Enter points and choose a rigid motion
Use one point per line as Label,x,y or simply x,y.
Example data table
Example: translate the rectangle by the vector (2, 1).
| Point | Original x | Original y | Transformed x | Transformed y |
|---|---|---|---|---|
| A | 0 | 0 | 2 | 1 |
| B | 4 | 0 | 6 | 1 |
| C | 4 | 3 | 6 | 4 |
| D | 0 | 3 | 2 | 4 |
Formula used
Translation: \((x', y') = (x + d_x, y + d_y)\)
Rotation about \((h,k)\): \(x' = h + (x-h)\cos\theta - (y-k)\sin\theta\), \(y' = k + (x-h)\sin\theta + (y-k)\cos\theta\)
Reflection: the calculator applies the matching reflection matrix for the chosen axis or mirror line.
Glide reflection: reflect first, then translate parallel to the mirror line by the glide distance.
Distance preservation check: rigid motions should keep every pairwise distance unchanged, so the maximum pairwise distance error should be near zero.
Area and perimeter: the shoelace formula computes polygon area, and Euclidean distance computes the path length or perimeter.
How to use this calculator
- Enter your points, one per line, as Label,x,y or x,y.
- Select the rigid motion type: translation, rotation, reflection, or glide reflection.
- Fill only the values needed for that motion, such as vector, angle, center, axis, or glide distance.
- Choose the number of decimal places and decide whether to close the shape.
- Press Calculate rigid motion to show results above the form.
- Review the transformed coordinates, matrix, preserved measurements, and Plotly graph.
- Use the export buttons to download a CSV file or a PDF copy of the result section.
FAQs
1. What is a rigid motion?
A rigid motion is a transformation that keeps all distances and angle measures unchanged. In the plane, common examples are translations, rotations, reflections, and glide reflections.
2. Why do area and perimeter stay the same?
Rigid motions preserve lengths between points. Because side lengths stay unchanged, the perimeter stays unchanged. Area magnitude also remains the same, though reflections can reverse orientation.
3. What does the determinant tell me?
The determinant of the linear part helps classify the motion. A positive determinant preserves orientation, while a negative determinant reverses orientation, as reflections and glide reflections do.
4. Why is my pairwise distance error not exactly zero?
Tiny nonzero values usually come from floating-point rounding. Very small errors are normal in numerical computing and still indicate a correct rigid motion.
5. Can I enter a single point instead of a polygon?
Yes. The calculator works with one point, multiple independent points, or an ordered polygon. Area needs at least three vertices, while perimeter needs at least two points.
6. What is a glide reflection?
A glide reflection combines a reflection in a line with a translation parallel to that line. It is still a rigid motion because distances remain preserved.
7. What format should I use for points?
Use one point per line. Good examples are A,2,5, B,-1,3, or simply 2,5 when you want automatic labels.
8. Does the graph show both original and image points?
Yes. The Plotly graph draws the original figure, the transformed figure, and dashed connector segments so you can compare movement and orientation visually.