A double integral teaching demonstration software algorithm based on three-dimensional visualization dynamic segmentation

Abstract

The application discloses a kind of based on three-dimensional visualization dynamic segmentation double integral teaching demonstration software algorithm, belong to computer-aided instruction technical field.The system includes mathematical modeling module, grid segmentation module, geometry construction module, rendering engine module, calculation analysis module, error visualization module, animation generation module and user interaction module.System is through gradual regional division strategy, the long rectangular cuboid column unit of the rectangular domain on the dynamic segmentation of the curved top cylinder corresponding to the integral function, using transparent curved surface rendering and entity cylinder color coding technology, intuitively show the riemann and approximation process of double integral.System supports multi-angle view angle animation synthesis, real-time calculation and display approximate volume, accurate integral value and error index, solve the technical problem that double integral concept is abstract and difficult to understand in traditional teaching, lack intuitive cognitive tool, significantly improve the visualization effect of mathematical teaching and student's learning efficiency.

Description

Technical Field

[0001] This invention relates to algorithms, and more particularly to the field of computer-aided instruction technology. Background Technology

[0002] The definition of a double integral is a core concept in advanced mathematics, and its geometric meaning is the volume of a prism with a curved top. However, because double integrals involve the calculation of the volume of a prism with a curved top and its division, traditional teaching methods mainly rely on static blackboard writing and formula derivation. Students cannot see the dynamic demonstration process, making it difficult to establish an intuitive geometric understanding. Summary of the Invention

[0003] To address the aforementioned technical problems, this invention provides a teaching demonstration software algorithm for double integrals based on three-dimensional visualization and dynamic segmentation. Through progressive region division, transparent surface rendering, solid cylinder color encoding, high and low color differentiation, visual angle conversion, and error visualization techniques, it solves the technical problems of abstract concepts and lack of intuitive cognitive tools in the teaching of double integrals.

[0004] To achieve the above objectives, the present invention adopts the following technical solution: A teaching demonstration software algorithm for double integrals based on 3D visualization and dynamic segmentation includes: The A1 Mathematical Modeling Module defines the integrand z = f(x, y) and the integration region D = [a,b]×[c,d]. The default function used is z = sin(x)·cos(y) + 2, with a range of [1,3] to avoid visual interference from negative values. This module calculates the function value, the exact integral value, and the area of ​​the integration region.

[0005] A2 Grid Segmentation Module: Implements a progressive region division strategy, using a non-uniformly progressive sequence of segmentation numbers [1,2,4,6,8,10,12], so that the segmentation density gradually increases as the teaching demonstration progresses, which is in line with the cognitive load theory.

[0006] A3 Geometry Building Module: Generates 3D vertex coordinates and facet topology for cylindrical elements based on the mesh. A gap coefficient of 0.02 times the mesh width is introduced to avoid visual clutter. Supports four sampling strategies: top right corner, center point, bottom left corner, and random points. Different color mapping encodings are applied based on the cylinder height.

[0007] A4 rendering engine module: Performs layered rendering, with the bottom layer being a semi-transparent reference surface with an opacity of 0.3, and the upper layer being a collection of solid cylinders with an opacity of 0.9 (based on height color coding), and draws bottom grid lines and boundary contour lines to enhance spatial recognition.

[0008] A5 Calculation and Analysis Module: Real-time calculation of Riemann sum as an approximate volume, comparison with the exact integral value to obtain absolute error, relative error and error percentage, and formatted output of teaching information.

[0009] A6 Error Visualization Module: When the number of segments n≤8, a red dashed error marker line (0.8 line width, 0.4 transparency) is drawn between the center point of each sub-rectangle and the real surface to visually display the approximate error. It is automatically hidden in fine grids to avoid visual confusion.

[0010] A7 Animation Generation Module: Synthesizes multi-angle perspective animation sequences. Each segment generates 3 frames with different azimuth angles (range 35°-55°), which are then synthesized into a GIF animation. Frame dwell time can be set.

[0011] A8 User Interaction Module: Provides parameter adjustment interfaces, including the number of segments (1-50), viewing angle (elevation angle, azimuth angle), display characteristics (surface / cylinder / error bar / grid switch), and sampling strategy selection.

[0012] (1) Enhanced intuitiveness of teaching: Through three-dimensional dynamic visualization, the abstract concept of double integral is transformed into an observable geometric process, and students can intuitively understand the mathematical idea of ​​"replacing curves with straight lines".

[0013] (2) Progressive cognitive construction: adopting a non-uniform progressive segmentation strategy, gradually transitioning from 1×1 coarse segmentation to 12×12 fine segmentation, which conforms to the cognitive law from simple to complex, and understands the volume segmentation of the curved top cylinder corresponding to the geometry of double integral.

[0014] (3) Multi-dimensional error perception: Through color coding and error marking lines, students can perceive approximate errors from both height differences (color) and distance differences (error lines).

[0015] (4) High interactivity support: real-time parameter adjustment and immediate feedback support inquiry-based learning, allowing students to independently verify the mathematical conclusion that "the finer the segmentation, the smaller the error".

[0016] (5) Optimization of teaching adaptability: It is specifically designed for classroom teaching scenarios. Teachers do not need programming skills and can use it directly or insert it into PPT, etc. Attached Figure Description

[0017] The present invention will be further described in detail below with reference to the accompanying drawings. The program uses Python 3.13, and takes the function z = sin(x)·cos(y) + 2, with the range of z being [1,3], as an example.

[0018] Figure 1 : This is a cutout of the wireframe animation corresponding to the color image generated by the algorithm of this invention. Figure 2: This is a cutout of the wireframe animation corresponding to the color image generated by the algorithm of this invention. Figure 3 : This is a cutout of the wireframe animation corresponding to the color image generated by the algorithm of this invention. Figure 4 : This is a cutout of the wireframe animation corresponding to the color image generated by the algorithm of this invention. Figure 5 : This is a cutout of the wireframe animation corresponding to the color image generated by the algorithm of this invention. Detailed Implementation

[0019] You can copy and paste it directly into PowerPoint or use it directly.

Claims

1. A teaching demonstration software algorithm for double integrals based on three-dimensional visualization and dynamic segmentation, characterized in that, Including the following aspects The A1 mathematical modeling module is used to define the integrand and the integration region, select the bounded region, and calculate the function value, the exact integral value, and the area of ​​the integration region. The A2 grid partitioning module is used to implement progressive region partitioning, generate an n×n grid, and provide a non-uniformly progressive sequence of partition numbers [1, 2, 4, 6, 8, 10, 12]. The A3 geometry construction module is used to construct a three-dimensional geometric model of the curved-top cylinder based on the mesh generated by the mesh segmentation module, generate the vertex coordinates and face topology of the cuboid cylinder unit, and calculate the cylinder height according to the sampling strategy. The A4 rendering engine module is used to perform transparent surface rendering and solid cylinder color encoding, including rendering semi-transparent reference surfaces, rendering cylinder sets, rendering bottom mesh and boundary lines, and continuously changing the visual angle as solid cylinders are added. The A5 calculation and analysis module is used to calculate the Riemann sum, precise integral value, and error indicators in real time, including absolute error, relative error, and error percentage. The A6 error visualization module is used to generate error marker lines between approximate geometry and real curved surfaces. When the number of segments is less than a preset threshold, the error lines are displayed. The A7 animation generation module is used to synthesize animation sequences from multiple perspectives and generate teaching demonstration animation files. The A8 user interaction module provides an interface for parameter adjustment, including setting the number of segments, adjusting the viewing angle, switching display characteristics, and selecting the sampling strategy.

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