Similarity Transformation Calculator

Math Geometry • Transformations and Symmetry

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

Similarity Transformation Calculator – Dilation & Rigid Motions

Apply a similarity transformation in 2D: a dilation by factor \(k\) combined with a rigid motion (rotation/reflection + translation). Visualize the overlay and see how lengths scale by \(|k|\) and areas by \(k^2\).

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

Similarity settings

Mode: Scale factor \(k\):

Dilation center \(c_x\): Dilation center \(c_y\):

Tolerance for similarity check \(M^T M \approx k^2 I\):

Rigid part (orthogonal \(Q\)):

Rotation angle \(\theta\) (degrees):

Translation \(t_x\): Translation \(t_y\):

Preset form: \[ x' = k\,Q\,(x-c)+c+t \] where \(Q\) is rotation/reflection (orthogonal).

Input geometry

Apply to: Points (one per line):

Format: (x,y) or x,y. Parentheses inside expressions (like sqrt(2)) are supported.

Close shape (connect last → first) when using polyline: Display precision:

View options

Axis units label: Show grid: Show axes: Show transformed overlay: Show dilation center: Animation progress:

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

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View (square units • interactive)

Original geometry is solid. Transformed geometry is dashed. Use the slider or Play to animate.

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Theory

What is a similarity transformation?

A similarity transformation preserves shape (angles stay the same) but can change size. In the plane, it can be built from:

a uniform dilation (scale) by factor \(k\),
an isometry (rotation/reflection + translation).

\[ x' = k\,Q\,(x-c) + c + t \]

Here \(Q\) is an orthogonal matrix (\(Q^TQ=I\)): it represents a rotation (\(\det Q=+1\)) or reflection-type motion (\(\det Q=-1\)). The point \(c\) is the dilation center.

How lengths and areas scale

Similarity transformations have very clean scaling rules:

\[ \text{Lengths} \times |k|, \qquad \text{Areas} \times k^2 \]

(In 3D, volumes scale by \(k^3\).) Rotations/reflections and translations do not change distances—only the dilation factor affects size.

Matrix form and the similarity test

If we write the linear part as \(M = kQ\), then:

\[ M^TM = (kQ)^T(kQ) = k^2 Q^TQ = k^2 I. \]

That gives a practical check:

If \(M^TM\) is (approximately) a scalar multiple of \(I\), the map behaves like a uniform scale + rigid motion.
The scale factor is \(k = \sqrt{\tfrac{1}{2}\mathrm{tr}(M^TM)}\) in 2D (average squared stretch).

\[ \hat{k}=\sqrt{\tfrac{1}{2}\mathrm{tr}(M^TM)}. \]

Worked example: dilate then rotate

Suppose a rectangle is dilated by \(k=0.5\) about the origin and then rotated by \(\theta=30^\circ\). Take a point \(x=(4,2)\) (one corner).

\[ R_{30^\circ}= \begin{bmatrix} \cos 30^\circ & -\sin 30^\circ\\ \sin 30^\circ & \cos 30^\circ \end{bmatrix} = \begin{bmatrix} \tfrac{\sqrt{3}}{2} & -\tfrac{1}{2}\\ \tfrac{1}{2} & \tfrac{\sqrt{3}}{2} \end{bmatrix}. \]

The similarity linear part is \(M = kR_{30^\circ} = 0.5R_{30^\circ}\). Then:

\[ x' = Mx = 0.5\,R_{30^\circ}\,(4,2) = 0.5\,(4\cos 30^\circ - 2\sin 30^\circ,\; 4\sin 30^\circ + 2\cos 30^\circ). \]

Your calculator plot overlays the original figure and the transformed one, showing they are similar (same angles, scaled size).

Map scaling intuition (website tip)

A classic similarity in real life is a map scale. If a map uses scale factor \(k=\tfrac{1}{50{,}000}\), then a 2 cm segment on the map corresponds to a real-world distance multiplied by \(\tfrac{1}{k}\). Areas on the map scale by \(k^2\), which is why area measurements on maps must be handled carefully.

Frequently Asked Questions

What is a similarity transformation in 2D geometry?

A similarity transformation preserves shape (angles stay the same) but changes size by a uniform scale factor k. It can be written as x' = k Q (x - c) + c + t, where Q is a rotation or reflection, c is the dilation center, and t is a translation.

How do lengths and areas change under a similarity transformation?

All lengths scale by |k| and all areas scale by k^2. Rotations, reflections, and translations do not change distances; only the dilation factor affects size.

What does the condition M^T M approx k^2 I mean?

For a true similarity linear part, M must have the form M = k Q with Q orthogonal, so M^T M equals k^2 times the identity matrix. The calculator uses a tolerance to decide how closely the entered matrix matches this similarity behavior.

How do I enter a polyline or polygon for this calculator?

Select the polyline/polygon option and enter one vertex per line in (x,y) or x,y format in the order you want them connected. Turn on Close shape if you want the last vertex connected back to the first.

Similarity Transformation Calculator – Dilation & Rigid Motions

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Frequently Asked Questions

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