Understanding the Transformation of 3D Points to 2D Coordinates on a Plane

Discover how to accurately convert 3D points to local 2D coordinates on a defined plane with practical explanations and insights.
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This video is based on the question https://stackoverflow.com/q/74207329/ asked by the user 'shadespirit' ( https://stackoverflow.com/u/7968355/ ) and on the answer https://stackoverflow.com/a/74207545/ provided by the user 'Scott Settlemier' ( https://stackoverflow.com/u/19727548/ ) at 'Stack Overflow' website. Thanks to these great users and Stackexchange community for their contributions.

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Transforming 3D Points to Local 2D Coordinates on a Plane

When working in 3D space, you might come across the challenge of projecting 3D points onto a 2D plane. This is a common requirement in fields like graphics programming, robotics, and spatial analysis. In this guide, we'll walk through the concepts and methods necessary to return local 2D coordinates from a 3D point on a defined plane.

The Problem at Hand

An individual is facing difficulties converting a point in 3D space to its corresponding 2D coordinates on a defined plane. The problem includes having a plane specified by its origin and direction (or boundaries), and they wish to determine where a point on this plane falls in local 2D coordinates.

The Starting Point

The provided code snippet attempts to perform this transformation, but there are a few key considerations and potential pitfalls that we need to address. Here's the code in question:

[[See Video to Reveal this Text or Code Snippet]]

Understanding the Solution

The code appears to be on the right track, but there are important conditions that must be met to ensure its success. Let's break down these critical aspects.

Orthogonality of Direction Vectors

Orthogonal Directions: For planeXDir and planeYDir to properly define a plane, they must be orthogonal (at right angles to each other). If they are not orthogonal, the projection may yield unexpected results.

Unit Vectors: Ideally, these direction vectors should be unit vectors (length of 1). This guarantees consistent scaling in the transformation. If they aren't unit vectors, you may unintentionally apply a scaling factor that distorts the resulting coordinates.

Steps for Accurate Transformation

Verify Direction Vectors:

Ensure planeXDir and planeYDir are orthogonal. A simple dot product check can confirm this:

[[See Video to Reveal this Text or Code Snippet]]

Normalization:

If the direction vectors are not of unit length, normalize them to ensure they represent unit vectors:

[[See Video to Reveal this Text or Code Snippet]]

Point Calculation:

Proceed with the original calculation for finding the local coordinates:

[[See Video to Reveal this Text or Code Snippet]]

Conclusion

By paying attention to the orthogonality and normalization of your direction vectors (planeXDir and planeYDir), you can successfully project a 3D point onto a 2D plane and obtain the correct local coordinates. These considerations help avoid common pitfalls and ensure that your projection works as intended.

If you find yourself still struggling, consider revisiting the definitions and properties of vectors and spatial geometry to strengthen your foundation in this area.

Feel free to leave your questions and thoughts below. Happy coding!

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