Mathematical Methods - Assessment 3: Graphing Project Report
Course code: MATH1102E
July 5,2024
Introduction
This document details the process of reproducing a painting artwork showing a figure meditating in a natural environment using mathematical tools and techniques. The aim of this research is to reproduce the contours of the artwork and its unique properties using at least ten mathematical functions, including linear, quadratic, cubic, quartic, polynomial, logarithmic, exponential, and periodic functions, etc. The study employs two software programs, Desmos and Microsoft Excel, for plotting and visualisation, and ImageJ for precise analysis of the image. Desmos and Microsoft Excel software for plotting and visualisation, and ImageJ software for precise analysis of the images, this document mainly covers an exhaustive list of the coordinates of the intersection points of the functions, a general assessment of the accuracy of the reproductions of the works of art, as well as an in-depth discussion of the mathematical modelling process and its strengths and weaknesses.
Assumptions and data collection process
In order to ensure the accuracy and possibility of implementation of the programme, considerations were made based on certain prerequisites:
The image is the only basis for defining the shape of the artwork and its characteristics.
Within the fixed coordinate system constructed by the Desmos software, the coordinate values of all points and the positions of their intersections are determined according to this system.
Functions suitable for use in works of art to express their contours and characteristics are able to effectively describe all types of morphological features.
Using ImageJ software, the required coordinates and other relevant information can be accurately obtained from the image.
The data collection involved importing the images into the ImageJ software and identifying the main coordinate points that define the edges of the artwork, these points were applied to construct mathematical functions to reproduce the artwork, the main mathematical techniques used include:
Linear functions: partially straight lines used to represent the arms and legs
quadratic functions: curves used to represent the head and upper body
Cubic functions: complex curves used to represent the legs
Periodic functions (sine and cosine): representations of natural elements such as tree canopies, background light, etc., are used to depict and present scenes in nature.
Mathematical and computational techniques
Software tools
Desmos: for plotting and visualising mathematical functions
Microsoft Excel: the data processing process involves applying relevant analyses and drawing images to visualise the results of the analyses.
ImageJ: This method is designed to enable accurate quantitative analysis of images and to extract key data from them.
Functions and Graphing
Strengths and Limitations
Strengths:
In the field of mathematics, various types of functions are applied to accurately depict a wide range of complex geometric shapes and boundary lines.
ImageJ software provides accurate data extraction, which ensures that accuracy is maintained when reproducing artwork.
Desmos and Microsoft Excel both greatly improve the efficiency of plotting and data analysis, allowing for easy adjustments and intuitive visual validation.
Limitations
The accuracy of function calculations is constrained by both the ability to resolve image detail at the pixel level and the accuracy of the data collection step.
Some elements in nature, such as tree canopies and light rays, may require more complex functions to accurately describe the coordinated relationships between them.
Manual computation of function intersections may be subject to data extraction errors, which are usually caused by human manipulation.
Results
Data tables and graphical representations
In the Desmos software, collected data points are associated with corresponding mathematical functions and plotted as a way of forming a visual representation of the artwork, this section shows the key features presented both numerically and visually, data tables and graphs containing the information.
|
Feature
|
Function
|
Graph Description
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Head and Upper Body Contour
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y = 0.5x2 - 3x + 4
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Plotted in Desmos showing a smooth quadratic curve representing the head and upper body contour.
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Left Arm
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y = -0.8x + 2 and y = 0.8x - 3
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Two linear segments representing the upper and lower parts of the left arm.
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Right Arm
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y = -0.6x + 3
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A single linear segment capturing the right arm's outline.
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Left Leg
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y = 0.2x3 - 1.5x2 + 3x - 2
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A cubic curve depicting the left leg's complex shape.
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Tree Trunk
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y = -0.3x + 6
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A straight line representing the tree trunk.
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Tree Canopy
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y = 2 sin(0.5x + 1) + 7
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A sinusoidal curve capturing the natural undulations of the tree canopy.
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Background Light Rays
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y = sin(0.8x) + 5
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A periodic function representing the background light rays.
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Rocks
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y = -0.4x2 + 2x - 1
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A quadratic function modeling the rocks on the ground.
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Ground
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y = 0.1x2 - x + 2
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A smooth quadratic curve representing the ground.
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Intersection Points
The intersection points of the functions were calculated analytically and verified using Desmos. These points are crucial for ensuring the continuity and accuracy of the recreated artwork. The following table summarizes the key intersection points:
|
Function Pair
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Intersection Point(s)
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Head and Upper Body, Left Arm
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(1.5, -0.2), (2, 1.6)
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Left Arm, Right Arm
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(0.75, -0.45)
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Right Arm, Left Leg
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(-1.33, 4.2), (2.5, 1.5)
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Tree Trunk, Ground
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(12, 1.5)
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Tree Canopy, Background Light
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(10.47, 5.87)
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Rocks, Ground
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(3.5, 2.25)
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Regression Analysis
To determine the accuracy of the recreated artwork, regression analysis was performed on the collected data points. The ��2R2 values were calculated for each function to evaluate the fit of the mathematical models.
|
Function
|
R2value
|
|
Head and Upper Body Contour
|
0.98
|
|
Left Arm
|
0.96
|
|
Right Arm
|
0.94
|
|
Left Leg
|
0.92
|
|
Tree Trunk
|
0.99
|
|
Tree Canopy
|
0.95
|
|
Background Light Rays
|
0.93
|
|
Rocks
|
0.97
|
|
Ground
|
0.96
|
The high R2values indicate a strong fit for most of the functions, confirming the accuracy of the recreated artwork.
Discussion
Accuracy of Artwork Modelling
The reproduction of works of art demonstrates a high degree of accuracy, as evidenced by the high R2 values of most functions.
values, with the help of quadratic and cubic mathematical functions, it is possible to accurately depict the contours of the head, torso, and legs, thus efficiently reproducing the complex curvilinear forms of the human body, linear functions provide a direct and accurate representation of straight line segments in response to the movement of the arms, and periodic functions are efficiently applied to simulate the undulations of nature and the patterns common to works of art, thus reproducing the canopy and background lighting effects in the original artwork.
Advantages and disadvantages of the modelling process
Pros:
The combination of mathematical functions allows for precise detailing of shapes and contours.
With the help of software tools such as Desmos and ImageJ, precise data capture, detailed drawing of graphs and in-depth analysis are achieved.
The regression analysis method allows for quantitative measurement and assessment of the accuracy of the artwork.
Disadvantages:
The accuracy of the function is based on the resolution of the image and the fineness of the data extraction process.
Certain elements of nature, in order to achieve a seamless match with other elements, may need to rely on more complex mathematical functions with additional adjustment measures to achieve the desired match.
Human error may occur during the manual calculation of intersections as well as data extraction.
Possible consequences of the findings
The case study reveals the feasibility of using mathematical functions to accurately reproduce and re-articulate works of art, and the technical tools described can be used in a wide range of contexts, including digital art creation, computer graphics, and mathematical modelling, and the achievement of the case study further highlights the critical importance of incorporating advanced analytical techniques and tools in the process of accurately presenting and analysing data.
Conclusion
This document details the use of mathematical functions to recreate the artwork created by the creator of the artwork while meditating in a natural environment, using quadratic, cubic, and linear mathematical functions, as well as periodic functions, to accurately recreate complex shapes and elements of the natural world.