Composition of Transformation Calculator

Math Geometry • Transformations and Symmetry

Written by STEM Calculators Team Published January 25, 2026 Updated February 24, 2026

Composition of Transformations Calculator – Sequence Application

Apply a sequence of 2D transformations to a point or shape. View step-by-step results and the net effect as one affine map using homogeneous coordinates.

Inputs accept 1e-3, pi, e, sqrt(2), sin(), cos(), tan(), ln(), log(), abs(). Use * for multiplication.

Input geometry

Apply to:

Point (x, y):

Format: (x,y) or x,y. Expressions allowed (e.g., (sqrt(2), 3*pi/4)).

Matching tolerance (numeric cleanup): Display precision:

Transformation sequence builder

Add steps in order. The calculator applies them top → bottom. Order matters: “rotate then translate” is usually not the same as “translate then rotate”.
Editing fix: click a step row to select it. When selected, changing parameters updates that step automatically.

Add / edit step type:

Translation tx:

Translation ty:

Point update: \(x' = x + t\).

Angle θ (degrees):

Right-hand convention in 2D: positive θ rotates counterclockwise.

Angle θ (degrees):

Center cx:

Center cy:

Rotate around \(c\): \(x' = c + R_\theta (x - c)\).

Scale sx:

Scale sy:

\(A = \begin{bmatrix}s_x&0\\0&s_y\end{bmatrix}\).

Scale sx:

Scale sy:

Center cx:

Center cy:

Scale around \(c\): \(x' = c + S(x-c)\).

Shear factor k (x' = x + k y):

\(A=\begin{bmatrix}1&k\\0&1\end{bmatrix}\).

Shear factor k (y' = y + k x):

\(A=\begin{bmatrix}1&0\\k&1\end{bmatrix}\).

Define \(x' = Ax + t\), where \(A\in\mathbb{R}^{2\times 2}\), \(t\in\mathbb{R}^2\).

a11

a12

a21

a22

Not editing

#	Step	Matrix / info
No steps yet. Add one above.

Tip: Click a row to edit. Selected row turns highlighted. Changing parameters updates that step instantly.

Plot options

Axis units label: Show grid: Show tick labels: Show transformed overlay: Show trace path (reference point): Animation progress:

Drag to pan • wheel/trackpad to zoom • pinch on touch • “Reset view” fits the geometry. Units are square.

Ready

2D view (square units • pan/zoom enabled)

Original geometry is solid. The transformed geometry is dashed. Use the slider/Play to animate the sequence.

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Composition of Transformations Calculator – Sequence Application — Theory

A 2D transformation step is modeled as an affine map: \[ x' = A x + t, \quad A \in \mathbb{R}^{2\times 2},\; t\in\mathbb{R}^2. \] A composition applies multiple steps in order: the output of one step becomes the input to the next.

Homogeneous coordinates (net effect matrix)

Using homogeneous coordinates, we combine linear and translation parts into one \(3\times 3\) matrix: \[ \begin{bmatrix} x' \\ 1 \end{bmatrix} = \underbrace{\begin{bmatrix} A & t \\ 0 & 1 \end{bmatrix}}_{M} \begin{bmatrix} x \\ 1 \end{bmatrix}. \] The calculator reports the net matrix \(M\) for the whole sequence.

Order matters

In general, transformations do not commute: \[ T_2(T_1(x)) \neq T_1(T_2(x)). \] For example, “translate then rotate” usually differs from “rotate then translate”.

Common 2D steps

Step	Matrix \(A\)	Translation \(t\)
Translation \((t_x,t_y)\)	\(\begin{bmatrix}1&0\\0&1\end{bmatrix}\)	\(\begin{bmatrix}t_x\\t_y\end{bmatrix}\)
Rotation by \(\theta\)	\(\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}\)	\(\begin{bmatrix}0\\0\end{bmatrix}\)
Scaling \((s_x,s_y)\)	\(\begin{bmatrix}s_x&0\\0&s_y\end{bmatrix}\)	\(\begin{bmatrix}0\\0\end{bmatrix}\)
Shear X with \(k\)	\(\begin{bmatrix}1&k\\0&1\end{bmatrix}\)	\(\begin{bmatrix}0\\0\end{bmatrix}\)
Reflect across y-axis	\(\begin{bmatrix}-1&0\\0&1\end{bmatrix}\)	\(\begin{bmatrix}0\\0\end{bmatrix}\)

Determinant meaning

The determinant \(\det(A)\) describes area scaling and orientation:

\(|\det(A)|\) = factor by which areas scale
\(\det(A) < 0\) indicates an orientation flip (reflection-like behavior)

Tip: Use the plot slider or Play to see the transformation sequence. The “net effect” matrix shows the single affine map equivalent to the whole sequence.

Frequently Asked Questions

What does composition of transformations mean in geometry?

Composition means applying multiple transformations in sequence, where the output of one step becomes the input to the next. The final result depends on the entire ordered list of steps.

Why does the order of transformations matter?

Most transformations do not commute, so applying T2 after T1 usually gives a different result than applying T1 after T2. For example, rotate then translate is typically not the same as translate then rotate.

What is the net effect matrix in the calculator?

The calculator combines all steps into a single 3x3 homogeneous matrix M that represents the same overall affine map as the whole sequence. This lets you see the one-step equivalent of the full composition.

How do I enter vertices for a polyline or closed loop?

Enter one vertex per line using (x,y) or x,y, and keep vertices in the order the plot should connect them. Use the Close shape option to connect the last vertex back to the first.

What do matching tolerance and display precision change?

Matching tolerance controls numeric cleanup when comparing or simplifying coordinates after many steps, which helps with rounding-heavy inputs. Display precision only changes how many decimals are shown, not the underlying calculation.

Composition of Transformations Calculator – Sequence Application

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Frequently Asked Questions

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