3D Transformation Matrix Calculator
Apply a 3D affine transformation matrix to a point. This educational calculator shows the formula, result, and step-by-step interpretation.
Calculator
What this calculator teaches
3D transformation matrices are used in graphics, CAD, robotics, cameras, and simulation.
Use the result as a learning aid. For classwork, still show the formula and intermediate reasoning so the final answer is not just a black-box number.
Understanding a 3D Affine Transformation
A 3D affine transformation combines a linear transformation and a translation. This calculator reads the first three columns as a 3×3 matrix A and the fourth column as a translation vector t, then evaluates p′=Ap+t.
The 3×3 part can represent scaling, rotation, reflection, shear, or combinations of those operations. Translation is not a linear transformation by itself because it does not keep the origin fixed, so it appears as the separate vector t. In homogeneous coordinates, the same operation can be written as one 4×4 matrix acting on [x,y,z,1]ᵀ.
| Matrix row | Output coordinate |
|---|---|
| Row 1 | x′=a₁₁x+a₁₂y+a₁₃z+tₓ |
| Row 2 | y′=a₂₁x+a₂₂y+a₂₃z+tᵧ |
| Row 3 | z′=a₃₁x+a₃₂y+a₃₃z+t_z |
Order matters when composing transformations. Rotating and then translating generally produces a different point from translating first and then rotating. A single affine matrix already encodes whichever order was used when the matrix was constructed.
Separate position changes from shape changes. Translation moves every point by the same vector, while the 3×3 matrix determines how differences between points are stretched, rotated, reflected, or sheared. Consequently, subtracting two transformed points removes the translation term: p′₂−p′₁=A(p₂−p₁).
This gives a useful check. If A is a pure rotation matrix, it should preserve distances and angles, so the length of a vector before and after transformation should match apart from rounding. A uniform scale by s multiplies every distance by |s|. A reflection preserves lengths but reverses orientation. These properties help diagnose incorrectly entered matrices.
Formula & Symbols
| Concept | Formula or rule |
|---|---|
| Affine transform | p′ = Ap + t |
Worked example
Common mistakes
Keep lists comma separated, matrices as rows separated by semicolons, and modular inputs as integers.
Some methods require positive probabilities, valid moduli, independent trials, or small educational input sizes.
Each output coordinate is a dot product of one matrix row with the input point, followed by the row’s translation value.
Matrix multiplication is generally not commutative, so composing rotation, scale, and translation in a different order can change the result.
Homogeneous-coordinate checks should also confirm that translation affects position points as intended while preserving the expected linear part.
FAQ
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Formula Explorer connections
Interpretation: This relationship connects dimensions, coordinates, angles or trigonometric ratios to a geometric measurement. Assumption: Use a consistent angle mode and length unit, identify the intended shape or coordinate system, and verify that the supplied dimensions form valid geometry.