How to calculate projection point?

Finding Your Footing: How to Calculate the Projection of a Point onto a Line

Projecting a point onto a line is a fundamental concept in geometry with applications ranging from computer graphics and robotics to physics and machine learning. While the concept might seem abstract, the calculation itself is surprisingly straightforward, relying on the power of vector decomposition. This article will guide you through the process, explaining the underlying principles and providing a clear step-by-step approach.

Forget complex formulas; the core idea is simple: we want to find the point on a line that’s closest to a given point. This closest point is the projection. We achieve this by decomposing the vector connecting the given point to any point on the line into two components: one parallel to the line and one perpendicular to it. The point on the line corresponding to the parallel component is our projection.

Let’s break down the process with a specific example. Suppose we have:

• Point P: (xₚ, yₚ) – The point we want to project.
• Line L: defined by a point A (xₐ, yₐ) on the line and a direction vector v = (vₓ, vᵧ). The direction vector indicates the line’s orientation.

Step 1: Form the Vector from A to P

First, we create a vector w that connects point A on the line to point P:

w = P – A = (xₚ – xₐ, yₚ – yₐ)

Step 2: Calculate the Projection of w onto v

This is where vector projection comes into play. The projection of w onto v is given by the formula:

*projᵥ(w) = ((w • v) / ||v||²) v**

Let’s dissect this formula:

• w • v: This represents the dot product of vectors w and v. The dot product is calculated as (wₓ vₓ) + (wᵧ vᵧ). It essentially measures how much the two vectors point in the same direction.
• ||v||²: This is the squared magnitude (length) of vector v, calculated as vₓ² + vᵧ².
• v: This is the direction vector of the line.

The result of this calculation, projᵥ(w), is a vector parallel to the line v.

Step 3: Find the Projected Point

Finally, to find the coordinates of the projected point, let’s call it P’, we add the projection vector projᵥ(w) to the point A on the line:

P’ = A + projᵥ(w)

This gives us the coordinates of the projected point P’.

Example:

Let’s say P = (3, 4), A = (1, 1), and v = (1, 2).

1. w = (3-1, 4-1) = (2, 3)
2. w • v = (21) + (32) = 8
3. ||v||² = 1² + 2² = 5
4. projᵥ(w) = (8/5) * (1, 2) = (8/5, 16/5)
5. P’ = (1, 1) + (8/5, 16/5) = (13/5, 21/5)

Therefore, the projection of point P onto the line defined by A and v is (13/5, 21/5).

This method provides a robust and accurate way to calculate the projection of a point onto a line. While the formulas may seem daunting at first glance, breaking them down step-by-step reveals the underlying simplicity and elegance of the process. Understanding this fundamental concept opens doors to a deeper appreciation of vector algebra and its numerous applications in various fields.