{"id":562,"date":"2020-02-29T15:53:51","date_gmt":"2020-02-29T15:53:51","guid":{"rendered":"https:\/\/www.parametriczoo.com\/?p=562"},"modified":"2020-03-20T22:40:51","modified_gmt":"2020-03-20T22:40:51","slug":"signed-distance-of-a-point-from-a-plane","status":"publish","type":"post","link":"https:\/\/www.parametriczoo.com\/index.php\/2020\/02\/29\/signed-distance-of-a-point-from-a-plane\/","title":{"rendered":"Signed distance of a point from a plane"},"content":{"rendered":"

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Today I was looking at the Project()<\/a> method in Revit API. The method signature offers a distance parameter which outputs the distance of the projected point to the input point.\u00a0<\/p>\n

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public<\/span> void<\/span> Project<\/span>(\n\tXYZ<\/a> point<\/span>,\n\tout<\/span> UV<\/a> uv<\/span>,\n\tout<\/span> double<\/a> distance<\/span>\n)<\/code><\/pre>\n
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\u00a0The value of distance parameter is always positive which means we have no clue if the point is above or below the plane. Apart from this issue , the method is implemented in Revit API 2018.1 and it is not available in earlier versions. In this post I discuss two different methods for finding the signed distance of a point from a plane, first using change of basis and trasformations and in the second method we use dot product<\/a> which in my opinion it is the essence of linear algebra!\u00a0<\/p>\n
In order find the signed distance of point p<\/strong><\/em> from the given plane Q\u00a0<\/b><\/em>we can simply evaluate the point with respect to the local coordinate system of the plane. This operation is called change of basis<\/strong>. Let us say the plane Q<\/em> <\/strong>is at origin O<\/strong> <\/em>with basis X<\/em><\/strong>,Y<\/em><\/strong> and Z<\/strong><\/em>, We can write the components of the plane Q<\/em><\/strong> in vector format :<\/p>\n
 <\/p>\n
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The transformation of any point from global coordinate system to the plane coordinate system can be define by the 4×4\u00a0 matrix A<\/strong> <\/em>which is composed of the basis vectors of the plane and the origin point (translation part).<\/p>\n
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The point p’<\/em><\/strong> is described in the plane local coordinate system, therefore by transforming p’<\/strong><\/em> to using plane transformation we obtain point p<\/em><\/strong> in global coordinate system. This operation can be reversed to find the point\u00a0p’<\/strong><\/em> which is the coordinates of point p<\/strong> <\/em>with resepct to the plane coordinate system.<\/p>\n
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Below snipped code demonstrates the implementation of change of basis and extracts the Z<\/strong><\/em> coordinate of point p’ as the signed distance of the point p<\/strong><\/em> with respect to the plane. Also note that we can output the projection point just by transforming the local projection of p’<\/em><\/strong> to the global coordsys.<\/p>\n
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        \/\/\/<\/span> <\/span><<\/span>summary<\/span>><\/span>
        \/\/\/<\/span> return signed distance of point from the plane. <\/span>
        \/\/\/<\/span> If the point is above the plane (in positive side of the plane) <\/span>
        \/\/\/<\/span> the result is positive and if the point<\/span>
        \/\/\/<\/span> is below the plane the result is negetive<\/span>
        \/\/\/<\/span> <\/span><\/<\/span>summary<\/span>><\/span>
        \/\/\/<\/span> <\/span><<\/span>param<\/span> name<\/span>=<\/span>\"<\/span>plane<\/span>\"<\/span>><\/span>Input plane<\/span><\/<\/span>param<\/span>><\/span>
        \/\/\/<\/span> <\/span><<\/span>param<\/span> name<\/span>=<\/span>\"<\/span>point<\/span>\"<\/span>><\/span>Point to measure the distance from<\/span><\/<\/span>param<\/span>><\/span>
        \/\/\/<\/span> <\/span><<\/span>param<\/span> name<\/span>=<\/span>\"<\/span>projection<\/span>\"<\/span>><\/span>Projection of input point on this plane<\/span><\/<\/span>param<\/span>><\/span>
        \/\/\/<\/span> <\/span><<\/span>returns<\/span>><\/<\/span>returns<\/span>><\/span>
        public<\/span> static<\/span> double<\/span> DistanceTo<\/span> (<\/span>this<\/span> Plane<\/span> plane<\/span>,<\/span>XYZ<\/span> point<\/span>,<\/span>out<\/span> XYZ<\/span> projection<\/span>)<\/span>
        {<\/span>
            \/\/build a transformation matrix based on the plane coordinate system<\/span>
            Transform<\/span> tr<\/span> =<\/span> Transform<\/span>.<\/span>CreateTranslation<\/span>(<\/span>plane<\/span>.<\/span>Origin<\/span>);<\/span> 
            tr<\/span>.<\/span>BasisX<\/span> =<\/span> plane<\/span>.<\/span>XVec<\/span>.<\/span>Normalize<\/span>();<\/span>
            tr<\/span>.<\/span>BasisY<\/span> =<\/span> plane<\/span>.<\/span>YVec<\/span>.<\/span>Normalize<\/span>()<\/span> ;<\/span>
            tr<\/span>.<\/span>BasisZ<\/span> =<\/span> plane<\/span>.<\/span>Normal<\/span>.<\/span>Normalize<\/span>();<\/span>
            \/\/Find the local coordinates of the given point with resepct to plabe coordinate system<\/span>
            XYZ<\/span> local<\/span> =<\/span> tr<\/span>.<\/span>Inverse<\/span>.<\/span>OfPoint<\/span>(<\/span>point<\/span>);<\/span>
            \/\/project the local point to the XY plane (Z=0)<\/span>
            projection<\/span> =<\/span> new<\/span> XYZ<\/span>(<\/span>local<\/span>.<\/span>X<\/span>,<\/span> local<\/span>.<\/span>Y<\/span>,<\/span> 0<\/span>);<\/span>
            \/\/ transform the projection back to the global coordSYS<\/span>
            projection<\/span> =<\/span> tr<\/span>.<\/span>OfPoint<\/span>(<\/span>projection<\/span>);<\/span>
            \/\/The Z coordinate of the local point is the signed distance from the plane <\/span>
            return<\/span> local<\/span>.<\/span>Z<\/span>;<\/span>
        }<\/span><\/pre>\n
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Well, above is not really an elegant method!, my aim was to discuss a bit of change of basis which is very important topic. The distance of a point from a plane is simply calculated from the equation (5)<\/strong><\/em> in below.\u00a0 Let us assume plane Q<\/strong><\/em> with normal n<\/strong><\/em>\u00a0 and a point on plane p0<\/strong><\/em>\u00a0 and a point p<\/strong><\/em> outside the plane with its normal projection on plane p’<\/strong><\/em>.\u00a0<\/p>\n
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Starting from equation (1)<\/em><\/strong> which is simply sum of vectors and taking dot product of both sides we obtain equation (2)<\/strong><\/em>.\u00a0\u00a0<\/p>\n
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Equation (3)<\/em><\/strong> is obtained by expanding (2)<\/strong><\/em> and rewriting (p-p’).n<\/strong><\/em> as d*n<\/strong> <\/em>where d<\/strong><\/em> is the distance between the p<\/strong><\/em> and p’<\/strong><\/em> (Note if p<\/strong><\/em> is below the plane then d<\/strong><\/em> must be negative therefore d<\/strong><\/em> is signed distance).\u00a0Then we can write (4)<\/strong><\/em> by replacing\u00a0(p\u2080-p’).n<\/strong><\/em> with zero since two vectors are perpendicular and their dot product is zero.<\/p>\n
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Finally in we replace n.n <\/strong><\/em>by 1 since n is normal vector. Equation (5)<\/em><\/strong> gives us the signed distance from the plane and by moving the point p<\/strong><\/em> in opposite direction of plane normal (n)<\/strong><\/em> by the magnitude of d we obtain p’<\/strong><\/em> in equation (6)<\/strong><\/em><\/p>\n
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Below is a simple and more elegant implementation of the same method.\u00a0<\/p>\n
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public<\/span> static<\/span> double<\/span> DistanceTo2<\/span>(<\/span>this<\/span> Plane<\/span> plane<\/span>,<\/span> XYZ<\/span> point<\/span>,<\/span> out<\/span> XYZ<\/span> projection<\/span>)<\/span>
{<\/span>
    \/\/ signed distance from equation (5)<\/span>
    double<\/span> d<\/span> =<\/span> (<\/span>point<\/span> -<\/span> plane<\/span>.<\/span>Origin<\/span>)<\/span>.<\/span>DotProduct<\/span>(<\/span>plane<\/span>.<\/span>Normal<\/span>);<\/span>
    \/\/ the projection point from equation (6)<\/span>
    projection<\/span> =<\/span> point<\/span> -<\/span> d<\/span> *<\/span> plane<\/span>.<\/span>Normal<\/span>;<\/span>
    return<\/span> d<\/span>;<\/span>
}<\/span><\/pre>\n
[\/et_pb_text][\/et_pb_column][\/et_pb_row][\/et_pb_section]<\/p>\n","protected":false},"excerpt":{"rendered":"
Today I was looking at the Project() method in Revit API. The method signature offers a distance parameter which outputs the distance of the projected point to the input point.\u00a0public void Project( XYZ point, out UV uv, out double distance ) \u00a0The value of distance parameter is always positive which means we have no clue […]<\/p>\n","protected":false},"author":1,"featured_media":591,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"on","_et_pb_old_content":"","_et_gb_content_width":""},"categories":[45],"tags":[49,33,47],"yoast_head":"\nSigned distance of a point from a plane - Parametric Zoo<\/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:\/\/www.parametriczoo.com\/index.php\/2020\/02\/29\/signed-distance-of-a-point-from-a-plane\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Signed distance of a point from a plane - Parametric Zoo\" \/>\n<meta property=\"og:description\" content=\"Today I was looking at the Project() method in Revit API. The method signature offers a distance parameter which outputs the distance of the projected point to the input point.\u00a0public void Project( XYZ point, out UV uv, out double distance ) \u00a0The value of distance parameter is always positive which means we have no clue […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.parametriczoo.com\/index.php\/2020\/02\/29\/signed-distance-of-a-point-from-a-plane\/\" \/>\n<meta property=\"og:site_name\" content=\"Parametric Zoo\" \/>\n<meta property=\"article:published_time\" content=\"2020-02-29T15:53:51+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2020-03-20T22:40:51+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.parametriczoo.com\/wp-content\/uploads\/2020\/02\/distance-to-plabe.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1680\" \/>\n\t<meta property=\"og:image:height\" content=\"1300\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"PARA\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"PARA\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"7 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebSite\",\"@id\":\"https:\/\/www.parametriczoo.com\/#website\",\"url\":\"https:\/\/www.parametriczoo.com\/\",\"name\":\"Parametric Zoo\",\"description\":\"\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/www.parametriczoo.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/www.parametriczoo.com\/index.php\/2020\/02\/29\/signed-distance-of-a-point-from-a-plane\/#primaryimage\",\"url\":\"https:\/\/www.parametriczoo.com\/wp-content\/uploads\/2020\/02\/distance-to-plabe.jpg\",\"contentUrl\":\"https:\/\/www.parametriczoo.com\/wp-content\/uploads\/2020\/02\/distance-to-plabe.jpg\",\"width\":1680,\"height\":1300},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.parametriczoo.com\/index.php\/2020\/02\/29\/signed-distance-of-a-point-from-a-plane\/\",\"url\":\"https:\/\/www.parametriczoo.com\/index.php\/2020\/02\/29\/signed-distance-of-a-point-from-a-plane\/\",\"name\":\"Signed distance of a point from a plane - Parametric Zoo\",\"isPartOf\":{\"@id\":\"https:\/\/www.parametriczoo.com\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/www.parametriczoo.com\/index.php\/2020\/02\/29\/signed-distance-of-a-point-from-a-plane\/#primaryimage\"},\"datePublished\":\"2020-02-29T15:53:51+00:00\",\"dateModified\":\"2020-03-20T22:40:51+00:00\",\"author\":{\"@id\":\"https:\/\/www.parametriczoo.com\/#\/schema\/person\/0368c6eb8bfe3a003504793be2a2e0e3\"},\"breadcrumb\":{\"@id\":\"https:\/\/www.parametriczoo.com\/index.php\/2020\/02\/29\/signed-distance-of-a-point-from-a-plane\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/www.parametriczoo.com\/index.php\/2020\/02\/29\/signed-distance-of-a-point-from-a-plane\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/www.parametriczoo.com\/index.php\/2020\/02\/29\/signed-distance-of-a-point-from-a-plane\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/www.parametriczoo.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Signed distance of a point from a plane\"}]},{\"@type\":\"Person\",\"@id\":\"https:\/\/www.parametriczoo.com\/#\/schema\/person\/0368c6eb8bfe3a003504793be2a2e0e3\",\"name\":\"PARA\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/www.parametriczoo.com\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/2b2ff0ff40493545df12d0b31a504675?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/2b2ff0ff40493545df12d0b31a504675?s=96&d=mm&r=g\",\"caption\":\"PARA\"},\"url\":\"https:\/\/www.parametriczoo.com\/index.php\/author\/para\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Signed distance of a point from a plane - Parametric Zoo","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.parametriczoo.com\/index.php\/2020\/02\/29\/signed-distance-of-a-point-from-a-plane\/","og_locale":"en_US","og_type":"article","og_title":"Signed distance of a point from a plane - Parametric Zoo","og_description":"Today I was looking at the Project() method in Revit API. 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