Geometric optics — rectilinear propagation, reflection, refraction: Foundations

Before you can follow the parent note Geometric optics, you must own a small toolbox of ideas. We build them in order — each one uses only what came before it.

1. A point and its coordinates $(x, y)$

Look at the first figure. The bottom line is the $x$ -axis, the side line is the $y$ -axis, and the corner where they cross is the origin $(0, 0)$ . The point $A$ marked in lavender sits at $(0, a)$ — zero steps right, $a$ steps up.

Figure — Geometric optics — rectilinear propagation, reflection, refraction

Why the topic needs this. Every derivation in the parent note (reflection, refraction) places light-sources and screen-points at spots like $A = (0, a)$ and $B = (d, b)$ . Those pairs of numbers are just points on this map. If you can't read $(x, y)$ , none of the algebra means anything.

2. Distance between two points — where the square root comes from

Why a square root, and why this tool? Because of the Pythagoras rule for right triangles: the sideways gap and the up-down gap are the two short sides of a right triangle, and the straight distance is the long slanted side (the hypotenuse). Pythagoras says $(long side)^{2} = (side_{1})^{2} + (side_{2})^{2}$ , so to undo the square we take the square root. Look at figure 2: the mint dashed lines are the two short sides, the coral line is the hypotenuse we want.

Why the topic needs this. The path a ray travels is a sum of straight segments; each segment's length is one of these square roots. The "least-time" idea works by making the total length (or time) as small as possible — so we must be able to write down each length first.

3. The right triangle, and the angle inside it

In figure 3 the angle $θ$ sits at the top corner. The opposite side (butter colour) lies across from it; the adjacent side (mint) touches it; the coral hypotenuse is the ray itself.

Why the topic needs this. In both reflection and refraction the ray, the boundary, and the normal form a right triangle. The angle the ray makes with the normal is our $θ$ . To measure "how slanted" the ray is, we need a number that captures that slant — that number is the sine, next.

4. The sine — a number that measures "slant"

Why this tool and not another? We want one number that says how tilted the ray is, that does not depend on how long we drew the ray. Dividing the opposite side by the hypotenuse gives exactly that: stretch the ray twice as long and both lengths double, so the ratio is unchanged. That ratio grows from $0$ (ray flat along the reference line) to $1$ (ray at right angles), so it perfectly encodes "amount of slant."

5. The normal — the line we measure angles from

Figure 4 shows a mirror (the flat surface) with the slate normal standing straight up. The incoming ray makes angle $θ_{i}$ with the normal, not with the mirror. Every angle in optics — incidence, reflection, refraction — is measured from this line.

Why the topic needs this. All three laws (reflection, Snell) are stated with angles-from-the-normal. Without the normal, "angle of incidence" has no fixed meaning.

6. Speed of light $c$ , speed in a medium $v$ , and the ratio $n$

Because $v$ can never beat $c$ , the fraction $c / v$ is always at least $1$ , so $n \geq 1$ always.

Why this ratio and not a subtraction? We care about how many times slower, not how much slower in km/s — a ratio compares fairly across materials and drops the units. That ratio is exactly the quantity that controls how sharply a ray bends, which is why the parent note builds Snell's law out of it.

Recall Quick check on speeds

If $n = 2$ for a diamond, is light faster or slower inside it, and by what factor? ::: Slower — exactly half of $c$ , since $v = c / n = c /2$ .

7. The tiny idea of a derivative (only what you need)

Why this tool? "Least time" is a minimisation question, and the one universal trick for finding a minimum is: find where the slope is zero. You don't need to master calculus here — just accept that "flat slope = valley bottom = least time," and that this is where the sines pop out.

The prerequisite map

Read it top-down: coordinates and Pythagoras give you lengths; triangles and the normal give you angles as sines; speeds give you the index; the least-time slope-is-zero idea ties them into the two big laws.

Equipment checklist

Recall Am I ready? (cover the answers)

Write $A = (0, a)$ in words. ::: The point zero steps right and $a$ steps up from the origin. Distance from $(x, 0)$ to $(0, a)$ ? ::: $x^{2} + a^{2}$ , straight from Pythagoras. Why a square root in that distance? ::: Pythagoras gives the square of the hypotenuse, so we take the root to undo it. $sin θ$ in words? ::: Opposite side divided by the hypotenuse of a right triangle. Why is $sin θ$ unit-free? ::: It is a length divided by a length, so the units cancel. What is the normal? ::: The line perpendicular to the surface at the point of contact — angles are measured from it. Meaning of $n = c / v$ ? ::: How many times slower light travels in the material than in vacuum; always $\geq 1$ . If $n$ is large, is $v$ large or small? ::: Small — big $n$ means slow light. What does $d T / d x = 0$ find? ::: The value of $x$ where total time is smallest — the bottom of the time-valley. Which single principle uses that flat-slope idea? ::: Fermat's principle — light takes the least-time path.

Connections

Parent: Geometric optics — the note these foundations unlock.
Fermat's principle — the least-time idea behind "slope = zero".
Total internal reflection and Optical fibres — where the index $n$ and Snell's law lead next.
Wavefronts and Huygens' principle — the wave-level picture beneath the ray.