Practical applications of the dot product
I recently started at Standard Cyborg where I’ve been ramping up on Computational Geometry. I’ve started diving into our lower level source code to see how it ticks. This post documents my learnings about the dot product.
What we’ll cover
- Projecting a vector onto a vector
- Finding the orthogonal component of a vector to another vector
- Finding the shortest distance from a point to a segment
Projecting a vector onto a vector
Projecting a vector is one of the simpler practical things we can do with a dot product
Below is a proof explaining how the demo works. After that is a few shorter related demos and proofs.
Math notation refresher
As a courtesy to those who haven’t touched Math in a while, here’s some notation Linear Algebra notation we’ll be using:
Let the “orange” to “blue” vertices be our axis vector and “orange” to “green” be our offshoot vector. This is the largest proof so we’ll take it on in 3 steps.
Step 1: Resolving cosine
We have an angle θ between our 2 vectors. This angle has 3 scenarios (assuming -π ≤ θ ≤ π):
In our first scenario (-π/2 ≤ θ < π/2), we first focus on 0 ≤ θ < π/2. For the sake of brevity, I’m going to handwave over the existence of a right triangle with our projection vector. Please see the gists at the end for a robust proof.
By the cosine trigonometric identity, we have:
For -π/2 < θ< 0, we build a similar triangle and yield the same cos(θ) equation.
For our second scenario (θ = -π/2, θ = π/2), our offshoot vector is orthogonal and thus our projection vector is the zero vector. We can still use the same cos(θ) equation.
For our third scenario (-π ≤ θ <-π/2, π/2 < θ ≤ π), we will focus on π/2 < θ ≤ π. We build a right triangle similar to above but ours is with φ:
For our -π ≤ θ <-π/2 case, we build a similar triangle and yield the same cos(θ) equation.
Now that we’ve resolve cosine for each of our scenarios, we can consolidate them. If cos(θ) ≥ 0, then let d = 1. Otherwise, let d = -1. This allows us to restate:
Additionally, we can define our projection unit vector using d:
Step 2: Restructure dot product
To make our final derivation easier, we’re going to restructure the dot product a little. First, let’s start by writing out more knowns:
With these knowns, we restructure our axis projection vector as follows:
Step 3: Calculate projection vector
And there we go, we have our formula for a projection vector:
Finding the orthogonal component of a vector to another vector
With the calculation above, we can derive an orthogonal component of a vector with respect to another vector
This is practical for computing an angle between 2 vectors around a given axis in 3 dimensional space (e.g. rotation around a fixed axis).
For the proof, we’ve already resolved this vector via our right triangle construction; it’s the “opposite” edge (i.e. our green line). As a result, we can use vector addition to resolve our orthogonal vector:
For some sanity, we can also think of our vector space with our axis as an actual axis:
Finding the shortest distance from a point to a segment
For this proof, we will solve it in a piece-wise fashion. There are 3 scenarios:
In the “behind” and “ahead of” scenarios, the shortest distance will be to the closest vertex. In the “between” scenario, the shortest distance will be a perpendicular line from the segment to the point.
To determine which of these scenarios our point falls into, we find the projection vector and compare it to our segment as a vector. We simplify this a little so it’s a comparison between their scalars, not their vectors
This ratio tells us how much of our offshoot vector is projecting onto our axis.
If the ratio is negative, then we know the point is “behind” our segment and our length is the distance from our “orange” point to “green” point:
Otherwise if the ratio is greater than 1, then we know our projection vector is longer than the axis vector itself. Thus, the point is “ahead” of our segment and our length is the distance from our “blue” point to our “green” point:
Otherwise, the ratio is between 0 and 1 inclusively, so our projection vector falls onto the segment. Thus, the shortest distance is a line perpendicular to the line. We’ve calculated this before via the orthogonal component of a vector:
We can shorten this a little bit with some variable reuse:
Here’s our final solution in a piece-wise format:
- Robust proof, KaTeX equations, and screenshots: https://gist.github.com/twolfson/207556d7ac0cd5d04fa283f6062841ab
- KaTeX screenshotting tool: https://github.com/StandardCyborg/katex-screenshot
- Projection vector demo gist: https://gist.github.com/twolfson/c6927d3c854087f74096e65c0db87279
- Orthogonal vector demo gist: https://gist.github.com/twolfson/687bda8e59b1785d7eb7c9efcb095c54
- Distance to segment demo gist: https://gist.github.com/twolfson/5b1a1ed21a77bc0fdd215e80ee7225cb