Glide Reflection Transformation Calculator

Math Geometry • Transformations and Symmetry

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

Glide Reflection Calculator – Combine Translation & Reflection

A glide reflection is the composition of a translation along a line and a reflection across the same line. In symbols: \[ G = R_{\ell}\circ T_{\vec{t}} \] where \(\vec{t}\) is parallel to the reflection line \(\ell\).

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

Glide settings

Reflection line:

Preset:

Glide distance \(d\) (signed) along the line:

We choose a unit direction vector along \(\ell: ax+by=c\) as \(\hat{u}=\dfrac{(b,-a)}{\sqrt{a^2+b^2}}\). Then \(\vec{t}=d\,\hat{u}\).

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 reflection line: Show translated + final overlays: Show “footprints” path: Animation progress:

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

Ready

View (square units • interactive)

Original geometry is solid. Translation (along the line) is dotted. Final glide reflection is dashed. The footprint shows the “walking reflection” path (translate → reflect).

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Theory

What is a glide reflection?

A glide reflection is an isometry formed by doing two actions in order:

Translate a shape by a vector that is parallel to a given line \(\ell\).
Reflect the translated shape across that same line \(\ell\).

Using a point \(p\), translation vector \(\vec t\), and reflection across line \(\ell\) denoted \(R_{\ell}\), the glide reflection map is

\[ G(p) = R_{\ell}\big(p+\vec t\big) \]

Key facts: A glide reflection preserves distances and angles (it’s an isometry), but it reverses orientation (like any reflection-type transformation).

Line model \(ax+by=c\) and translation along the line

In this calculator, the reflection line is written as

\[ \ell:\; a x + b y = c \]

A normal vector to the line is \(\vec n=(a,b)\), and its length is \(\|\vec n\|=\sqrt{a^2+b^2}\). A direction vector along the line (perpendicular to \(\vec n\)) is

\[ \hat u = \frac{(b,-a)}{\sqrt{a^2+b^2}} \]

If the glide distance is the signed scalar \(d\), then the translation vector is

\[ \vec t = d\,\hat u \]

Because \(\hat u\) is parallel to the line, it is perpendicular to the normal: \(\vec n\cdot \hat u = 0\). This is important: it guarantees the translation is truly “along the line.”

Reflection formula across \(ax+by=c\)

To reflect a point \(q=(x,y)\) across \(\ell:ax+by=c\), use the standard projection onto the normal direction. Define

\[ s=\frac{a x + b y - c}{a^2+b^2}. \]

Then the reflection is

\[ R_{\ell}(x,y)=(x,y)-2s\,(a,b). \]

Geometric meaning: \(a x + b y - c\) measures signed “how far” the point is from the line along the normal direction.

Putting it together: translate then reflect (the glide)

For an input point \(p\):

\[ q = p+\vec t \quad\text{(translate along \(\ell\))}, \qquad r = R_{\ell}(q) \quad\text{(reflect across \(\ell\))}. \]

So the glide reflection output is \(G(p)=r\).

Animation tip (“walking reflection”): the footprint path for a point is the polyline \(p \to q \to r\) (translate step, then reflect step).

Special lines (common presets)

Line	Reflection \(R_{\ell}(x,y)\)	Glide with distance \(d\) along the line
\(\;y=0\) (x-axis)	\((x,-y)\)	\((x+d,\,-y)\)
\(\;x=0\) (y-axis)	\((-x,y)\)	\((-x,\,y+d)\)
\(\;y=x\)	\((y,x)\)	Translate along \((1,1)\), then swap
\(\;y=-x\)	\((-y,-x)\)	Translate along \((1,-1)\), then reflect

The “glide distance” \(d\) is signed: switching \(d\to -d\) glides the other way along the same line.

Isometry properties

Distance preserved: for any points \(p_1,p_2\), \(\|G(p_1)-G(p_2)\|=\|p_1-p_2\|\).
Angle preserved: the angle between any two segments is unchanged.
Orientation reversed: a reflection is orientation-reversing, so glide reflections are too.
No fixed points (usually): if \(d\neq 0\), there is typically no point with \(G(p)=p\).
Square is a translation: applying a glide reflection twice yields a pure translation along the line by \(2\vec t\).

\[ G^2 \;\text{acts like}\; T_{2\vec t}. \]

Worked example (matches the calculator)

Example: take \(p=(2,3)\), glide along the x-axis by \(d=4\), then reflect across \(y=0\).

\[ q = p + (4,0) = (6,3), \qquad r = (6,-3). \]

Final answer: \(\;G(2,3)=(6,-3)\).

Frequently Asked Questions

What is a glide reflection transformation?

A glide reflection is the composition of a translation along a line and a reflection across that same line. It is an isometry, so it preserves distances and angles, but it reverses orientation.

How is the glide translation vector computed for ax+by=c?

A unit direction vector along the line is u = (b,-a)/sqrt(a^2+b^2). The calculator uses t = d u, where d is the signed glide distance.

How do you reflect a point across the line ax+by=c?

For q=(x,y), compute s = (a x + b y - c)/(a^2 + b^2), then R(q) = (x,y) - 2 s (a,b). The glide reflection applies this after first translating the point.

Why does the footprints path show two steps?

A glide reflection is done in two actions: first translate along the mirror line, then reflect across it. The footprints visualize the path p -> p+t -> R(p+t).

Glide Reflection Calculator – Combine Translation & Reflection

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Frequently Asked Questions

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