Reference Frames (Fundamentals)

Defines the origin, axes, and orientation used to measure position and motion.

Conceptual context

Every measurement of position requires a starting point and a reference direction. When we say an object is "3 meters away", the mandatory question is: "3 meters from where and in which direction?". A reference frame is exactly that answer: an origin, a set of axes, and an orientation that turn the question into a concrete calculation. The central physical question of this leaf is: how does the position of an object change when transforming coordinates from one reference frame to another?

🟢 Essential level

Imagine you are standing in the center of a square and your friend is at a corner. You say they are "5 meters to the right and 3 meters ahead".

Your friend, standing at the corner, would say they are "at their own origin". Both of you are right: each uses a different reference frame. A reference frame is simply the choice of where zero is and which direction you look. $x$ is the position of the object measured from your origin; $x'$ is the same position measured from your friend's origin. The two numbers are different but describe the same physical point. The displacement $x_0$ between the two origins is the only difference between the two descriptions. When there is also a rotation, the angle $\theta$ indicates how much frame S' rotates relative to the original.

This same reasoning applies equally in two, three, or more dimensions: only the number of coordinates being transformed changes.

🔵 Formal level

Basic formulas

The three fundamental formulas of this leaf transform the coordinates of a point when the reference frame changes.

1D Translation:

1D Position Transformation

$x'$ is the coordinate in frame S'; $x$ is the coordinate in the original frame S; $x_0$ is the position of the S' origin measured in S. The relation is linear with slope 1 and intercept -$x_0$ in the $x'$ vs $x$ graph. The sign is critical: adding $x_0$ instead of subtracting produces an error of 2·$x_0$.

2D Translation:

2D Position Transformation (pure translation)

Each axis transforms independently: $x'$ depends only on $x$ and $x_0$; $y'$ depends only on $y$ and $y_0$. The translation vector ($x_0$, $y_0$) connects the origin of S to the origin of S'. Both components must have the same units as $x$ and $y$.

Real student questions

Why can $x'$ be negative if $x$ is positive? Because $x_0$ can be larger than $x$. If the S' origin is 8 m from S and the point is 5 m from S, then $x'$ is equal to 5 - 8 is equal to -3 m: the point is to the left of the S' origin, even though it is to the right of the S origin.

Does rotation move the physical point? No. Rotation changes the reference frame S', not the position of the point in space. The point remains in the same place; only how we describe it with coordinates changes.

What happens if translation and rotation are applied in the wrong order? The result is different, except in special cases. The composition of transformations is not commutative: translating then rotating is not the same as rotating then translating, because the rotation acts from the origin of the intermediate system.

Final synthesis

A reference frame is not a property of physical space but a choice by the observer. Position $x'$ in S' and position $x$ in S describe the same physical point with different numbers. The formula for $x'$ converts between descriptions: changing the reference frame does not move the point, only its numerical representation.