lesson
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When video game animators make a character run across the screen, the sprite doesn't just slide — it takes a step forward and flips its legs with each stride.
Combining multiple moves into one sequence is called a composition of transformations.
What is a Composition?
A rigid transformation (also called an isometry) is any movement that preserves the exact size and shape of a figure, including translations (slides), reflections (flips), and rotations (turns).
When you perform one rigid transformation and then apply a second one to the result, the starting shape is the pre-image, the middle shape is the intermediate image (A′), and the final shape is the image (A′′).
📊Interactive step-by-step diagram showing a 2-step transformation on a coordinate grid. Step 1: Pre-image triangle ABC (A(1,1), B(3,1), C(1,4)) in vibrant blue. Step 2: Translates right 4, up 1 to become intermediate triangle A'B'C' in dashed amber. Step 3: Reflects over the x-axis to become final triangle A''B''C'' in solid green. Includes a three-state slider or next/prev buttons to step through: 'Start (Pre-image)', 'Step 1: Translate (A')', 'Step 2: Reflect (A'')'. Responsive layout, coordinates clearly labeled next to vertices.
How do we write this mathematically without writing out full sentences each time?
Reading Composition Notation
In math, the small circle ∘ represents composition, and we read the expression (R∘T)(P) as "R of T of P", meaning R(T(P)).
Just like evaluating parentheses in algebra, you always perform the transformation on the inside right first, then apply the outside left transformation to that result.
📊A visual breakdown diagram explaining the notation (r_{y-axis} ∘ T_{(3,-2)})(x, y). Left box highlights T_{(3,-2)} in orange with an arrow labeled '1st: Apply right rule first'. Right box highlights r_{y-axis} in purple with an arrow labeled '2nd: Apply left rule to the new result'. A step arrow shows: (x, y) -> (x+3, y-2) -> (-(x+3), y-2).
What happens when we trace a specific shape point-by-point through a full composition?