<\/mo>[<\/mo>1<\/mn><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>1<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>\u2212<\/mo>x<\/mi>s<\/mi><\/msub><\/mrow><\/mtd>\u2212<\/mo>y<\/mi>s<\/mi><\/msub><\/mrow><\/mtd>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow>T_1=\n\\begin{gathered}\n\\begin{bmatrix} \n1 & 0 & 0 \\\\ \n0 & 1 & 0 \\\\\n-x_s & -y_s & 1\n\\end{bmatrix}\n\\end{gathered}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
    R<\/mi>=<\/mo>[<\/mo>c<\/mi>o<\/mi>s<\/mi>(<\/mo>\u03b8<\/mi>)<\/mo><\/mrow><\/mtd>\u2212<\/mo>s<\/mi>i<\/mi>n<\/mi>(<\/mo>\u03b8<\/mi>)<\/mo><\/mrow><\/mtd>0<\/mn><\/mtd><\/mtr>s<\/mi>i<\/mi>n<\/mi>(<\/mo>\u03b8<\/mi>)<\/mo><\/mrow><\/mtd>c<\/mi>o<\/mi>s<\/mi>(<\/mo>\u03b8<\/mi>)<\/mo><\/mrow><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>0<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow>R=\n\\begin{gathered}\n\\begin{bmatrix} \ncos(\\theta) & -sin(\\theta) & 0 \\\\ \nsin(\\theta) & cos(\\theta) & 0 \\\\\n0 & 0 & 1\n\\end{bmatrix}\n\\end{gathered}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
    T<\/mi>2<\/mn><\/msub>=<\/mo>[<\/mo>1<\/mn><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>1<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>x<\/mi>s<\/mi><\/msub><\/mtd>y<\/mi>s<\/mi><\/msub><\/mtd>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow>T_2=\n\\begin{gathered}\n\\begin{bmatrix} \n1 & 0 & 0 \\\\ \n0 & 1 & 0 \\\\\nx_s & y_s & 1\n\\end{bmatrix}\n\\end{gathered}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
    [<\/mo>x<\/mi>1<\/mn><\/msub><\/mtd><\/mtr>y<\/mi>1<\/mn><\/msub><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable>=<\/mo>[<\/mo>x<\/mi>0<\/mn><\/msub><\/mtd><\/mtr>y<\/mi>0<\/mn><\/msub><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable>\u22c5<\/mo>T<\/mi>1<\/mn><\/msub>\u22c5<\/mo>R<\/mi>\u22c5<\/mo>T<\/mi>2<\/mn><\/msub><\/mrow>\\begin{gathered}\n\\begin{bmatrix} \nx_1 \\\\ \ny_1 \\\\\n1\n\\end{bmatrix}\n\\end{gathered}\n=\n\\begin{gathered}\n\\begin{bmatrix} \nx_0 \\\\ \ny_0 \\\\\n1\n\\end{bmatrix}\n\\end{gathered}\n\\cdot\nT_1 \\cdot R \\cdot T_2<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
    Translation<\/h2>\n\n\n\n
    [<\/mo>x<\/mi>1<\/mn><\/msub><\/mtd><\/mtr>y<\/mi>1<\/mn><\/msub><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable>=<\/mo>[<\/mo>1<\/mn><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>1<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>d<\/mi>x<\/mi><\/mrow><\/mtd>d<\/mi>y<\/mi><\/mrow><\/mtd>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable>\u22c5<\/mo>[<\/mo>x<\/mi>0<\/mn><\/msub><\/mtd><\/mtr>y<\/mi>0<\/mn><\/msub><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow>\\begin{gathered}\n\\begin{bmatrix} \nx_1 \\\\ \ny_1 \\\\\n1\n\\end{bmatrix}\n\\end{gathered}\n=\n\\begin{gathered}\n\\begin{bmatrix} \n1 & 0 & 0 \\\\ \n0 & 1 & 0 \\\\\ndx & dy & 1\n\\end{bmatrix}\n\\end{gathered}\n\\cdot\n\\begin{gathered}\n\\begin{bmatrix} \nx_0 \\\\ \ny_0 \\\\\n1\n\\end{bmatrix}\n\\end{gathered}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
    Zoom<\/h2>\n\n\n\n
    [<\/mo>x<\/mi>1<\/mn><\/msub><\/mtd><\/mtr>y<\/mi>1<\/mn><\/msub><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable>=<\/mo>[<\/mo>s<\/mi>x<\/mi><\/msub><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>s<\/mi>y<\/mi><\/msub><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>0<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable>\u22c5<\/mo>[<\/mo>x<\/mi>0<\/mn><\/msub><\/mtd><\/mtr>y<\/mi>0<\/mn><\/msub><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow>\\begin{gathered}\n\\begin{bmatrix} \nx_1 \\\\ \ny_1 \\\\\n1\n\\end{bmatrix}\n\\end{gathered}\n=\n\\begin{gathered}\n\\begin{bmatrix} \ns_x & 0 & 0 \\\\ \n0 & s_y & 0 \\\\\n0 & 0 & 1\n\\end{bmatrix}\n\\end{gathered}\n\\cdot\n\\begin{gathered}\n\\begin{bmatrix} \nx_0 \\\\ \ny_0 \\\\\n1\n\\end{bmatrix}\n\\end{gathered}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
    Rust demo code<\/h3>\n\n\n\n
    use<\/strong> ndarray::arr2;\n\nfn<\/strong> main<\/strong>() {\n    zoom<\/strong>(10.0, 10.0, 1.2, 1.2);\n}\n\nfn<\/strong> zoom<\/strong>(x_0: f64, y_0: f64, s_x: f64, s_y: f64) {\n    let<\/strong> point =<\/strong> arr2<\/strong>(&<\/strong>[[x_0], [y_0], [1.0]]);\n\n    let<\/strong> s =<\/strong> arr2<\/strong>(&<\/strong>[\n        [s_x, 0.0, 0.0],\n        [0.0, s_y, 0.0],\n        [0.0, 0.0, 1.0],\n    ]);\n\n    let<\/strong> new_point =<\/strong> s.dot<\/strong>(&<\/strong>point);\n    println!(\"{}\", &<\/strong>new_point);\n}\n<\/code><\/pre>\n\n\n\n
        Finished dev [<\/strong>unoptimized + debuginfo] target(<\/strong>s)<\/strong> in <\/strong>0.34s\n     Running `target\/debug\/two_d_cover`\n[[<\/strong>12],\n [<\/strong>12],\n [<\/strong>1]]<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"
    Record the formulas commonly used for the conversion of 2D coordinate systems.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7],"tags":[],"class_list":["post-216","post","type-post","status-publish","format-standard","hentry","category-develop"],"_links":{"self":[{"href":"https:\/\/blog.bexonbai.com\/index.php?rest_route=\/wp\/v2\/posts\/216","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.bexonbai.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.bexonbai.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.bexonbai.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.bexonbai.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=216"}],"version-history":[{"count":1,"href":"https:\/\/blog.bexonbai.com\/index.php?rest_route=\/wp\/v2\/posts\/216\/revisions"}],"predecessor-version":[{"id":217,"href":"https:\/\/blog.bexonbai.com\/index.php?rest_route=\/wp\/v2\/posts\/216\/revisions\/217"}],"wp:attachment":[{"href":"https:\/\/blog.bexonbai.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=216"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.bexonbai.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=216"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.bexonbai.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=216"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}