MHB Quadratic Forms: Beyond Sketching Conics

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Quadratic forms have significant real-life applications beyond sketching conics, particularly in optimization problems like the linear-quadratic regulator in control theory, which involves solving linear differential equations with quadratic constraints. They are also utilized in image processing, such as matching digital images of pupils with ellipses and aligning 3D MRI scans of hearts with ellipsoids. These applications highlight the versatility of quadratic forms in both theoretical and practical contexts. The discussion emphasizes their importance in various fields, including engineering and medical imaging. Overall, quadratic forms play a crucial role in solving complex problems across multiple disciplines.
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What are the real life applications of quadratic forms? I have used them to sketch conics but are there any other applications?
 
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One of the more important (fundamental) optimization problems in control theory is the linear-quadratic regulator problem, in which you try to solve a linear DE subject to a quadratic constraint. The quadratic constraint involves quadratic forms.
 
I have used quadratic forms to match the digital image of a pupil (in perspective) with an ellipse.

A friend of mine worked on an application of quadric forms (3 dimensional) to match a heart in an MRI scan (3D image) with an ellipsoid.
 

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