{"id":12120,"date":"2026-03-18T12:00:59","date_gmt":"2026-03-18T08:00:59","guid":{"rendered":"https:\/\/beegraphy.com\/blog\/?p=12120"},"modified":"2026-03-19T20:20:54","modified_gmt":"2026-03-19T16:20:54","slug":"parameterizations-in-mathematics-a-case-study-of-zayed-national-museum","status":"publish","type":"post","link":"https:\/\/beegraphy.com\/blog\/parameterizations-in-mathematics-a-case-study-of-zayed-national-museum\/","title":{"rendered":"Parameterizations in Mathematics: A case study of Zayed National Museum"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_75 ez-toc-wrap-left counter-hierarchy ez-toc-counter ez-toc-custom ez-toc-container-direction\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<label for=\"ez-toc-cssicon-toggle-item-69d19d355a42d\" class=\"ez-toc-cssicon-toggle-label\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #757575;color:#757575\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #757575;color:#757575\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/label><input type=\"checkbox\"  id=\"ez-toc-cssicon-toggle-item-69d19d355a42d\" checked aria-label=\"Toggle\" \/><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/beegraphy.com\/blog\/parameterizations-in-mathematics-a-case-study-of-zayed-national-museum\/#Overview\" >Overview<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/beegraphy.com\/blog\/parameterizations-in-mathematics-a-case-study-of-zayed-national-museum\/#Parametrization_in_Mathematics\" >Parametrization in Mathematics<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/beegraphy.com\/blog\/parameterizations-in-mathematics-a-case-study-of-zayed-national-museum\/#Parameters_in_Architecture\" >Parameters in Architecture<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/beegraphy.com\/blog\/parameterizations-in-mathematics-a-case-study-of-zayed-national-museum\/#Introduction_to_Conic_Sections\" >Introduction to Conic Sections<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/beegraphy.com\/blog\/parameterizations-in-mathematics-a-case-study-of-zayed-national-museum\/#Creating_Conic_Sections_With_Beegraphy\" >Creating Conic Sections With Beegraphy<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/beegraphy.com\/blog\/parameterizations-in-mathematics-a-case-study-of-zayed-national-museum\/#Parametric_Equation_of_Line\" >Parametric Equation of Line<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/beegraphy.com\/blog\/parameterizations-in-mathematics-a-case-study-of-zayed-national-museum\/#Parametric_Equation_of_Circle\" >Parametric Equation of Circle<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/beegraphy.com\/blog\/parameterizations-in-mathematics-a-case-study-of-zayed-national-museum\/#Parametric_Equation_of_Ellipse\" >Parametric Equation of Ellipse<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/beegraphy.com\/blog\/parameterizations-in-mathematics-a-case-study-of-zayed-national-museum\/#Parametric_Equation_of_Parabola\" >Parametric Equation of Parabola<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/beegraphy.com\/blog\/parameterizations-in-mathematics-a-case-study-of-zayed-national-museum\/#Parametric_Equation_of_Hyperbola\" >Parametric Equation of Hyperbola<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/beegraphy.com\/blog\/parameterizations-in-mathematics-a-case-study-of-zayed-national-museum\/#Zayed_National_Museum\" >Zayed National Museum<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/beegraphy.com\/blog\/parameterizations-in-mathematics-a-case-study-of-zayed-national-museum\/#Step_By_Step_Tutorial_in_BeeGraphy\" >Step By Step Tutorial in BeeGraphy<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/beegraphy.com\/blog\/parameterizations-in-mathematics-a-case-study-of-zayed-national-museum\/#Base_Structure\" >Base Structure<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/beegraphy.com\/blog\/parameterizations-in-mathematics-a-case-study-of-zayed-national-museum\/#Top_Structure\" >Top Structure<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/beegraphy.com\/blog\/parameterizations-in-mathematics-a-case-study-of-zayed-national-museum\/#Convert_the_Top_Structure_into_Feather_like_Structure\" >Convert the Top Structure into Feather like Structure<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/beegraphy.com\/blog\/parameterizations-in-mathematics-a-case-study-of-zayed-national-museum\/#About_Ms_Florina_Pantilimonescu-Cioanca\" >About Ms Florina Pantilimonescu-Cioanc\u0103<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Overview\"><\/span><b>Overview<\/b><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">This blog is inspired by one of the lectures by <a href=\"https:\/\/www.linkedin.com\/in\/florina-pantilimonescu\/\" target=\"_blank\" rel=\"noopener\"><strong>Ms Florina<b> Pantilimonescu-Cioanc\u0103<\/b><\/strong><\/a><\/span><b>. <\/b><span style=\"font-weight: 400;\">Her idea is to <\/span><span style=\"font-weight: 400;\">prepare tutorial-style posts introducing mathematical concepts and then present them through BeeGraphy components. As a result to bring forth application for those notions<\/span><b>.<\/b><span style=\"font-weight: 400;\"> This blog explores the synergy between architecture and mathematics through her vision. And in the end we conclude with a small applied example of the Zayed Museum in Abu Dhabi.<br \/>\n<\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Parametrization_in_Mathematics\"><\/span><strong>Parametrization in Mathematics<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">In mathematics<\/span><b>, <\/b><span style=\"font-weight: 400;\">we encounter two fundamental quantities when working with functions: <\/span><i><span style=\"font-weight: 400;\">variables and parameters<\/span><\/i><span style=\"font-weight: 400;\">.\u00a0<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-weight: 400;\">The <\/span><b>variable<\/b><span style=\"font-weight: 400;\"> of a function, typically denoted as <\/span><i><span style=\"font-weight: 400;\">x<\/span><\/i><span style=\"font-weight: 400;\">, represents an independent quantity that is manipulated to evaluate the function. Consider a function of variable x,<\/span><span style=\"font-weight: 400;\"><br \/>\n<\/span> <b><i>f(x) = 2x\u00b2 + 7x + 1<\/i><\/b><\/p>\n<p><span style=\"font-weight: 400;\">We substitute different values of <\/span><i><span style=\"font-weight: 400;\">x<\/span><\/i><span style=\"font-weight: 400;\"> into the function to calculate the corresponding value of f(x). For instance, we might find f(0), f(1), or f(2) by plugging these values into our expression.\u00a0<\/span><\/p>\n<p style=\"text-align: left;\"><span style=\"font-weight: 400;\">In contrast, <\/span><b>parameters<\/b><span style=\"font-weight: 400;\"> are the coefficients or constants that define the specific form of a function, commonly represented by letters like <\/span><i><span style=\"font-weight: 400;\">a<\/span><\/i><span style=\"font-weight: 400;\">, <\/span><i><span style=\"font-weight: 400;\">b<\/span><\/i><span style=\"font-weight: 400;\">, and <\/span><i><span style=\"font-weight: 400;\">c<\/span><\/i><span style=\"font-weight: 400;\">. <\/span><b>The function <\/b><b><i>f(x) = 2x\u00b2 + 7x + 1<\/i><\/b><b> can be generalized as,<\/b><\/p>\n<p style=\"text-align: center;\"><b><i>f(x) = ax\u00b2 + bx + c,<\/i><\/b><\/p>\n<p><span style=\"font-weight: 400;\">Parameters are usually given implicitly and remain fixed. Though they don&#8217;t vary in typical problems, they serve a crucial role by allowing us to explore entire <\/span><b>families of functions<\/b><span style=\"font-weight: 400;\"> rather than just a single function.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">By varying <\/span><i><span style=\"font-weight: 400;\">parameters<\/span><\/i><span style=\"font-weight: 400;\">, we can study how functions behave as a collective class, revealing new patterns and relationships.<\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Parameters_in_Architecture\"><\/span><b>Parameters in Architecture<\/b><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">In architecture, <\/span><b>variables<\/b><span style=\"font-weight: 400;\"> refer to changing conditions, such as how many people use a space at different times. A room may be empty, lightly occupied, or crowded, and architects consider these variations.<\/span><\/p>\n<p><b>Parameters<\/b><span style=\"font-weight: 400;\">, in contrast, are fixed design constraints that define the framework. Elements like room size, ceiling height, column spacing, and structural grids remain constant once built. A parameter does not describe shape directly; instead, it sets the conditions under which shape is generated, determining what forms and uses are possible within the space.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Take the example of <\/span><a href=\"https:\/\/www.fosterandpartners.com\/projects\/zayed-national-museum\" target=\"_blank\" rel=\"noopener\"><b>Zayed National Museum<\/b><span style=\"font-weight: 400;\">, designed by <\/span><b>Foster + Partners<\/b><\/a><span style=\"font-weight: 400;\">. It showcases parametric architecture through its wing-like towers, whose height, curvature, and spacing are treated as adjustable parameters to respond to climate, structure, and cultural expression.<\/span><\/p>\n<p><img fetchpriority=\"high\" decoding=\"async\" class=\"size-large wp-image-12121 aligncenter lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/072825-Abu-Dhabi-Zayed-National-Museum-GettyImages-2182441689-1024x576.webp\" alt=\"\" width=\"1024\" height=\"576\" data-sizes=\"auto\" data-srcset=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/072825-Abu-Dhabi-Zayed-National-Museum-GettyImages-2182441689-1024x576.webp 1024w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/072825-Abu-Dhabi-Zayed-National-Museum-GettyImages-2182441689-300x169.webp 300w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/072825-Abu-Dhabi-Zayed-National-Museum-GettyImages-2182441689-768x432.webp 768w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/072825-Abu-Dhabi-Zayed-National-Museum-GettyImages-2182441689-1536x864.webp 1536w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/072825-Abu-Dhabi-Zayed-National-Museum-GettyImages-2182441689-2048x1152.webp 2048w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/072825-Abu-Dhabi-Zayed-National-Museum-GettyImages-2182441689-800x450.webp 800w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/072825-Abu-Dhabi-Zayed-National-Museum-GettyImages-2182441689-1920x1080.webp 1920w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/072825-Abu-Dhabi-Zayed-National-Museum-GettyImages-2182441689-20x11.webp 20w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Introduction_to_Conic_Sections\"><\/span><b>Introduction to Conic Sections<\/b><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">Conic sections are one of the most important families of curves in mathematics. They are formed by the intersection of a plane with a double cone (two identical cones joined at their vertices) as shown in figure below.<\/span><\/p>\n<p><img decoding=\"async\" class=\"size-full wp-image-12122 aligncenter lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/ellipse_hyperbola-1.gif\" alt=\"\" width=\"1000\" height=\"550\" \/><\/p>\n<p><span style=\"font-weight: 400;\">Depending on the angle at which a plane slices through the cone, we obtain different members of the conic family:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Circle<\/b><span style=\"font-weight: 400;\">: When the plane cuts perpendicular to the cone&#8217;s axis<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Ellipse<\/b><span style=\"font-weight: 400;\">: When the plane cuts at an angle, intersecting only one cone<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Parabola<\/b><span style=\"font-weight: 400;\">: When the plane is parallel to the slant of the cone<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Hyperbola<\/b><span style=\"font-weight: 400;\">: When the plane cuts through both cones<\/span><\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"Creating_Conic_Sections_With_Beegraphy\"><\/span><b>Creating Conic Sections With Beegraphy<\/b><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">Consider a cone with its center (vertex) at point O = (\u221220, 0, 0). We can explore different conic sections by intersecting this cone with three distinct planes, each defined by a point M on the plane and a normal vector N perpendicular to it:<\/span><\/p>\n<p><b>Plane P\u2081:<\/b><span style=\"font-weight: 400;\"> Point M\u2081 = (\u221220, 0, 5), Normal N\u2081 = (0, 0, 1)<\/span><span style=\"font-weight: 400;\"><br \/>\n<\/span><span style=\"font-weight: 400;\">This plane is horizontal (perpendicular to the z-axis), producing a <\/span><b>circle<\/b><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><b>Plane P\u2082:<\/b><span style=\"font-weight: 400;\"> Point M\u2082 = (\u221220, 0, 7), Normal N\u2082 = (0, 1, 1)<\/span><span style=\"font-weight: 400;\"><br \/>\n<\/span><span style=\"font-weight: 400;\">This tilted plane may produce an <\/span><b>ellipse<\/b><span style=\"font-weight: 400;\"> or <\/span><b>parabola<\/b><span style=\"font-weight: 400;\">, depending on its angle relative to the cone.<\/span><\/p>\n<p><b>Plane P\u2083:<\/b><span style=\"font-weight: 400;\"> Point M\u2083 = (\u221220, 3, 0), Normal N\u2083 = (0, 1, 0)<\/span><span style=\"font-weight: 400;\"><br \/>\n<\/span><span style=\"font-weight: 400;\">This plane is perpendicular to the y-axis and could yield a <\/span><b>hyperbola<\/b><span style=\"font-weight: 400;\"> if it intersects both cones.<\/span><\/p>\n<p><b>The General Form of Conic Equations\u00a0<\/b><\/p>\n<p><span style=\"font-weight: 400;\">All conic sections can be described by a single general second-degree equation in two variables:<\/span><\/p>\n<p style=\"text-align: center;\"><b>Ax\u00b2 + Bxy + Cy\u00b2 + Dx + Ey + F = 0<\/b><\/p>\n<p><span style=\"font-weight: 400;\">Here, A, B, C, D, E, and F are the <\/span><b>parameters<\/b><span style=\"font-weight: 400;\"> that determine which specific conic we obtain. This formulation shows that circles, ellipses, parabolas, and hyperbolas are all variations of the same fundamental equation. Thus giving us a parameterized family of curves.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">To learn more about conic sections and how they behave. Check out this article. <\/span><a href=\"https:\/\/en.wikipedia.org\/wiki\/Conic_section\" target=\"_blank\" rel=\"noopener\"><span style=\"font-weight: 400;\">Conic section<\/span><\/a><\/p>\n<p><iframe id=\"model-695cb1e12af40a35a8b8019a\" width=\"1210\" height=\"860\" src=\"https:\/\/beegraphy.com\/embed\/695cb1e12af40a35a8b8019a\" style=\"border: none; border-radius: 12px;\" allowfullscreen\/><\/iframe><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Parametric_Equation_of_Line\"><\/span><strong>Parametric Equation of Line<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">Consider a line in the plane that passes through the point <\/span><b>M\u2080 = (2, 5)<\/b><span style=\"font-weight: 400;\"> with direction vector <\/span><b>v = 3i + 10j<\/b><span style=\"font-weight: 400;\">. We define this line parametrically using BeeGraphy.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">The <\/span><b>canonical equation<\/b><span style=\"font-weight: 400;\"> of the line expresses the relationship between coordinates without an explicit parameter:<\/span><\/p>\n<p style=\"text-align: center;\"><b>(x \u2212 x\u2080) \/ l = (y \u2212 y\u2080) \/ m<\/b><\/p>\n<p><span style=\"font-weight: 400;\">We begin with <\/span><b>parametric equation of a line <\/b><span style=\"font-weight: 400;\">by introducing a parameter \u03bb (lambda):<\/span><\/p>\n<p style=\"text-align: center;\"><b>x = x\u2080 + \u03bbl<\/b><b><br \/>\n<\/b><b>y = y\u2080 + \u03bbm<\/b><\/p>\n<p><span style=\"font-weight: 400;\">Here, \u03bb \u2208 \u211d is the <\/span><b>parameter<\/b><span style=\"font-weight: 400;\"> that traces out the entire line. Each value of \u03bb corresponds to exactly one point on the line. \u03bb shifts the line as we change its values in the direction of the vector.\u00a0<\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Parametric_Equation_of_Circle\"><\/span><iframe id=\"model-697c5870c0a9f74b6f8b3234\" style=\"border: none; border-radius: 12px;\" src=\"https:\/\/beegraphy.com\/embed\/697c5870c0a9f74b6f8b3234\" width=\"1210\" height=\"860\" allowfullscreen=\"allowfullscreen\"><\/iframe><strong>Parametric Equation of Circle<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">We will define this circle parametrically using BeeGraphy and observe how varying the parameter <\/span><i><span style=\"font-weight: 400;\">t<\/span><\/i><span style=\"font-weight: 400;\"> traces the circular path.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Consider a circle such that <\/span><b>C(a, b)<\/b><span style=\"font-weight: 400;\"> be a fixed point and<\/span><b> r &gt; 0<\/b><span style=\"font-weight: 400;\"> a fixed real number. The circle with center C and radius r is the geometric locus of all points <\/span><b>M(x, y)<\/b><span style=\"font-weight: 400;\"> that satisfy the equality:<\/span><\/p>\n<p style=\"text-align: center;\"><b>|CM| = r<\/b> <span style=\"font-weight: 400;\">\u2026(i)<\/span><\/p>\n<p><span style=\"font-weight: 400;\">This definition states that a circle consists of all points at a constant distance r from the center C.<\/span> <span style=\"font-weight: 400;\">Equation (i) is also equivalent to the <\/span><b>parametric equations<\/b><span style=\"font-weight: 400;\">:<\/span><\/p>\n<p style=\"text-align: center;\"><b>x = a + r cos t<\/b><b><br \/>\n<\/b> <b>y = b + r sin t<\/b><\/p>\n<p><span style=\"font-weight: 400;\">where <\/span><b>t \u2208 [0, 2\u03c0)<\/b><span style=\"font-weight: 400;\"> is the parameter.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">The parameter <\/span><i><span style=\"font-weight: 400;\">t<\/span><\/i><span style=\"font-weight: 400;\"> represents the <\/span><b>angle in radians<\/b><span style=\"font-weight: 400;\"> measured counterclockwise from the positive x-axis. As <\/span><i><span style=\"font-weight: 400;\">t<\/span><\/i><span style=\"font-weight: 400;\"> varies continuously from 0 to 2\u03c0, the point M(x, y) traces the entire circle exactly once in a counterclockwise direction. This is the geometric beauty of parametric representation, where circular motion emerges from the periodic nature of sine and cosine.<\/span><\/p>\n<p><iframe id=\"model-695e12e22af40a35a8b80d32\" style=\"border: none; border-radius: 12px;\" src=\"https:\/\/beegraphy.com\/embed\/695e12e22af40a35a8b80d32\" width=\"1210\" height=\"860\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Parametric_Equation_of_Ellipse\"><\/span><strong>Parametric Equation of Ellipse<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">Now we will define this ellipse parametrically using BeeGraphy and observe how varying the parameter <\/span><i><span style=\"font-weight: 400;\">t<\/span><\/i><span style=\"font-weight: 400;\"> traces the elliptical path.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Let C(h, k) be a fixed point and a &gt; b &gt; 0 be fixed real numbers.\u00a0<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">C(h, k) be a fixed point (center)<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">a is the semi-major axis (horizontal radius)<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">b is the semi-minor axis (vertical radius)<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">The ellipse is the geometric locus of all points M(x, y) that satisfy:<\/span><\/p>\n<p style=\"text-align: center;\"><b>[(x \u2212 h)\u00b2 \/ a\u00b2] + [(y \u2212 k)\u00b2 \/ b\u00b2] = 1<\/b><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">The Cartesian equation is equivalent to the <\/span><b>parametric equations<\/b><span style=\"font-weight: 400;\">:<\/span><\/p>\n<p style=\"text-align: center;\"><b>x = h + a cos t<\/b><b><br \/>\n<\/b> <b>y = k + b sin t<\/b><\/p>\n<p><span style=\"font-weight: 400;\">where <\/span><b>t \u2208 [0, 2\u03c0)<\/b><span style=\"font-weight: 400;\"> is the parameter.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">The parameter <\/span><i><span style=\"font-weight: 400;\">t<\/span><\/i><span style=\"font-weight: 400;\"> represents the <\/span><b>angle parameter<\/b><span style=\"font-weight: 400;\"> (not the geometric angle from the center, but the parameter of the ellipse). As <\/span><i><span style=\"font-weight: 400;\">t<\/span><\/i><span style=\"font-weight: 400;\"> varies continuously from 0 to 2\u03c0, the point M(x, y) traces the entire ellipse exactly once in a counterclockwise direction.\u00a0<\/span><\/p>\n<p><iframe id=\"model-695e2d512af40a35a8b80eef\" style=\"border: none; border-radius: 12px;\" src=\"https:\/\/beegraphy.com\/embed\/695e2d512af40a35a8b80eef\" width=\"1210\" height=\"860\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Parametric_Equation_of_Parabola\"><\/span><strong>Parametric Equation of Parabola<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">Consider a parabola with a vertex at <\/span><b>O = (0, 0)<\/b><span style=\"font-weight: 400;\"> and a directrix axis along <\/span><b>OY<\/b><span style=\"font-weight: 400;\"> (the y-axis). The parabola is the geometric locus of all points M in the plane with the property that the distance from M to the point F equals the distance from M to the line <\/span><i><span style=\"font-weight: 400;\">d<\/span><\/i><span style=\"font-weight: 400;\">. We will define this parabola parametrically in BeeGraphy and explore its geometric properties.<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>The focus F:<\/b><span style=\"font-weight: 400;\"> The fixed point from which distances are measured<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>The directrix d:<\/b><span style=\"font-weight: 400;\"> The fixed line from which distances are measured<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>The parameter p:<\/b><span style=\"font-weight: 400;\"> The distance from the focus to the directrix, denoted by <\/span><i><span style=\"font-weight: 400;\">p<\/span><\/i><span style=\"font-weight: 400;\"> &gt; 0<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">From the geometric definition, the <\/span><b>equation<\/b><span style=\"font-weight: 400;\"> of the parabola is derived:<\/span><\/p>\n<p style=\"text-align: center;\"><b>y\u00b2 = 4px<\/b><\/p>\n<p><span style=\"font-weight: 400;\">The equation y\u00b2 = 4px can be expressed in <\/span><b>parametric form<\/b><span style=\"font-weight: 400;\">. Let <\/span><i><span style=\"font-weight: 400;\">t<\/span><\/i><span style=\"font-weight: 400;\"> be a parameter representing a vertical displacement. Then:<\/span><\/p>\n<p style=\"text-align: center;\"><b>x = t\u00b2<\/b><b><br \/>\n<\/b> <b>y = 2pt<\/b><\/p>\n<p><span style=\"font-weight: 400;\">where <\/span><b>t \u2208 (\u2212\u221e, +\u221e)<\/b><span style=\"font-weight: 400;\"> is the parameter.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">The parameter <\/span><i><span style=\"font-weight: 400;\">t<\/span><\/i><span style=\"font-weight: 400;\"> represents a scaled vertical coordinate. As <\/span><i><span style=\"font-weight: 400;\">t<\/span><\/i><span style=\"font-weight: 400;\"> varies continuously from \u2212\u221e to +\u221e, the point M(x, y) traces the entire parabola. Notice that both positive and negative values of <\/span><i><span style=\"font-weight: 400;\">t<\/span><\/i><span style=\"font-weight: 400;\"> produce the same x-coordinate (since x = t\u00b2), generating the symmetric upper and lower branches simultaneously.<\/span><\/p>\n<p><iframe id=\"model-695e2e5b2af40a35a8b80f18\" style=\"border: none; border-radius: 12px;\" src=\"https:\/\/beegraphy.com\/embed\/695e2e5b2af40a35a8b80f18\" width=\"1210\" height=\"860\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Parametric_Equation_of_Hyperbola\"><\/span><strong>Parametric Equation of Hyperbola<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">The hyperbola is the geometric locus of all points M in the plane such that the absolute difference of the distances from M to the two foci is constant and equal to 2a. We will define this hyperbola parametrically in BeeGraphy and explore its geometric properties.<\/span><\/p>\n<p style=\"text-align: center;\"><b>||MF\u2081| \u2212 |MF\u2082|| = 2a<\/b><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>The foci F\u2081 and F\u2082:<\/b><span style=\"font-weight: 400;\"> Two fixed points symmetric about the center<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>The center O:<\/b><span style=\"font-weight: 400;\"> The midpoint between the foci<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Semi-transverse axis a:<\/b><span style=\"font-weight: 400;\"> Half the distance between the vertices<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Semi-conjugate axis b:<\/b><span style=\"font-weight: 400;\"> Related to the &#8220;width&#8221; of the hyperbola<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">From the geometric definition, the <\/span><b>standard Cartesian equation<\/b><span style=\"font-weight: 400;\"> of the hyperbola with center at O(0, 0) and transverse axis along the x-axis is:<\/span><\/p>\n<p style=\"text-align: center;\"><b>x\u00b2\/a\u00b2 \u2212 y\u00b2\/b\u00b2 = 1<\/b><\/p>\n<p><span style=\"font-weight: 400;\">The Cartesian equation can be expressed in <\/span><b>parametric form<\/b><span style=\"font-weight: 400;\"> using hyperbolic trigonometric functions:<\/span><\/p>\n<p style=\"text-align: center;\"><b>x = a sec \u03b8<\/b><b><br \/>\n<\/b><b>y = b tan \u03b8<\/b><\/p>\n<p><span style=\"font-weight: 400;\">where \u03b8 \u2208 [0, 2\u03c0) with \u03b8 \u2260 \u03c0\/2, 3\u03c0\/2<\/span><\/p>\n<p><span style=\"font-weight: 400;\">As <\/span><i><span style=\"font-weight: 400;\">t<\/span><\/i><span style=\"font-weight: 400;\"> varies continuously from \u2212\u221e to +\u221e, the point M(x, y) traces the entire branches of the hyperbola.\u00a0<\/span><\/p>\n<p><iframe id=\"model-695e46bb2af40a35a8b8120f\" style=\"border: none; border-radius: 12px;\" src=\"https:\/\/beegraphy.com\/embed\/695e46bb2af40a35a8b8120f\" width=\"1210\" height=\"860\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Zayed_National_Museum\"><\/span><b>Zayed National Museum<\/b><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">The Zayed National Museum can be read as a parametric system in which architectural form is created through geometric and spatial parameters.\u00a0<\/span><\/p>\n<p><b>The Five Falcon Wings: <\/b><span style=\"font-weight: 400;\">The museum&#8217;s most distinctive feature consists of five lightweight steel wings modeled after falcon feathers. They are defined by controllable variables such as height, curvature, orientation, and spacing, allowing the overall form to emerge from mathematical relationships between these parameters.<\/span><\/p>\n<p><b>The Mound Structure: <\/b><span style=\"font-weight: 400;\">The museum spaces are housed within a faceted mound that abstracts the UAE&#8217;s desert topography. The geometric parameters of the mound &#8211; its slope angles, facet sizes, and material composition &#8211; were optimized to balance cultural symbolism with environmental efficiency.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Together, the wings and the mound form a unified geometric framework where changes in one parameter affect the entire system, leading to variations in spatial proportion and structural configuration.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Environmental performance is embedded within this parametric logic. Parameters such as tower height, vent placement, and section depth influence airflow and thermal behavior, showing how physical performance emerges from geometric control.<\/span><\/p>\n<p><i><span style=\"font-weight: 400;\">Key parameters of the museum include:<\/span><\/i><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Geometric:<\/b><span style=\"font-weight: 400;\"> wing height, curvature, orientation<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Environmental<\/b><span style=\"font-weight: 400;\">: ventilation rates, cooling depths, thermal mass<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Spatial: <\/b><span style=\"font-weight: 400;\">gallery size, atrium volume, circulation patterns<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Material: <\/b><span style=\"font-weight: 400;\">white concrete with crushed marble, patinated bronze<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">By adjusting these parameters during design development, the architects explored multiple configurations, treating architecture as a flexible system rather than a single fixed form. Tryout with the Beegraphy model below to get a clearer understanding of how it works.<\/span><\/p>\n<p><iframe id=\"model-6964deb421f9b4e6ccb0dbe1\" style=\"border: none; border-radius: 12px;\" src=\"https:\/\/beegraphy.com\/embed\/6964deb421f9b4e6ccb0dbe1\" width=\"1210\" height=\"860\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Step_By_Step_Tutorial_in_BeeGraphy\"><\/span>Step By Step Tutorial in BeeGraphy<b><br \/>\n<\/b><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3><span class=\"ez-toc-section\" id=\"Base_Structure\"><\/span><strong>Base Structure<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ol>\n<li><b>Add Point A<\/b><span style=\"font-weight: 400;\"> on the canvas. Treat this point as the <\/span><b>origin<\/b><span style=\"font-weight: 400;\"> to keep the setup simple.<\/span><\/li>\n<li><b>Add Point B<\/b><span style=\"font-weight: 400;\"> along the <\/span><b>x-axis only<\/b><span style=\"font-weight: 400;\">, at a chosen distance from Point A.<\/span><\/li>\n<li><b>Define Point C<\/b><span style=\"font-weight: 400;\"> such that its <\/span><b>x-coordinate lies exactly midway<\/b><span style=\"font-weight: 400;\"> between Point A and Point B and its <\/span><b>y-coordinate is offset<\/b><span style=\"font-weight: 400;\"> by a chosen value in the same plane.\u00a0<\/span>\n<ul>\n<li><span style=\"font-weight: 400;\">To obtain the midpoint in the x-direction, <\/span><b>use a Division node<\/b><span style=\"font-weight: 400;\"> and divide the <\/span><b>x-coordinate of Point B by 2<\/b><span style=\"font-weight: 400;\">.<\/span><\/li>\n<li><b>Assign this divided value<\/b><span style=\"font-weight: 400;\"> as the x-coordinate of Point C.<\/span><\/li>\n<li><b>Specify the value of y-coordinate<\/b><span style=\"font-weight: 400;\"> for Point C to complete its placement.<\/span><\/li>\n<\/ul>\n<\/li>\n<li><b>Add a NURBS Curve<\/b><span style=\"font-weight: 400;\"> component to the canvas. Define the curve to smoothly pass smoothly through <\/span><b>A \u2192 C \u2192 B<\/b><span style=\"font-weight: 400;\">.<\/span>\n<ul>\n<li><b>Set Point A<\/b><span style=\"font-weight: 400;\"> as <\/span><b>Point 1<\/b><span style=\"font-weight: 400;\"> of the curve.<\/span><\/li>\n<li><b>Set Point C<\/b><span style=\"font-weight: 400;\"> as <\/span><b>Point 2<\/b><span style=\"font-weight: 400;\"> of the curve.<\/span><\/li>\n<li><b>Set Point B<\/b><span style=\"font-weight: 400;\"> as <\/span><b>Point 3<\/b><span style=\"font-weight: 400;\"> of the curve.<\/span><span style=\"font-weight: 400;\"><img decoding=\"async\" class=\"wp-image-12132 size-large alignnone lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/1-1024x466.png\" alt=\"\" width=\"1024\" height=\"466\" data-sizes=\"auto\" data-srcset=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/1-1024x466.png 1024w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/1-300x136.png 300w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/1-768x349.png 768w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/1-800x364.png 800w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/1-20x9.png 20w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/1.png 1432w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/span><\/li>\n<\/ul>\n<\/li>\n<li><b>Add Point D<\/b><span style=\"font-weight: 400;\"> to the canvas using the <\/span><b>Construct Point<\/b><span style=\"font-weight: 400;\"> component.<\/span>\n<ul>\n<li><span style=\"font-weight: 400;\">Define the <\/span><b>x-coordinate<\/b><span style=\"font-weight: 400;\"> of Point D <\/span><b>identically to Point C <\/b><span style=\"font-weight: 400;\">(i.e., halfway between Point A and Point B).<\/span><\/li>\n<li><span style=\"font-weight: 400;\">For the <\/span><b>y-coordinate<\/b><span style=\"font-weight: 400;\">, take the offset in the <\/span><b>opposite direction<\/b><span style=\"font-weight: 400;\">:<\/span>\n<ul>\n<li><span style=\"font-weight: 400;\">Add a <\/span><b>Subtraction node<\/b><span style=\"font-weight: 400;\">.<\/span><\/li>\n<li><span style=\"font-weight: 400;\">Set the <\/span><b>minuend to 0<\/b><span style=\"font-weight: 400;\">.<\/span><\/li>\n<li><span style=\"font-weight: 400;\">Subtract the y-coordinate value to obtain a negative y value.<\/span><\/li>\n<\/ul>\n<\/li>\n<li><span style=\"font-weight: 400;\">To lift the point into <\/span><b>3D space<\/b><span style=\"font-weight: 400;\">, define the <\/span><b>z-coordinate<\/b><span style=\"font-weight: 400;\"> of Point D. Point D is now positioned symmetrically in y and elevated along the z-axis.<\/span><\/li>\n<\/ul>\n<\/li>\n<li><b>Add a second NURBS Curve<\/b><span style=\"font-weight: 400;\"> component to the canvas. The second curve is now defined, passing through <\/span><b>A \u2192 D \u2192 B<\/b><span style=\"font-weight: 400;\">, forming a spatial (3D) curve.<\/span>\n<ul>\n<li><b>Set Point A<\/b><span style=\"font-weight: 400;\"> as <\/span><b>Point 1<\/b><span style=\"font-weight: 400;\"> of the curve.<\/span><\/li>\n<li><b>Set Point D<\/b><span style=\"font-weight: 400;\"> as <\/span><b>Point 2<\/b><span style=\"font-weight: 400;\"> of the curve.<\/span><\/li>\n<li><b>Set Point B<\/b><span style=\"font-weight: 400;\"> as <\/span><b>Point 3<\/b><span style=\"font-weight: 400;\"> of the curve.<img loading=\"lazy\" decoding=\"async\" class=\"size-large wp-image-12131 aligncenter lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/12-1024x474.png\" alt=\"\" width=\"1024\" height=\"474\" data-sizes=\"auto\" data-srcset=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/12-1024x474.png 1024w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/12-300x139.png 300w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/12-768x355.png 768w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/12-800x370.png 800w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/12-20x9.png 20w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/12.png 1427w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/span><\/li>\n<\/ul>\n<\/li>\n<li><b>Add a Loft<\/b><span style=\"font-weight: 400;\"> surface component to the canvas. <\/span><b>Connect both NURBS curves<\/b><span style=\"font-weight: 400;\"> as inputs to the Loft node. The Loft component <\/span><b>generates a surface<\/b><span style=\"font-weight: 400;\"> by smoothly interpolating <\/span><b>between the two curves<\/b><span style=\"font-weight: 400;\"> along their length.<\/span><\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-large wp-image-12130 aligncenter lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/13-1024x485.png\" alt=\"\" width=\"1024\" height=\"485\" data-sizes=\"auto\" data-srcset=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/13-1024x485.png 1024w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/13-300x142.png 300w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/13-768x364.png 768w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/13-800x379.png 800w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/13-20x9.png 20w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/13.png 1428w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Top_Structure\"><\/span><strong>Top Structure<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><b>To Create the feather-like extension<\/b><span style=\"font-weight: 400;\"> we will use the same base points as before.\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">The feather is simply an <\/span><b>extension of the existing curve A\u2013D\u2013B<\/b><span style=\"font-weight: 400;\">. Therefore, <\/span><b>D\u2032 must lie in the same direction as D<\/b><span style=\"font-weight: 400;\">, not in a new direction. The <\/span><b>direction of D<\/b><span style=\"font-weight: 400;\"> relative to the base curve is already defined by its <\/span><b>y<\/b><span style=\"font-weight: 400;\"> and <\/span><b>z<\/b><span style=\"font-weight: 400;\"> coordinates. <\/span><span style=\"font-weight: 400;\">To extend the curve <\/span><b>without changing its direction<\/b><span style=\"font-weight: 400;\">, we <\/span><b>do not redefine<\/b><span style=\"font-weight: 400;\"> these coordinates. Instead, we <\/span><b>scale them<\/b><span style=\"font-weight: 400;\">. This is done by multiplying the <\/span><b>y<\/b><span style=\"font-weight: 400;\"> and <\/span><b>z<\/b><span style=\"font-weight: 400;\"> values of <\/span><b>D<\/b><span style=\"font-weight: 400;\"> by an <\/span><b>amplitude factor <\/b><b>a<\/b><span style=\"font-weight: 400;\">. The <\/span><b>x-coordinate is kept the same<\/b><span style=\"font-weight: 400;\">, so the extension happens <\/span><b>outward<\/b><span style=\"font-weight: 400;\">, not along the curve length.<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-weight: 400;\">D\u2032=(x,\u2005\u200aay,\u2005\u200aaz)<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Since both coordinates are scaled by the <\/span><b>same factor<\/b><span style=\"font-weight: 400;\">, the point moves <\/span><b>further along the same line of action<\/b><span style=\"font-weight: 400;\">.<\/span><\/p>\n<ol>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Add<b>\u00a0Construct Point<\/b><span style=\"font-weight: 400;\"> component to define <\/span><b>Point D\u2032<\/b><span style=\"font-weight: 400;\">.\u00a0<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Keep the <\/span><b>x-coordinate<\/b><span style=\"font-weight: 400;\"> of <\/span><b>D\u2032<\/b><span style=\"font-weight: 400;\"> the <\/span><b>same as Point D<\/b><span style=\"font-weight: 400;\">.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Introduce an <\/span><b>amplitude factor<\/b><span style=\"font-weight: 400;\">, call it <\/span><b>a<\/b><span style=\"font-weight: 400;\">.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Apply this amplitude to both the <\/span><b>y-coordinate<\/b><span style=\"font-weight: 400;\"> and <\/span><b>z-coordinate<\/b><span style=\"font-weight: 400;\">:<\/span>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"2\"><span style=\"font-weight: 400;\">Multiply the original <\/span><b>y-value by <\/b><b>a<\/b><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\"><span style=\"font-weight: 400;\">Multiply the original <\/span><b>z-value by <\/b><b>a<\/b><\/li>\n<\/ul>\n<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Add NURBS Curve<\/b><span style=\"font-weight: 400;\"> component to the canvas. The second curve is now defined, passing through <\/span><b>A \u2192 D\u2019 \u2192 B<\/b><span style=\"font-weight: 400;\">, forming a spatial (3D) curve.<\/span>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"2\"><b>Set Point A<\/b><span style=\"font-weight: 400;\"> as <\/span><b>Point 1<\/b><span style=\"font-weight: 400;\"> of the curve.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\"><b>Set Point D\u2019<\/b><span style=\"font-weight: 400;\"> as <\/span><b>Point 2<\/b><span style=\"font-weight: 400;\"> of the curve.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\"><b>Set Point B<\/b><span style=\"font-weight: 400;\"> as <\/span><b>Point 3<\/b><span style=\"font-weight: 400;\"> of the curve.<img loading=\"lazy\" decoding=\"async\" class=\"size-large wp-image-12129 aligncenter lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/14-1024x243.png\" alt=\"\" width=\"1024\" height=\"243\" data-sizes=\"auto\" data-srcset=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/14-1024x243.png 1024w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/14-300x71.png 300w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/14-768x182.png 768w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/14-800x190.png 800w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/14-20x5.png 20w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/14.png 1502w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/span><\/li>\n<\/ul>\n<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Add a <strong>Loft<\/strong><span style=\"font-weight: 400;\"><strong> surface node<\/strong>. <\/span><b>Connect Curves ACB and AD\u2019B<\/b><span style=\"font-weight: 400;\"> as inputs to the Loft node. The Loft component <\/span><b>generates a surface<\/b><span style=\"font-weight: 400;\"> by smoothly interpolating <\/span><b>between the two curves<\/b><span style=\"font-weight: 400;\"> along their length. (<\/span><b>Hide the Loft node<\/b><span style=\"font-weight: 400;\">, as the surface is only used as a reference and <\/span><b>we now want to establish a net like structure<\/b><span style=\"font-weight: 400;\"> from the underlying geometry.)<\/span><\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-large wp-image-12128 aligncenter lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/15-1024x537.png\" alt=\"\" width=\"1024\" height=\"537\" data-sizes=\"auto\" data-srcset=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/15-1024x537.png 1024w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/15-300x157.png 300w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/15-768x403.png 768w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/15-800x419.png 800w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/15-20x10.png 20w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/15.png 1356w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Convert_the_Top_Structure_into_Feather_like_Structure\"><\/span><strong>Convert the Top Structure into Feather like Structure<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><b>To create a net or feather-like structure<\/b><span style=\"font-weight: 400;\">, add a <\/span><b>Contour<\/b><span style=\"font-weight: 400;\"> component to the canvas.<\/span><\/p>\n<ol>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Connect the Surface<\/b><span style=\"font-weight: 400;\"> (from the loft surface outport) to the <\/span><b>Contour<\/b><span style=\"font-weight: 400;\"> input.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Define a distance value<\/b><span style=\"font-weight: 400;\"> for the contour spacing.<\/span><span style=\"font-weight: 400;\"><br \/>\n<\/span><span style=\"font-weight: 400;\"> This generates <\/span><b>horizontal contour lines<\/b><span style=\"font-weight: 400;\"> across the geometry, following its form.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">For <\/span><b>vertical contour lines<\/b><span style=\"font-weight: 400;\">, add <\/span><b>another Contour<\/b><span style=\"font-weight: 400;\"> component.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Set the contour plane to XZ plane<\/b><span style=\"font-weight: 400;\"> using <\/span><b>PlaneXZ node<\/b><span style=\"font-weight: 400;\"> for this second contour node.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Connect the same geometry<\/b><span style=\"font-weight: 400;\"> to this Contour component.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Adjust the <\/span><b>spacing distance<\/b><span style=\"font-weight: 400;\"> until the desired net pattern appears.<img loading=\"lazy\" decoding=\"async\" class=\"size-large wp-image-12127 aligncenter lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/16-1024x315.png\" alt=\"\" width=\"1024\" height=\"315\" data-sizes=\"auto\" data-srcset=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/16-1024x315.png 1024w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/16-300x92.png 300w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/16-768x236.png 768w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/16-800x246.png 800w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/16-20x6.png 20w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/16.png 1502w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">To enhance the structure, <\/span><b>add a Pipe component<\/b><span style=\"font-weight: 400;\">.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Connect the contour curves <\/b><span style=\"font-weight: 400;\">\u00a0to the Pipe node and <\/span><b>set the radius<\/b><span style=\"font-weight: 400;\"> appropriately to give thickness.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">For a more <\/span><b>presentable and readable structure<\/b><span style=\"font-weight: 400;\">, also <\/span><b>connect the main curves ACB<\/b><span style=\"font-weight: 400;\">, <\/span><b>ADB<\/b><span style=\"font-weight: 400;\">, and <\/span><b>AD\u2032B<\/b><span style=\"font-weight: 400;\"> to the <\/span><b>Pipe<\/b><span style=\"font-weight: 400;\"> component.<img loading=\"lazy\" decoding=\"async\" class=\"size-large wp-image-12126 aligncenter lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/17-1024x463.png\" alt=\"\" width=\"1024\" height=\"463\" data-sizes=\"auto\" data-srcset=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/17-1024x463.png 1024w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/17-300x136.png 300w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/17-768x348.png 768w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/17-800x362.png 800w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/17-20x9.png 20w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/01\/17.png 1496w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/span><\/li>\n<\/ol>\n<p><span style=\"font-weight: 400;\">The result is a <\/span><b>netted, feather-like spatial structure<\/b><span style=\"font-weight: 400;\"> defined by intersecting contour lines. You can find the Model <a href=\"https:\/\/beegraphy.com\/market\/product\/zayed-museum-single-structure-4e2\">here.<\/a><\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"About_Ms_Florina_Pantilimonescu-Cioanca\"><\/span><b>About Ms Florina Pantilimonescu-Cioanc\u0103<\/b><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">Ms Florina Pantilimonescu-Cioanc\u0103 is an architect, dedicated to blending creativity with data-driven approaches. She has a vast mathematical background centered around analytic and differential geometry, probability, and statistics. Today, her research aims at bridging the gap between architecture and mathematical knowledge, with a particular focus on machine learning.\u00a0<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12257 aligncenter lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/03\/1655542403820.jpg\" alt=\"\" width=\"286\" height=\"286\" data-sizes=\"auto\" data-srcset=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/03\/1655542403820.jpg 800w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/03\/1655542403820-300x300.jpg 300w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/03\/1655542403820-150x150.jpg 150w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/03\/1655542403820-768x768.jpg 768w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/03\/1655542403820-120x120.jpg 120w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/03\/1655542403820-450x450.jpg 450w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2026\/03\/1655542403820-20x20.jpg 20w\" sizes=\"(max-width: 286px) 100vw, 286px\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Overview This blog is inspired by one of the lectures by Ms Florina Pantilimonescu-Cioanc\u0103. Her idea is to prepare tutorial-style posts introducing mathematical concepts and then present them through BeeGraphy components. As a result to bring forth application for those notions. This blog explores the synergy between architecture and mathematics through her vision. And in [&hellip;]<\/p>\n","protected":false},"author":22,"featured_media":12148,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,99,417],"tags":[109,494,36],"class_list":["post-12120","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-featured","category-parametric-education","category-tutorials","tag-beegraphy","tag-mathematics","tag-parametric-design"],"_links":{"self":[{"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/posts\/12120","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/users\/22"}],"replies":[{"embeddable":true,"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/comments?post=12120"}],"version-history":[{"count":26,"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/posts\/12120\/revisions"}],"predecessor-version":[{"id":12266,"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/posts\/12120\/revisions\/12266"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/media\/12148"}],"wp:attachment":[{"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/media?parent=12120"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/categories?post=12120"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/tags?post=12120"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}