Every point on a graph can be moved, turned, flipped or resized with a short rule. Choose the transformation, type the points, and set the angle, mirror line, shift or scale factor; this calculator applies the exact rule to each point, shows the working, and draws the before and after picture on one set of axes so you can see the shape move.
How to use this calculator
- Choose the transformation: rotation turns points around a centre, translation slides them by a fixed vector, reflection flips them in a mirror line, and dilation resizes them from a centre by a scale factor.
- Type the points, one per line as x, y. A single point works; a shape is just its corners listed in order, and the plot joins them in that order.
- Set the parameters for the chosen transformation: the angle and centre, the shift vector, the mirror line, or the scale factor and centre.
- Press Calculate. Every point is transformed, the rule and the numbers are shown, and the plot draws the original shape in grey and the image in green on the same axes.
The four transformation rules
Each transformation is a rule that sends every point (x, y) to a new point (x′, y′). The rules below are the ones this calculator applies, in the same form taught in geometry courses.
Translation
A slide: every point moves by the same vector. Size, shape and orientation are unchanged.
Rotation about a centre (cx, cy) by angle θ
y′ = cy + (x − cx) sin θ + (y − cy) cos θ
Positive angles turn counterclockwise, the mathematical convention. Written as a matrix about the origin, the rotation is [cos θ −sin θ; sin θ cos θ]. Turning about another centre means: move the centre to the origin, rotate, then move back, which is exactly what the formula does. A quarter turn counterclockwise has the memorable rule (x, y) → (−y, x).
Reflection
Line y = x: (x, y) → (y, x) Line y = −x: (x, y) → (−y, −x)
Line y = k: (x, y) → (x, 2k − y) Line x = k: (x, y) → (2k − x, y)
A flip: the mirror line is the perpendicular bisector of the segment joining each point to its image, so the point and its image sit the same distance from the line on opposite sides. Reflection across the x-axis is the most searched case: it simply negates the y coordinate.
Dilation from a centre (cx, cy) with scale factor k
A resize: each point moves along the ray from the centre so its distance from the centre multiplies by k. A factor between 0 and 1 shrinks the shape, a factor above 1 enlarges it, a factor of 1 leaves it alone, and a negative factor also turns it through half a turn about the centre. A dilation calculator in geometry is this rule; the centre matters as much as the factor.
Worked examples
Example 1: rotate (2, 3) by 90 degrees about the origin
The quarter-turn rule (x, y) → (−y, x) gives (2, 3) → (−3, 2). This is the preloaded case, and the plot shows the point jump from the first quadrant to the second, the same distance from the origin as before: rotation never changes distances from the centre.
Example 2: rotate (2, 3) by 90 degrees about the centre (1, 1)
Move relative to the centre: (2 − 1, 3 − 1) = (1, 2). Quarter turn: (−2, 1). Move back: (1 − 2, 1 + 1) = (−1, 2). Same point, same angle, different centre, different answer: the centre is part of the transformation.
Example 3: reflect (4, 1) in three lines
Across the x-axis: (4, −1). In the line y = x: (1, 4), the coordinates swap. In the horizontal line y = 2: (4, 3), because the point sits 1 unit below the line, so its image sits 1 unit above it.
Example 4: dilate a triangle by 2 from the origin
The triangle (0, 0), (4, 0), (0, 3) with k = 2 becomes (0, 0), (8, 0), (0, 6): every coordinate doubles, the sides double (3, 4, 5 becomes 6, 8, 10), and the area multiplies by k² = 4, from 6 to 24. This is the second example button.
Reading your results
The table lists each point beside its image, in the order you typed them. For a shape, read down the image column to get the new corners in order.
The rule line shows the exact rule with your numbers substituted, including the cos and sin values for a rotation, so the arithmetic can be checked by hand.
The plot is the real check. Rotation and reflection keep the shape’s size: the green image should look congruent to the grey original. Translation keeps its orientation too; reflection flips it (a left hand becomes a right hand); dilation keeps the shape but changes the size, and every image point should sit on the ray from the centre through its original.
Distances: under rotation, reflection and translation, distances between points are preserved, so side lengths in the image equal those in the original. Under dilation they multiply by |k|.
Common mistakes
- Turning the wrong way. Positive angles are counterclockwise by mathematical convention, but many graphics and games use clockwise. If your image turned the wrong way, negate the angle rather than swapping the coordinates.
- Forgetting the centre. A rotation or dilation about (1, 1) is not the same as about the origin. The worked examples show the same point landing in two different places purely because the centre changed.
- Reflecting across the x-axis and changing x. The x-axis reflection negates y only: (4, 1) → (4, −1). If the x coordinate changed, the mirror line was probably the y-axis.
- Scaling coordinates when the centre is not the origin. Doubling (x, y) only works for a dilation centred at the origin. For any other centre, the offsets from the centre double, not the coordinates themselves.
- Expecting dilation by a negative factor to be a plain shrink. A negative k sends each point through the centre to the opposite side, a resize combined with a half turn. k = −1 about the origin is exactly a 180 degree rotation.
Limitations
- One transformation at a time. Chains (for example rotate then translate) are done by running the calculator again with the image points as the new input; the order of a chain matters, because transformations do not generally commute.
- Points are two-dimensional and the lines offered are the coordinate axes, the two diagonals, and horizontal or vertical lines. Reflections in slanted lines other than y = ±x are not offered.
- Up to 20 points per run. Coordinates and parameters are plain numbers in the same unit throughout; results are shown to 4 decimal places with trailing zeros removed, which can make a mathematically exact 0 appear from a value smaller than 0.00005.
- The plot is a schematic on auto-fitted axes, drawn to help you check the movement; it is not a measured survey drawing.
Frequently asked questions
How do you rotate a point 90 degrees counterclockwise?
About the origin, swap the coordinates and negate the new x: (x, y) → (−y, x). So (2, 3) becomes (−3, 2). Clockwise 90 degrees is the mirror rule: (x, y) → (y, −x). For other angles the calculator uses cos and sin in the matrix formula above.
What is the rule for reflection across the x-axis?
(x, y) → (x, −y): the x coordinate stays, the y coordinate changes sign. It is the single most common reflection, and the preloaded mirror line in the reflection panel.
What does a dilation with scale factor 0.5 do?
It halves every distance from the centre: the shape shrinks towards the centre, keeping its proportions. With the centre at the origin it simply halves both coordinates. Areas multiply by k² = 0.25.
Does the order of transformations matter?
Yes. Translate then rotate generally gives a different result from rotate then translate. This calculator applies one transformation per run, exactly as entered, so a chain keeps its order when you feed the image points into the next run.
What stays the same under each transformation?
Translation, rotation and reflection are rigid motions: lengths, angles and areas are all preserved, and reflection alone reverses orientation. Dilation preserves angles and proportions but scales lengths by |k| and areas by k².
Can this calculator do time dilation?
No. Time dilation is a relativity formula about clocks and speeds, a different topic that shares only the word. This page covers geometry: dilations of points and shapes on a graph, with a centre and a scale factor.
Related calculators
- Math calculators
- Regular Polygon Calculator: shapes from their centre, radii and angles, plotted the same way.
- Scientific Calculator: the cos and sin values behind the rotation matrix.
Method: translation (x + dx, y + dy); rotation matrix about a centre with cos and sin of the entered angle; reflections in the axes, the lines y = ±x and the lines y = k and x = k; dilation (c + k(p − c)) from the entered centre. Figures checked against independently calculated fixtures, including off-centre rotation and negative scale factors. Last reviewed: 21 October 2026.
Disclaimer: This calculator gives estimates for general information only. It is not financial, medical or other professional advice.