Rotate like a compass
Rotation is just turning every point around a center—like a compass needle swinging about its base.
Spin move
Coordinate Transformations — Translate, Rotate, Reflect, Scale (2D)
Points & Transform
Tip: Paste CSV pairs or type values. Use the Examples button to load shapes.
Transformed points will appear here…
Rotation: (x',y') = (cₓ + cosθ(x−cₓ) − sinθ(y−c_y), c_y + sinθ(x−cₓ) + cosθ(y−c_y));
Scaling about (cₓ,c_y): (x',y') = (cₓ + sₓ(x−cₓ), c_y + s_y(y−c_y)).
Reflection across a line at angle θ uses rotate → flip x → rotate back. Press Enter to apply.
Preview
Blue = original, Red = transformed. The view auto-scales to fit all points. Grid and axes are illustrative.
Coordinate Transformations in 2D: A Quick Guide
This tool applies four core geometric transformations—translation, rotation, reflection, and scaling—to one or many points in the plane. These operations are ubiquitous in computer graphics, robotics, CAD, GIS, and classroom geometry, and they can be composed to produce complex motions and mappings.
- Translation adds a fixed offset:
(x',y') = (x + Δx, y + Δy). - Rotation by angle
θabout a center(cₓ,c_y):x' = cₓ + cosθ(x−cₓ) − sinθ(y−c_y),y' = c_y + sinθ(x−cₓ) + cosθ(y−c_y). - Scaling about
(cₓ,c_y)with factorssₓ, s_y:x' = cₓ + sₓ(x−cₓ),y' = c_y + s_y(y−c_y)(uniform ifsₓ=s_y). - Reflection over common lines (axes,
y=x,y=−x) or over a line through the origin at angleθvia rotate–flip–rotate back.
Under the hood, these are affine transformations—linear maps optionally combined with translation. In matrix
form, a point is extended to homogeneous coordinates (x,y,1) and multiplied by a 3×3 matrix,
which makes chaining operations straightforward. This page focuses on direct formulas for clarity, while
producing the same results as matrix multiplication.
Pro tip: For “rotate/scale about a point”, set the center to the shape’s centroid if you want to keep it roughly in place as you transform.
5 Fun Facts about Coordinate Transformations
Reflections are mirror worlds
Flipping across a line acts like holding a mirror there—each point gets a twin the same distance on the other side.
Mirror magic
Scaling keeps directions
Scaling stretches or shrinks lengths but doesn’t change angles when it’s uniform—your shape grows without getting “tilted.”
Stretch & shrink
Translation = sliding
Translate just slides every point the same way—no spinning, no stretching. Think of moving a sticker across a page.
Simple slide
Combos make anything
Games and animations chain these moves—slide, spin, flip, stretch—to build dances, camera moves, and on-screen magic.
Action recipe
Frequently Asked Questions
How do I enter multiple points?
Enter one pair per line as x,y. Example: 0,0 then 2,1 then 4,0.
What angle convention is used?
Angles are in degrees, positive counter-clockwise.
Why can’t I reflect across any arbitrary line y=mx+b?
To keep the UI simple, we support axes, y=±x, and any line through the origin at angle θ.
For a general line, translate points so the line passes through the origin, apply the angle reflection, then translate back.
Is my data private?
Yes. Everything runs locally in your browser; nothing is uploaded.