y′ = k + (x − h)·sin(θ) + (y − k)·cos(θ)
- Pick Single point or Bulk points.
- Enter the pivot (h, k) and the rotation angle.
- Select unit and direction, then set precision.
- Press Submit to view a results table above.
- Use Download CSV/PDF to export the computed values.
| Input X | Input Y | Pivot H | Pivot K | Angle (deg) | Direction | Rotated X | Rotated Y |
|---|---|---|---|---|---|---|---|
| 3 | 2 | 1 | 1 | 45 | Counterclockwise | 2.414214 | 2.414214 |
| -1 | 4 | 1 | 1 | 45 | Counterclockwise | -2.535534 | 1.707107 |
| 0 | 0 | 1 | 1 | 45 | Counterclockwise | 1 | -0.414214 |
Coordinate transformation context
Rotation about a pivot is a rigid motion that preserves distance. If a point P(x,y) is rotated about C(h,k), the displacement vector (x−h, y−k) changes direction while keeping its length r. This calculator reports r implicitly through the rotated coordinates and shows optional intermediate values.
Angle handling and units
Angles can be entered in degrees or radians. The internal computation uses radians, with θ(rad)=θ(deg)×π/180. Clockwise motion is represented by −θ, matching the standard counterclockwise-positive convention used in analytic geometry.
Matrix model and numerical stability
The transformation applies the 2×2 rotation matrix R=[ [cosθ, −sinθ], [sinθ, cosθ] ] to the displacement vector, then translates back to the pivot. Because cosθ and sinθ are bounded, rounding error mainly comes from floating-point representation and your chosen output precision.
Bulk processing and auditability
For coordinate lists, each row is computed independently, enabling quick verification of drafts, exam problems, or design sketches. Exported CSV preserves input and output pairs for spreadsheet checks, while the PDF summary provides a lightweight record suitable for attachments.
Typical classroom dataset: rotating (3,2) about (1,1) by 45° yields approximately (2.414214, 2.414214). Rotating (0,0) about (1,1) by 45° gives (1, −0.414214). These values illustrate how both coordinates shift when the pivot is not the origin.
Interpretation checks with sample metrics
A practical check is distance preservation: √((x′−h)²+(y′−k)²)=√((x−h)²+(y−k)²). Another is special angles: at 90°, (dx,dy) maps to (−dy,dx). At 180°, results become (2h−x, 2k−y). Use these to sanity-check outputs.
Visualization and decision support
The Plotly scatter view overlays original and rotated points and draws connectors between each pair. Clusters indicate how a shape moves around the pivot, which helps compare alternative centers, explore symmetry, and confirm direction choices before exporting final values.
Precision guidance: For most worksheets, 4–6 decimals are sufficient. If you are chaining transformations, keep a higher precision during intermediate steps and round only at the end. When comparing two candidate pivots, focus on relative changes in x′ and y′ rather than tiny differences caused by rounding. In real time.
FAQs
1) What does “rotation about a point” mean?
It moves each point around a fixed pivot while keeping the distance to the pivot unchanged. Only the direction of the point relative to the pivot changes by the chosen angle.
2) How do I choose clockwise versus counterclockwise?
Counterclockwise is the standard positive direction on Cartesian axes. Choose clockwise when the problem statement explicitly says clockwise, or when your diagram’s motion is visually clockwise around the pivot.
3) Why do my coordinates change even for small angles?
Any nonzero angle changes both x and y unless the point lies exactly on a symmetry line relative to the pivot. Small angles still produce measurable shifts, especially when the point is far from the pivot.
4) Can I rotate multiple points at once?
Yes. Use Bulk points and enter one coordinate pair per line. The results table and exports will include a row for each point, calculated using the same pivot and angle.
5) How can I verify the output quickly?
Check distance preservation: the distance from the pivot to the point should match before and after rotation. Also test special angles like 90° or 180° where expected patterns are easy to recognize.
6) What does the PDF export contain?
The PDF is a single-page text summary listing the pivot, angle, direction, and each input point mapped to its rotated coordinates. It’s designed for quick sharing and record keeping.