How is the perpendicular distance from a 3D point to a projection plane mathematically defined in the video?

Conditions

• A target plane $\mathcal{P}$ has been chosen.
• Projection is perpendicular to that plane.
• The distance is measured using a vector norm.

Reasoning, step by step

1. Identify the original data point $\mathbf{x}$ in the ambient 3D space.
2. Identify the target plane $\mathcal{P}$ onto which the data is projected.
3. Formulate the distance as the minimum over all candidate points $\mathbf{p}$ on the plane.
4. Apply the vector norm to the difference between the original point and the candidate point.
5. Conclude that the minimizing point is the orthogonal projection of $\mathbf{x}$ onto $\mathcal{P}$.

Example

The video displays the formula $d_\perp = \min_{\mathbf{p} \in \mathcal{P}} \|\mathbf{x} - \mathbf{p}\|$ while animating blue points dropping vertically to yellow markers on a green plane.

Common misconceptions

• Assuming the distance is measured along an arbitrary axis rather than the shortest perpendicular path.
• Believing that the formula calculates the distance to the origin rather than to the plane.

Watch the explanation

Explore next

• What does the perpendicular distance represent in the context of dimensionality reduction?
• How does the video visualize the collapse of a 3D cloud onto a 2D surface?