2 posts de una; estamos que nos salimos

Author: agarnung

_posts/2025-01-27-geometric-fitting-intuition-pt1.md

@@ -0,0 +1,73 @@
+---
+title: Geometric fitting intuition pt. I
+tags: [math, geometry]
+style: fill
+color: info
+description: A friendly introduction to geometric fitting algorithms.
+---
+
+<img src="../assets/blog_images/2025-01-27-geometric-fitting-intuition-pt1/geompt1.png" alt="geompt1" width=500>
+
+## Introduction
+
+Many concepts for analyzing a geometric problem are applicable to other contexts. For example, in 3D problems involving the computation of distances between pairs of objects (points, line segments, triangles, tetrahedra, etc.), each type of object can be analyzed using specific knowledge about its shape. The unifying idea that brings these problems under a common framework is that objects can be parameterized using zero (point), one (line segment), two (triangle, rectangle), or three (tetrahedron, cube) parameters.
+The squared distance between any two points is a quadratic polynomial in the appropriate parameters. A search within the domain of this quadratic function will lead to the parameters corresponding to the closest points on the objects, and thus to the minimum squared distance between them. This provides a powerful and generalizable approach for solving geometric fitting problems in a least-squares sense.
+
+## Paper
+
+To the benefict of better visualization, you can access the post main content in the LaTex PDF attached below:
+
+<script src="/assets/js/pdf.js"></script>
+
+<div class="container text-center" id="pdf-container" style="min-height: 100%;">
+  <div id="viewerContainer align-items-center">
+    <div id="pdf-viewer" class="mt-6"></div>
+  </div>
+  <h4 class="font-weight-bold" style="text-align: right; margin-top: 5px"><a target="_blank" href="{{ '/assets/blog_pdfs/2025-01-27-geometric-fitting-intuition-pt1/geometric-fitting-intuition-pt1.pdf' }}">Open as PDF</a></h4>
+</div>
+
+<script>
+  var url = '../assets/blog_pdfs/2025-01-27-geometric-fitting-intuition-pt1/geometric-fitting-intuition-pt1.pdf';
+
+  pdfjsLib.getDocument(url).promise.then(function (pdf) {
+    var viewer = document.getElementById('pdf-viewer');
+
+    for (var pageNumber = 1; pageNumber <= pdf.numPages; pageNumber++) {
+      var pageContainer = document.createElement('div');
+      pageContainer.className = 'pdf-page';
+
+      var canvas = document.createElement('canvas');
+      canvas.className = 'pdf-page-canvas';
+      pageContainer.appendChild(canvas);
+
+      viewer.appendChild(pageContainer);
+
+      renderPage(pageNumber, canvas, pdf);
+    }
+  });
+
+  function renderPage(pageNumber, canvas, pdf) {
+    pdf.getPage(pageNumber).then(function (page) {
+      var viewport = page.getViewport({ scale: 0.2 });
+      var scale = canvas.clientWidth / viewport.width;
+
+      var scaledViewport = page.getViewport({ scale: scale });
+
+      var context = canvas.getContext('2d');
+      canvas.height = scaledViewport.height;
+      canvas.width = scaledViewport.width;
+
+      var renderContext = {
+        canvasContext: context,
+        viewport: scaledViewport,
+      };
+
+      page.render(renderContext);
+    });
+  }
+</script>
+
+## References
+
+[1] https://understandinglinearalgebra.org/sec-least-squares.html
+

_posts/2025-01-27-geometric-fitting-intuition-pt2.md

@@ -91,9 +91,7 @@ To the benefict of better visualization, you can access the post main content in
 
 ## Conclusions and Future Work
 
-We now have a slightly more formed intuition about this topic, although so far we have only glimpsed the world of geometric fitting.
-
-__A work in progress...__
+A cone? A parabola? An unaligned torus? What else can we test under the least-squares gauntlet?... We now have a slightly more formed intuition about this topic, although so far we have only glimpsed the world of geometric fitting.
 
 ## References
 

_posts/2025-01-28-geometric-fitting-intuition-pt0.md

@@ -0,0 +1,75 @@
+---
+title: Geometric fitting intuition pt. 0
+tags: [math, geometry]
+style: fill
+color: danger
+description: A friendly introduction to geometric fitting algorithms.
+---
+
+<img src="../assets/blog_images/2025-01-28-geometric-fitting-intuition-pt0/ls.png" alt="ls" width=500>
+
+## Introduction
+
+Whether you're interested in designing an analytical classification model, calibrating a measurement or instrumentation system quickly and easily, performing linear regression for your chemistry tasks, fitting a simple prediction model to your observed data, tracking 3D eye movements and needing a real-time mathematical model for further processing, or measuring the distance between two walls or any other related task, we will explore and learn, with not much introduction, very interesting concepts about the world of geometric fitting and linear algebra, oriented towards a practical approach.
+
+## Paper
+
+To the benefict of better visualization, you can access the post main content in the LaTex PDF attached below:
+
+<script src="/assets/js/pdf.js"></script>
+
+<div class="container text-center" id="pdf-container" style="min-height: 100%;">
+  <div id="viewerContainer align-items-center">
+    <div id="pdf-viewer" class="mt-6"></div>
+  </div>
+  <h4 class="font-weight-bold" style="text-align: right; margin-top: 5px"><a target="_blank" href="{{ '/assets/blog_pdfs/2025-01-28-geometric-fitting-intuition-pt0/geometric-fitting-intuition-pt0.pdf' }}">Open as PDF</a></h4>
+</div>
+
+<script>
+  var url = '../assets/blog_pdfs/2025-01-28-geometric-fitting-intuition-pt0/geometric-fitting-intuition-pt0.pdf';
+
+  pdfjsLib.getDocument(url).promise.then(function (pdf) {
+    var viewer = document.getElementById('pdf-viewer');
+
+    for (var pageNumber = 1; pageNumber <= pdf.numPages; pageNumber++) {
+      var pageContainer = document.createElement('div');
+      pageContainer.className = 'pdf-page';
+
+      var canvas = document.createElement('canvas');
+      canvas.className = 'pdf-page-canvas';
+      pageContainer.appendChild(canvas);
+
+      viewer.appendChild(pageContainer);
+
+      renderPage(pageNumber, canvas, pdf);
+    }
+  });
+
+  function renderPage(pageNumber, canvas, pdf) {
+    pdf.getPage(pageNumber).then(function (page) {
+      var viewport = page.getViewport({ scale: 0.2 });
+      var scale = canvas.clientWidth / viewport.width;
+
+      var scaledViewport = page.getViewport({ scale: scale });
+
+      var context = canvas.getContext('2d');
+      canvas.height = scaledViewport.height;
+      canvas.width = scaledViewport.width;
+
+      var renderContext = {
+        canvasContext: context,
+        viewport: scaledViewport,
+      };
+
+      page.render(renderContext);
+    });
+  }
+</script>
+
+## References
+
+[1] Schneider, P., & Eberly, D. H. (2002). *Geometric Tools for Computer Graphics*. 1st Edition. Morgan Kaufmann. ISBN 978-1558605947.
+
+[2] Eberly, D. (2020). *Robust and Error-Free Geometric Computing*. 1st Edition. CRC Press. ISBN 978-0367352943.
+
+[3] https://mathworld.wolfram.com/VandermondeMatrix.html