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    This study introduces two new algorithms for direct parameterization of quadrilateral meshes, crucial for real-world applications. These angle-preserving methods efficiently map surfaces to planes and spheres.

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    Area of Science:

    • Computer Graphics
    • Computational Geometry
    • Applied Mathematics

    Background:

    • Quadrilateral meshes are increasingly used in various real-world applications.
    • Efficient and accurate parameterization methods are essential for utilizing these meshes.

    Purpose of the Study:

    • To present two novel algorithms for the direct parameterization of quadrilateral meshes.
    • To develop angle-preserving mappings for surface parameterization.

    Main Methods:

    • For disk surfaces: Developed a discrete conformal energy function with a length-preserving boundary condition to flatten quadrilateral meshes.
    • For sphere surfaces: Utilized a derived Tuette energy function for mesh initialization, followed by minimizing a devised harmonic energy function for spherical parameterization.

    Main Results:

    • The proposed algorithms provide direct parameterization of quadrilateral meshes.
    • Experimental results validate the efficiency of both developed methods.

    Conclusions:

    • The presented algorithms offer effective solutions for quadrilateral mesh parameterization.
    • These methods are suitable for applications requiring angle-preserving mappings.