{"id":4634,"date":"2022-08-23T17:30:37","date_gmt":"2022-08-23T12:00:37","guid":{"rendered":"https:\/\/jassweb.com\/solved\/solved-coordinate-system-transformation-3d-projection-to-2d-plane\/"},"modified":"2022-08-23T17:30:37","modified_gmt":"2022-08-23T12:00:37","slug":"solved-coordinate-system-transformation-3d-projection-to-2d-plane","status":"publish","type":"post","link":"https:\/\/jassweb.com\/solved\/solved-coordinate-system-transformation-3d-projection-to-2d-plane\/","title":{"rendered":"[Solved] Coordinate system transformation, 3d projection to 2d plane"},"content":{"rendered":"<p> [ad_1]<br \/>\n<\/p>\n<div id=\"answer-38589828\" class=\"answer js-answer accepted-answer js-accepted-answer\" data-answerid=\"38589828\" data-parentid=\"38588842\" data-score=\"2\" data-position-on-page=\"1\" data-highest-scored=\"1\" data-question-has-accepted-highest-score=\"1\" itemprop=\"acceptedAnswer\" itemscope itemtype=\"https:\/\/schema.org\/Answer\">\n<div class=\"post-layout\">\n<div class=\"votecell post-layout--left\"><\/div>\n<div class=\"answercell post-layout--right\">\n<div class=\"s-prose js-post-body\" itemprop=\"text\">\n<p>The problem can be expressed as finding the 3-by-3 matrix <code>M<\/code> such that the coordinates of a point <code>P<\/code> can be converted between the old coordinate system (<code>P_old<\/code>, 3 rows) and the new coordinate system (<code>P_new<\/code>, 3 rows). This is an affine transformation:<\/p>\n<pre><code>P_old = Center + M * P_new     (1)\n<\/code><\/pre>\n<p>The (matrix-vector) multiplication with <code>M<\/code> orientates it back to the old system, and adding <code>Center<\/code>&#8216;s coordinates translates it back to the old origin.<\/p>\n<p>The equation (1) can then be turned into:<\/p>\n<pre><code>P_new = M^{-1} * (P_old - Center)     (2)\n<\/code><\/pre>\n<p>where <code>M^{-1}<\/code> is the inverse of <code>M<\/code>, to compute the new coordinates from the old ones (the third row will have a 0 if the point belongs to the plane of the triangle).<\/p>\n<p>The matrix <code>M<\/code> is made of the coordinates of the new basis in the old system, one basis vector in each column. One must now find such a basis.<\/p>\n<p>This basis can be taken from (this is all pseudo-code):<\/p>\n<ol>\n<li>\n<p>Renormalizing <code>AB<\/code><\/p>\n<pre><code>       AB\nV1 = ______\n    || AB ||\n<\/code><\/pre>\n<ul>\n<li>\n<p><code>AB<\/code> here is meant as the vector <code>AB<\/code> (with an arrow on top):<\/p>\n<pre><code>|b_x - a_x|\n|b_y - a_y|\n|b_z - a_z|\n<\/code><\/pre>\n<\/li>\n<li>\n<p><code>|| . ||<\/code> is the Euclidian norm (<code>^2<\/code> means the square, not <em>xor<\/em>):<\/p>\n<pre><code>|| V || = sqrt(V_x^2 + V_y^2 + V_z^2)\n<\/code><\/pre>\n<\/li>\n<\/ul>\n<\/li>\n<li>\n<p><code>AC<\/code> (also a vector, defined like <code>AB<\/code>), but minus its projection on <code>V1<\/code> to make it orthogonal to <code>V1<\/code>, and renormalized (this will fail with a division by zero if the triangle is not really a triangle):<\/p>\n<pre><code>        AC - (AC.V_1) V1\nV2 = _______________________\n     || AC - (AC.V_1) V1 ||\n<\/code><\/pre>\n<ul>\n<li>\n<p><code>M.N<\/code> is the scalar product:<\/p>\n<pre><code>M.N = M_x * N_x + M_y * N_y + M_z * N_z\n<\/code><\/pre>\n<\/li>\n<li>\n<p><code>(AC.V_1) V1<\/code> is simply the product of a scalar, <code>(AC.V_1)<\/code>, with a vector, <code>V1<\/code><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>A third vector that can be taken as the cross product to get a Cartesian coordinate system:<\/p>\n<pre><code>V3 = V1 x V2\n<\/code><\/pre>\n<ul>\n<li>\n<p>The cross product is defined as:<\/p>\n<pre><code>          |V1_y*V2_z - V1_z*V2_y|\nV1 x V2 = |V1_z*V2_x - V1_x*V2_z|\n          |V1_x*V2_y - V1_y*V2_x|\n<\/code><\/pre>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p>Then M can be taken as <code>|V1 V2 V3|<\/code> (each <code>Vx<\/code> is on 3 rows) and, then its inverse computed to use formula (2).<\/p>\n<p>This transformation (with the inverted M) should both generate new coordinates for the points on the plane of the triangle that have 0 on the third axis (which makes it 2D-coordinates on that plane), and preserve the size in terms of Euclidian norm.<\/p>\n<\/p><\/div>\n<div class=\"mt24\"><\/div>\n<\/div>\n<p>            <span class=\"d-none\" itemprop=\"commentCount\">8<\/span> <\/p><\/div>\n<\/div>\n<p>[ad_2]<\/p>\n<p>solved Coordinate system transformation, 3d projection to 2d plane <\/p>\n","protected":false},"excerpt":{"rendered":"<p>[ad_1] The problem can be expressed as finding the 3-by-3 matrix M such that the coordinates of a point P can be converted between the old coordinate system (P_old, 3 rows) and the new coordinate system (P_new, 3 rows). This is an affine transformation: P_old = Center + M * P_new (1) The (matrix-vector) multiplication &#8230; <a title=\"[Solved] Coordinate system transformation, 3d projection to 2d plane\" class=\"read-more\" href=\"https:\/\/jassweb.com\/solved\/solved-coordinate-system-transformation-3d-projection-to-2d-plane\/\" aria-label=\"More on [Solved] Coordinate system transformation, 3d projection to 2d plane\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[320],"tags":[324,987,986,357,516],"class_list":["post-4634","post","type-post","status-publish","format-standard","hentry","category-solved","tag-c","tag-coordinate-systems","tag-coordinates","tag-math","tag-matlab"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.5 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>[Solved] Coordinate system transformation, 3d projection to 2d plane - JassWeb<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/jassweb.com\/solved\/solved-coordinate-system-transformation-3d-projection-to-2d-plane\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"[Solved] Coordinate system transformation, 3d projection to 2d plane - JassWeb\" \/>\n<meta property=\"og:description\" content=\"[ad_1] The problem can be expressed as finding the 3-by-3 matrix M such that the coordinates of a point P can be converted between the old coordinate system (P_old, 3 rows) and the new coordinate system (P_new, 3 rows). 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