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Consider a component AB undergoing a linear motion. Along with a linear motion, point B also rotates around point A. To comprehend this complex movement, position vectors for both points A and B are established using a stationary reference frame.
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Angular resolution and scanning density estimation based on a 3D to 2D Lissajous transition.

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    This study explores 3D Lissajous scanning, introducing characteristic phase differences to define trajectory rules. It details methods for calculating scanning density and angular resolution, achieving high-density and high-resolution Lissajous imaging.

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    Area of Science:

    • Physics
    • Image Processing
    • Applied Mathematics

    Background:

    • Lissajous scanning quality depends on angular resolution and scanning density.
    • Understanding the transition from 3D to 2D Lissajous scanning is crucial for image quality.

    Purpose of the Study:

    • To investigate the transition characteristics from 3D to 2D Lissajous scanning.
    • To establish the relationship between 3D Lissajous scanning rotation and phase change.
    • To define rules for 3D Lissajous scanning trajectories and explore 2D phenomena from a 3D perspective.

    Main Methods:

    • Introduced the concept of characteristic phase difference for 2D and 3D Lissajous scanning trajectories.
    • Developed an extension method for multidimensional Lissajous scanning characteristic phase difference.
    • Defined three key rules for 3D Lissajous scanning trajectories.
    • Proposed methods for calculating scanning density and angular resolution using double-junction circles.
    • Analyzed curve characteristics for specific frequency ratios (9:7, 13:10, 17:13).

    Main Results:

    • Established three rules for 3D Lissajous scanning trajectories.
    • Derived quadrant patterns for 2D Lissajous scanning with "non-closed endpoints".
    • Calculated trajectory densities of 110, 234, and 412 for frequency ratios 9:7, 13:10, and 17:13, respectively.
    • Achieved an angular resolution of 0.0089 for a 17:13 frequency ratio with high accuracy (≤0.009).

    Conclusions:

    • The study provides a framework for understanding and optimizing 3D Lissajous scanning.
    • The developed methods enable high-resolution and high-density Lissajous imaging.
    • Experimental verification confirms the theoretical findings on trajectory density and angular resolution.