Transform one point through mixed operations confidently. See intermediate coordinates, matrices, vectors, and movement clearly. Export results, compare examples, and understand formulas without confusion.
Enter one point, select mixed transformations, then calculate the final coordinate after applying each operation in sequence.
The chart displays the original point, every intermediate step, and the final position after the mixed transformation sequence.
| Example | Start Point | Sequence | Expected Final Point |
|---|---|---|---|
| 1 | (2, 1) | Translate (3, 2) → Rotate 90° about origin | (-3, 5) |
| 2 | (4, -1) | Scale (2, 3) about origin → Reflect y-axis | (-8, -3) |
| 3 | (1, 2) | Shear (1, 0) → Translate (-2, 1) | (1, 3) |
| 4 | (3, 3) | Reflect y = x → Translate (1, -2) | (4, 1) |
Mixed point transformations are applied with homogeneous coordinates. A point becomes a column vector:
The composite result uses sequential matrix multiplication:
Common matrices used in this calculator:
For pivot-based rotation or scaling, the calculator translates the point to the pivot, applies the matrix, then translates back.
It means you apply different transformation types to one point in sequence. You can combine translation, rotation, scaling, reflection, shear, or a custom affine matrix in one workflow.
Matrix multiplication is not generally commutative. Rotating then translating usually gives a different final point than translating then rotating, even when the same values are used.
Yes. Enter the angle, then supply pivot X and pivot Y. The calculator handles the internal translate-rotate-translate sequence automatically.
Scaling changes size along the x and y directions. Shear slants the coordinate system by pushing one axis proportionally to the other axis.
Use it when you already know the six affine coefficients. It is useful for advanced geometry, graphics pipelines, or verifying external transformation matrices.
It is the single matrix that reproduces the full sequence. Applying that one matrix to the original point gives the same final result as all steps combined.
The connected path helps visualize the movement of the point after each transformation. It is useful for checking direction, step order, and overall displacement.
Yes. The page includes a CSV export for step data and a PDF export for the main result section, making reporting and sharing easier.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.