Fonction homographique d'un système optique

Introduction

This tutorial explains a new method for studying the image position of an object through an optical system using a homographic function. Understanding this concept is essential for anyone looking to analyze optical systems in fields such as physics, engineering, or photography.

Step 1: Understand the Homographic Function

• A homographic function relates the coordinates of points in the object and image spaces of an optical system.
• This function can be represented mathematically, allowing for the prediction of image positions based on object coordinates.

Step 2: Identify Key Parameters

Define the following parameters critical for your analysis

• Object Distance (do): Distance from the object to the optical system.
• Image Distance (di): Distance from the optical system to the image.
• Focal Length (f): The distance over which initially collimated rays are brought to a focus.

Step 3: Formulate the Homographic Equation

The homographic relationship can be expressed as

[ \frac{di}{do} = \frac{f + di}{f} ]
• Rearranging this equation allows you to solve for either the image distance or object distance based on known values.

Step 4: Apply the Method

Follow these steps to apply the homographic function

1. Measure the object distance (do).
2. Use the focal length (f) of your optical system.
3. Substitute these values into the homographic equation to find the image distance (di).

Step 5: Visualize the Results

• Create a diagram showing the object, optical system, and image.
• Label the distances (do, di, f) to better understand the relationships and verify your calculations visually.

Step 6: Test with Different Scenarios

Experiment with varying object distances and focal lengths

• Examine how changes affect the image position.
• Use real-world objects and optical systems for practical insights.

Conclusion

By applying the homographic function to optical systems, you can effectively predict image positions based on object distances and focal lengths. This tutorial provides a foundational understanding and practical steps to apply the method. For further exploration, consider experimenting with different optical setups or extending this method to more complex systems.