Geometrische-Transformations-Rechner

The Geometric Transformation Calculator applies the four foundational transformations of plane geometry — translation, reflection, rotation, and dilation — to a point (x, y) and returns the image point. This tutorial walks through what each transformation does to the point, what it does to a whole shape, and which transformations preserve which properties (length, angle, orientation).

The four transformations at a glance

Transformations that preserve length and angle are called isometries (also called rigid transformations) — they move a shape without distorting it. Translation, reflection, and rotation are isometries. Dilation is not an isometry — it scales the shape larger or smaller. Dilation does preserve angles, so it produces a similar figure (same shape, different size).

Translation — sliding without changing

A translation slides every point of the figure by a fixed amount h horizontally and k vertically. The rule:

(x, y) → (x + h, y + k)

Examples:

• Translate (3, 5) by (h, k) = (2, −1): new point is (3 + 2, 5 + (−1)) = (5, 4).
• Translate (−2, 0) by (4, 7): new point is (2, 7).

Translation is the simplest transformation: every point moves the same way. Shapes keep their size, orientation, and proportions — they just appear in a new location on the coordinate plane. If you translate a triangle, the new triangle is congruent (identical) to the original.

Reflection — mirror flip

A reflection flips the figure across a line called the axis of reflection. The most common axes are the x-axis, y-axis, and the lines y = x and y = −x.

• Reflection over x-axis: (x, y) → (x, −y). The y-coordinate flips sign.
• Reflection over y-axis: (x, y) → (−x, y). The x-coordinate flips sign.
• Reflection over y = x: (x, y) → (y, x). Swap the coordinates.
• Reflection over y = −x: (x, y) → (−y, −x). Swap and negate both.

Reflection preserves distances and angles but reverses orientation — if the original figure has a clockwise order, the reflected figure has counterclockwise (or vice versa). In coordinate geometry this matters: a right-handed coordinate system reflected becomes left-handed.

Examples:

• Reflect (3, 5) over x-axis: (3, −5).
• Reflect (3, 5) over y-axis: (−3, 5).
• Reflect (3, 5) over y = x: (5, 3).

Rotation — turning around a point

A rotation turns the figure around a fixed point (the center of rotation) by a given angle. The most common rotations are about the origin (0, 0) by 90°, 180°, and 270°:

• Rotation 90° CCW (counterclockwise): (x, y) → (−y, x).
• Rotation 180°: (x, y) → (−x, −y).
• Rotation 270° CCW (= 90° CW): (x, y) → (y, −x).

For a general rotation by angle θ about the origin, the formula uses trigonometry: (x, y) → (x·cosθ − y·sinθ, x·sinθ + y·cosθ). The three "nice" cases above come from plugging θ = 90°, 180°, 270° (where cos and sin are 0 and ±1).

Rotation preserves distances, angles, and orientation — it is the only nontrivial isometry that does. Two figures related by rotation are directly congruent (same shape, same size, same handedness, just turned).

Examples:

• Rotate (3, 5) by 90° CCW: (−5, 3).
• Rotate (3, 5) by 180°: (−3, −5).
• Rotate (3, 5) by 270° CCW: (5, −3).

Dilation — scaling

A dilation scales the figure by a constant factor k about a center (usually the origin). The rule for dilation about the origin:

(x, y) → (kx, ky)

Where:

• k > 1: enlargement (figure gets bigger)
• 0 < k < 1: reduction (figure gets smaller)
• k < 0: combined dilation + 180° rotation
• k = 1: identity (no change)
• k = −1: same as a 180° rotation

Dilation preserves angles (similar figures have congruent angles) but does not preserve distances. If you dilate by k = 2, every length doubles, every area quadruples (k²), and if it were a 3D dilation, every volume would be 8 × (k³).

Examples:

• Dilate (3, 5) by k = 2: (6, 10).
• Dilate (3, 5) by k = 0.5: (1.5, 2.5).
• Dilate (3, 5) by k = −1: (−3, −5). Same as rotation 180°.

Composing transformations

You can apply transformations one after another. The order usually matters:

• Translate then rotate is different from rotate then translate (because rotation around the origin uses the origin as a fixed point — translating first moves your figure away from the origin first).
• Reflect then reflect over two parallel lines = a translation perpendicular to those lines by twice the distance between them.
• Reflect then reflect over two intersecting lines = a rotation about their intersection point by twice the angle between them.
• A glide reflection is a reflection followed by a translation parallel to the axis of reflection — it produces footprints in sand pattern.

Real-world applications

• Computer graphics — every 2D / 3D game and CAD tool uses transformation matrices to translate, rotate, and scale models on screen.
• Physics — reference-frame changes are coordinate transformations (Galilean for classical, Lorentz for relativistic).
• Pattern design — wallpaper patterns, tiling, and textile design rely on systematic combinations of these four transformations (the 17 wallpaper groups classify every possible repeating 2D pattern).
• Symmetry — a figure has a symmetry if some non-identity transformation maps it onto itself. A square has 8 symmetries (4 rotations + 4 reflections).

Common mistakes

• Mixing up CCW and CW for rotations. In math, rotation angles are measured counterclockwise by convention. A "rotation by 90°" means 90° CCW unless specified otherwise.
• Confusing reflection over y = x with reflection over the x-axis. The first swaps coordinates: (x,y) → (y,x). The second flips the y sign: (x,y) → (x,−y). These give very different images.
• Forgetting that dilation scales area as k², not k. Doubling all lengths quadruples the area. Many real-world estimation errors stem from this.
• Assuming all transformations preserve orientation. Reflection reverses orientation; the others preserve it.