“Using Tensor Diagrams to Represent and Solve Geometric Problems” by Blinn

  • ©James (Jim) F. Blinn

Conference:


Entry Number: 18

Title:

    Using Tensor Diagrams to Represent and Solve Geometric Problems

Course Organizer(s):



Presenter(s):



Abstract:


    Prerequisites
    Familiarity with homogeneous-coordinate geometry and basic matrix operations. Distaste for page-long algebraic expressions.

    Topics
    Review of homogeneous-coordinate math. Notational problems with matrices. Einstein Index notation. How tensor diagrams represent basic operations. Application of tensor diagrams to: 1D homogeneous equations (polynomials), 2D homogeneous equations (curves), 3D homogeneous equations (surfaces). Unsolved (at least to the knowledge of the speaker) problems.

    Description
    Conventional matrix notation for algebraic geometry does not adequately represent some concepts that are important in understanding and manipulating geometrical quantities. Better notational tools can be appropriated from the field of mathematical physics. This course focuses on one such tool, the tensor diagram (which is related to the Feynman diagram), and shows how it can improve notational convenience, solve many geometrical problems that would otherwise be very complicated, and facilitate understanding of the algebraic structure of such problems.


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