Calculator
Example Data Table
| Starting Point | Pivot | Sequence | Expected Use |
|---|---|---|---|
| (2, 3) | (0, 0) | S(1.5,0.75); R(30); T(4,-2) | General graph movement |
| (5, -1) | (1, 1) | R(90); RefX; T(3,2) | Rotation around a custom point |
| (4, 2) | (0, 0) | H(0.5,0); S(2,2); RefY | Shear and resize model |
| (1, 7) | (0, 0) | M(1,0,6,0,1,-4); RefYeqX | Custom affine matrix test |
Formula Used
The calculator uses affine transformation matrices. A point is written as a column vector with a third coordinate equal to one.
[x'] [a b e] [x] [y'] = [c d f] [y] [1 ] [0 0 1] [1]
Translation uses this matrix:
[1 0 dx] [0 1 dy] [0 0 1 ]
Rotation uses cosine and sine. The angle is converted from degrees to radians.
[cos θ -sin θ 0] [sin θ cos θ 0] [0 0 1]
Scale, shear, and reflection also use matrices. When several transformations are combined, matrix multiplication preserves the chosen order. The determinant gives the area scale factor. A negative determinant means orientation has been reversed.
How to Use This Calculator
Enter the starting x and y coordinates. Add a pivot point when rotation, scale, shear, or reflection should happen around another point. Write each transformation in the sequence box. Separate operations with semicolons or new lines.
Use T(dx,dy) for translation. Use R(angle) for rotation. Use S(sx,sy) for scale. Use H(shx,shy) for shear. Use reflection codes such as RefX, RefY, RefOrigin, RefYeqX, and RefYeqNegX.
Press Calculate to view the final coordinate, composite matrix, area scale, orientation, and step table. Use Download CSV for spreadsheet work. Use Download PDF for a compact report.
Combinations of Transformations Guide
What Combined Transformations Mean
Combined transformations apply several movements to the same point, shape, or graph. A single action may translate a figure. Another action may rotate it. A third action may stretch, reflect, or shear it. The final location depends on every action in the list. The order matters because most transformations do not commute. Rotation then translation can give a different answer from translation then rotation.
Why Matrix Order Matters
This calculator treats each point as an affine vector. It multiplies matrices in the same order that the operations are applied. That makes the result useful for coordinate geometry, computer graphics, animation, classroom checks, and mapping tasks. The step table helps you find where a mistake enters the sequence. It also makes long problems easier to review.
Using a Pivot Point
A pivot point is important for rotation, scaling, shearing, and reflection. Without a custom pivot, the origin is used. With a pivot, the calculator first moves the pivot to the origin. It then applies the selected operation. Finally, it moves the point back. This method keeps the transformation consistent and easier to verify.
Reading the Composite Matrix
The composite matrix is a compact summary of the whole sequence. Its left part stores rotation, scale, shear, and reflection effects. Its right column stores translation effects. You can reuse this matrix on many points from the same shape. That saves time when a triangle, polygon, icon, or plotted design needs identical movement.
Area Scale and Orientation
The determinant gives the area scale factor. A value of two means areas become twice as large. A value of one means area is preserved. A negative determinant shows a reversal. Reflections usually reverse orientation. This is useful when checking mirrored designs or graph transformations.
Function Transformation Option
The function fields describe rules in the form g(x) = a f(b(x - h)) + k. The value a controls vertical stretch and reflection. The value b controls horizontal scale and horizontal reflection. The values h and k move the graph horizontally and vertically. This makes the calculator useful for coordinate and function work.
FAQs
What is a combination of transformations?
It is a sequence of two or more transformations. These may include translation, rotation, scale, shear, and reflection. The final result depends on the exact order used.
Does transformation order matter?
Yes. Many transformations produce different results when their order changes. For example, rotating before translating often differs from translating before rotating.
What does T(dx,dy) mean?
It means translation. The value dx moves the point horizontally. The value dy moves the point vertically. Positive and negative values control direction.
What does R(90) mean?
It means rotate the point by 90 degrees. The rotation uses the selected pivot. If no custom pivot is entered, the origin is used.
What does S(2,3) mean?
It means scale x-values by 2 and y-values by 3. Scaling can enlarge, shrink, or reflect coordinates when negative factors are used.
What is a composite matrix?
It is one matrix that represents the whole transformation sequence. You can apply it to other points from the same figure.
What does the determinant show?
The determinant shows the area scale factor. Its sign also shows orientation. A negative value means the transformation reverses orientation.
Can I export the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple report with the main result and steps.