Coordinate Transformation Calculator

Coordinate transformations on the plane, one point at a time

Introduction: what a coordinate transformation actually does

A coordinate transformation is a function that takes every point of the plane to a new point. In school geometry the four you meet first are translations (slides), reflections (mirror flips), rotations (turns), and dilations (rescalings). Each one has a purely algebraic rule — a recipe for turning the original coordinates $(x,y)$ into image coordinates $(x^{\prime },y^{\prime })$ — and a geometric meaning you can see on a graph. This calculator gives you both: the computed image point and an animated coordinate grid that shows the original point P, its image P′, and the guide geometry (mirror line, rotation arc, or dilation ray) that explains the motion.

Transforming a single point is the building block for everything else. A triangle, a quadrilateral, or any polygon transforms vertex by vertex: apply the same rule to each corner, then reconnect the corners in the same order. That is why homework problems, standardized-test questions, and computer-graphics pipelines all reduce to the per-point rules implemented here. All arithmetic runs locally in your browser — nothing is uploaded, so you can use the tool during class, test review, or offline study.

How to use the coordinate transformation calculator

1. Type the original coordinates into Original x and Original y. Decimals and negative numbers are fine.
2. Select a Transformation type. Extra inputs appear when the rule needs parameters: dx and dy for a translation, an angle for a rotation, a scale factor k for a dilation.
3. Press Transform. The result panel states the rule that was applied and the image coordinates, and the graph animates the point from P to P′.
4. To compose transformations, press Use result as new input: the image coordinates are copied back into the input fields so the next transformation starts where the last one ended. Chaining like this is the quickest way to see that order matters — translate-then-rotate usually lands somewhere different from rotate-then-translate.
5. Press Copy link to this transformation to put a shareable URL on your clipboard. Anyone opening the link sees the same inputs already filled in and calculated — handy for teachers writing answer keys or students asking for help.

Transformation formulas this calculator applies

The rules below act on an input point $(x,y)$. Reflections and rotations are performed about the origin (or an axis through it), rotation angles are in degrees with counterclockwise positive, and dilations are centered at the origin. These are the textbook conventions used in coordinate-geometry units and on most standardized exams.

• Translation by (a, b): $(x+a,y+b)$ — adds the same horizontal and vertical shift to every point.
• Reflection across the x-axis: $(x,-y)$ — keeps x and flips the sign of y.
• Reflection across the y-axis: $(-x,y)$ — keeps y and flips the sign of x.
• Reflection through the origin: $(-x,-y)$ — flips both signs; identical to a 180° rotation.
• Reflection across the line $y=x$: $(y,x)$ — swaps the coordinates.
• Rotation about the origin by angle θ: $(x\cos \theta -y\sin \theta ,x\sin \theta +y\cos \theta )$ — positive angles turn counterclockwise; for exact multiples of 90° the calculator uses the exact values of sine and cosine rather than floating-point approximations.
• Dilation from the origin by factor k: $(kx,ky)$ — multiplies both coordinates by the same factor; negative k also flips the point through the origin.

The rotation rule is just the 2×2 rotation matrix applied to the column vector $(x,y)$:

Seeing the matrix form early pays off later: compositions of rotations become matrix products, and the same machinery generalizes to 3D graphics and robotics. Here is the compact reference table for every rule on this page.

Sources: every rule on this page was checked against Wolfram MathWorld, which gives the standard definitions used in coordinate geometry: Rotation Matrix for the 2×2 rotation rule, Translation, Reflection, Dilation, and Rigid Motion for the classification of which transformations preserve distance and orientation. The exact values sin 90° = 1, cos 90° = 0, sin 180° = 0, and cos 180° = −1 are used for quarter-turn rotations so that integer inputs return integer outputs. Last reviewed August 2026.

Worked examples with real coordinates

Each example below can be reproduced by typing the same values into the form. Watching the graph while you do so is a fast way to build intuition for the sign changes.

Example 1: Translation of (3, −2) by dx = 5, dy = 1

The rule $(x+5,y+1)$ gives new x = 3 + 5 = 8 and new y = −2 + 1 = −1, so the image is (8, −1). On the graph the guide arrow runs 5 units right and 1 unit up.

Example 2: Reflection of (−4, 7) across the x-axis

Keep x, negate y: the image is (−4, −7). Sanity check: a point 7 units above the mirror line must land exactly 7 units below it, and the segment joining P to P′ must cross the x-axis at a right angle.

Example 3: Reflection of (2, 9) across the line y = x

Swap the coordinates: the image is (9, 2). This rule is the reason the graph of an inverse function is the mirror image of the original function across $y=x$.

Example 4: Rotation of (1, 0) by 90° counterclockwise

With the exact values cos 90° = 0 and sin 90° = 1, the rule gives (1·0 − 0·1, 1·1 + 0·0) = (0, 1): the point moves from the positive x-axis to the positive y-axis along a quarter-circle arc. Rotating by 180° instead would give (−1, 0).

Example 5: Rotation of (4, 3) by 30° counterclockwise

Now the trigonometric values are irrational: cos 30° ≈ 0.8660 and sin 30° = 0.5. The image is x′ = 4(0.8660) − 3(0.5) = 3.4641 − 1.5 = 1.9641 and y′ = 4(0.5) + 3(0.8660) = 2 + 2.5981 = 4.5981. The distance from the origin, 5, is unchanged — only the direction rotates.

Example 6: Dilation of (−3, 4) by k = 2 and by k = 0.5

Multiplying both coordinates by 2 gives (−6, 8): the point slides outward along the ray from the origin to twice its distance. With k = 0.5 the image is (−1.5, 2), halfway to the origin. A negative factor such as k = −1.5 applied to (2, −3) gives (−3, 4.5): scaled and flipped through the origin.

Rigid motions, dilations, and what each preserves

Translations, reflections, and rotations are rigid motions: they preserve every distance and every angle, so the image of any figure is congruent to the original. Two of them differ in one subtle respect — orientation. Rotations and translations keep the cyclic order of a polygon's vertices (clockwise stays clockwise); a single reflection reverses it. That is why the letter R still reads as an R after a rotation but appears mirror-written after a reflection.

Dilations preserve angles and keep every line parallel to its image, but they scale all distances by |k|, producing similar rather than congruent figures. If |k| is greater than 1 the image grows; if 0 < |k| < 1 it shrinks toward the center; k = −1 is exactly the reflection through the origin. Because the checkable invariants differ — congruence for rigid motions, similarity for dilations — knowing what should not change is the fastest way to audit an answer.

Common sign errors when transforming coordinates

• Swapping dx and dy: dx shifts left/right (x), dy shifts down/up (y). If the guide arrow on the graph moves the wrong way, the two are probably switched.
• Negating the wrong coordinate in a reflection: across the x-axis only y changes sign; across the y-axis only x does; through the origin both do.
• Confusing y = x reflection with axis reflections: across y = x the coordinates swap — nothing is negated.
• Rotation direction: positive angles are counterclockwise by convention. If your answer is "rotated the wrong way," re-enter the angle with the opposite sign.
• Expecting integers after arbitrary rotations: only quarter-turn multiples map integer points to integer points; 30°, 45°, or 60° almost always produce decimals.

Limitations of a single-point transformation tool

• One step, one point: the calculator applies exactly one transformation to one point per calculation. Compositions are supported by chaining with the "Use result as new input" button, but the intermediate steps are up to you to track.
• Fixed centers and mirror lines: rotations and dilations are centered at the origin, and reflections are limited to the x-axis, y-axis, the origin, and the line y = x. For any other center, translate so the center moves to the origin, transform, then translate back (the FAQ below works a full example).
• Rounding: results are rounded to 4 decimal places. Quarter-turn rotations are computed exactly, but other angles rely on floating-point sine and cosine, so exact radical forms like the square root of 2 divided by 2 must be kept symbolically in written work.
• Magnitude limits: extremely large coordinates (beyond about 15 significant digits) lose precision in IEEE-754 arithmetic. Classroom-scale numbers are unaffected.
• No shape input: to transform a polygon, run each vertex through the same rule and reconnect them in order; the calculator does not plot multi-vertex figures.

Where coordinate transformations show up beyond the classroom

The same four rules run the visual world. Every sprite that moves across a game screen is translated; every camera pan in a 3D engine is a rotation matrix; every pinch-to-zoom is a dilation about the pinch point (a translation, a scale, and a translation back — exactly the recentering trick described above). Robotics uses rotation matrices to convert between a robot's body frame and the world frame, and mapmakers compose scalings and translations to fit survey coordinates onto paper. Learning to read $(x,y)\to (-y,x)$ as "a quarter turn" is the first step toward reading any transformation pipeline.

Two different things are called a coordinate transformation

The phrase is used for two genuinely different operations, and landing on the wrong one wastes a lot of time. This calculator does the first.

1. Transforming points inside one plane coordinate system — the translations, reflections, rotations, and dilations above. The axes stay put and the points move.
2. Converting a position between coordinate reference systems — latitude and longitude to UTM easting and northing, or one geodetic datum to another. Here the point on the ground never moves; the frame used to describe it changes.

The second kind is a surveying and mapping problem, not a plane-geometry one, because the Earth is modelled as an ellipsoid rather than a flat sheet. If that is what you came for, the numbers below are the ones that matter, and the authoritative tools are linked at the end of this section.

Degrees, minutes, seconds to decimal degrees. The conversion itself is simple, but the sign is where almost everyone slips. Convert the magnitude first and apply the hemisphere sign to the whole result:

where $s$ is −1 for west longitude or south latitude and +1 otherwise. So 122° 30′ 00″ W is −(122 + 30/60) = −122.5°. Adding the minutes to a negative degree value instead gives −122 + 0.5 = −121.5°, which is a full degree of longitude away — about 88 km at the latitude of San Francisco, and 111 km at the equator. That single sign slip is the most common bug in hand-rolled conversion code.

The ellipsoid and the UTM grid. The WGS 84 reference ellipsoid uses a semi-major axis of exactly $a=6378137$ m with a flattening of $1/f=298.257223563$. Universal Transverse Mercator divides the world into 60 zones 6° wide, each with a central-meridian scale factor of $k_0=0.9996$, a false easting of 500 000 m, and — for southern-hemisphere zones only — a false northing of 10 000 000 m. The zone number for a longitude $\lambda$ in decimal degrees is

with two traps. At exactly $\lambda =+180$ the formula returns 61, which is not a zone — it has to wrap back to zone 1. And the grid is not perfectly regular: zone 32 is widened to cover southwestern Norway, and around Svalbard only zones 31, 33, 35, and 37 are used, with 32, 34, and 36 omitted. Any converter that ignores those exceptions will place points in the wrong zone in exactly those regions.

Why the datum matters. The same latitude and longitude expressed on NAD 27 and on WGS 84 describe two different spots on the ground, because the two datums use different ellipsoids fitted in different ways. Across the contiguous United States the discrepancy commonly runs to tens of metres, and it is larger in Alaska and Hawaii — far more than enough to move a boundary marker onto the neighbour's land. A coordinate is therefore not fully specified until you also state the datum it belongs to. For real conversions use the NOAA National Geodetic Survey coordinate conversion tool (NCAT), and see the NGS explanation of geodetic datums.

Practising composition with the Datum Drift game

The interactive stage below the calculator turns the same rules into a puzzle. Each round hides a transformation and shows you only its result: an amber target triad of three points. Your job is to recover the transformation by setting four dials — the mirror line, the dilation factor $k$, the rotation angle $\theta$, and the translation $(\mathrm{dx},\mathrm{dy})$ — which are always applied in that fixed order. The whole overlay grid warps as you turn the dials, so you can see a reflection flip the lattice, a dilation stretch it, and a rotation shear it across the reference grid underneath.

Scoring is the registration error: the root-mean-square distance, in grid units, between your three points and the target's. Three habits transfer straight back to homework. Read orientation first, because only a reflection reverses it. Read size next, because only a dilation changes it. Whatever offset is left over is the translation. You will also discover that different dial settings sometimes register perfectly onto the same target — a reflection followed by a rotation is itself a reflection in a different line — which is the clearest possible demonstration that a composition can be written in more than one way.

Coordinate transformation questions students ask