Research and Development Assignment — AI Curve Fitting

🎯 Objective

To estimate the unknown parameters θ, M, and X in the given parametric curve:

[ x = \left(t*\cos(\theta) - e^{M|t|}\sin(0.3t)\sin(\theta) + X\right) ] [ y = \left(42 + t*\sin(\theta) + e^{M|t|}\sin(0.3t)\cos(\theta)\right) ]

using the provided dataset of points (x, y) for ( 6 < t < 60 ).

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🧩 Data

Dataset provided as: xy_data.csv

Variable	Range
( 0° < \theta < 50° )	( -0.05 < M < 0.05 )
( 6 < t < 60 )

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🧠 Approach

1. Model Definition
   Defined a Python function for (x(t)) and (y(t)) based on given parametric equations.

2. Parameter Initialization
   Started with random guesses for θ, M, and X within the allowed ranges.

3. Optimization
   Used SciPy’s minimize (L-BFGS-B) algorithm to minimize the L1 distance between the predicted and observed points.

4. Evaluation
   Compared the fitted curve against actual data points using Matplotlib visualization.

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🧮 Final Results

Parameter	Symbol	Optimal Value
Angle	( \theta )	28.1275°
Exponential Coeff	( M )	0.02133
Offset	( X )	54.901

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✅ Final Parametric Equation (LaTeX Form)

[ \left(t*\cos(0.491) - e^{0.02133|t|}\sin(0.3t)\sin(0.491) + 54.901,\ 42 + t*\sin(0.491) + e^{0.02133|t|}\sin(0.3t)\cos(0.491)\right) ]

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🔗 Desmos Visualization

You can view the curve directly here:
Desmos Link

Paste the equation and set value of t (6<t<60)

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💻 Code

The full implementation (data loading, visualization, optimization, and plotting) is available in
curve_fitting.ipynb.

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📈 Libraries Used

• Python 3.x
• NumPy
• Pandas
• Matplotlib
• SciPy

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✍️ Author

Siddharth B — Research and Development / AI Assignment (Curve Parameter Estimation)